TTTrrraaannnsssaaaccctttiiiooonnnsss ooofff ttthhheee AAAOOOCCC October 2004 Volume 1, Number 1 ISSN 1151-83-9279

Remarks by the President of the AOC………………………….Austin K. Thomas, Jr. 2 Introduction to the AOC Transactions…………………….…Richard G. Wiley, Editor 2 About the Authors…………………………………………………………………….. 3

Papers The Farey Series in Synchronization and Intercept-Time Analysis for Electronic Support (Invited Paper)………………I. Vaughn L. Clarkson 7 Higher Order Shift Register Sequences……………………………….John Fielding 29 Implementation and Testing of Fuzzy Adaptive Resonance Theory Algorithm to Analyze Complex Radar Data Sets…Michael J. Thompson and John C. Sciortino Jr. 34 The Shannon-Hartley Theorem as a Unifying Principle in Electronic Warfare and Information Warfare……………………………………………………………..Andrew Borden 43 Effects of Radio Wave Propagation in Urbanized Areas on UAV-GCS Command and Control………….Lock Wai Lek and David C. Jenn 50 Impact of Low Altitude Propagation Losses on Aircraft Survival and Sensor Design……………………………………………….Richard W. (Bill) Bambrick 65 New Channelized Receivers…………………….……………….Ming-Chiang Li 82

The Design and Analysis of a Space-Based Experiment for Inflatable Structures. ……….David C. Moody, Richard A. Raines, Richard Cobb, Anthony Palazotto 106 A Comparison Between the Spectral Purity of Three Digital Radio Frequency Memory Architectures……………… T. W. Küsel, M. R. Inggs and J. E. Pienaar 135 Expendables and Ship Protection--Asset Coordination is Critical...….Arthur G. Self 145 Information for Authors 155

Transactions printing subsidized by Research Associates of Syracuse, Inc.

President’s Remarks

With the first issue of The Transactions of the AOC, we hope to not only increase the visibility of work being done in our mission area but to facilitate the sharing of innovative ideas, concepts, theories, and afford recognition to those involved in this important work. The procedures and review process set up by Dr. Wiley provide both breadth and depth of selected papers and ensure quality throughout the publication. Also, as you can see, included are several papers from our international community. My thanks and appreciation to our former AOC president, Dave Adamy, Dick Wiley, the AOC staff and most importantly to those who contributed papers for this charter issue. I believe it will set the course and provide the impetus to publish future Transactions of the AOC.

Austin K. Thomas, Jr., President of the AOC

Introduction

The Transactions of the AOC is a new publication designed for communicating technical information across the Information Operations and Electronic Warfare Community. This is the first issue of what the AOC hopes will become a standard reference for the entire international community in the years to come. The concept for the Transactions originated with David Adamy during his term as president of the AOC during 2001-2003. The invited paper by Vaughn Clarkson sets the tone for this first issue with analysis of problems related to probability of interception. He applies the Farey series to this problem to obtain some new and useful results. This is followed by the paper by John Fielding related to shift register sequences and then a paper by Michael J. Thompson and John C. Sciortino Jr. on a method for sorting radar pulses. The paper by Andrew Borden on relating information theory to Electronic Warfare attempts to provide a theroretical framework for the field of EW. This is followed by two papers related to propagation by Lock Wai Lek and Richard W. (Bill) Bambrick. The next paper by Ming-Chiang Li describes a new implementation of channelized receivers using optical techniques. A discussion of the design of an inflatable antenna structure for space-based antennas follows by David C. Moody, Richard A. Raines, Richard Cobb and Anthony Palazotto. The next paper covers some of the major design issues of DRFMs and is the work of T. W. Küsel, M. R. Inggs and J. E. Pienaar. The final paper by Arthur Self discusses important aspects of using expedables in ship protection. Biographical information about the authors is given below These papers represent a good cross section of the interests of the community. They were all reviewed by qualified AOC members. I want to thank the reviewers and especially the authors for making this first issue possible. I also want to acknowledge the support of the AOC Board of Directors and the AOC President, Austin Thomas, for their support. Finally, I am thankful for the work of the Headquarters Staff, in particular the Executive Director, Gene Bartlow, and all who provided support for the layout and reproduction of these Transactions. Richard G. Wiley, Editor Transactions of the AOC is published annually by the Association of Old Crows (AOC), 1000 North Payne Street, Alexandria, Virginia 22314. Telephone 703-549-1600. Responsibility for the contents rests upon the authors and not upon the AOC. Price/Publication Information: Individual copies are $20.00 for AOC members, nonmembers $50.00. Copyright and Reprint Permissions: Abstracting is permitted with reference to the source. For all other copying or republication permission, contact the Association. Copyright 2004 by the Association of Old Crows. All rights reserved.

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About the Authors I. Vaughan L. Clarkson received the B.Sc. degree in mathematics in 1989 and the B.E. degree (first-class honors) in computer systems engineering from the University of Queensland in 1990, and the Ph.D. degree from Australian National University in 1997. He then joined the Defense Science and Technology Organization. From 1998 to 2000, he was Lecturer at The University of Melbourne. He is now a Senior Lecturer at The University of Queensland. His research includes signal/image processing, information theory and geometry of numbers.

John E. Fielding received his BA degree in mathematics from the University of Texas, Austin, in 1966 and an MS degree in applied mathematics from the Southeastern Institute of Technology, Huntsville, AL, in 1979. Mr. Fielding is a member of the technical staff of CACI in Austin, TX. His primary interests include modulation, signal processing, pulsed Doppler radar, and phased array antenna design.

Mr. Michael J. Thompson is a member of the Systems Automation Section of the Electronic Support Measures branch of the Tactical Electronic Warfare Division of the US Naval Research Laboratory. His previous research has been in the fields of Neural Networks and Fuzzy Logic. He holds a BS and MS in Electrical Engineering from Western New England College in Springfield, Massachusetts. He is a member of IEEE and the Association of Old Crows.

Dr. John C. Sciortino, Jr. was a senior physicist in the ESM Branch, Tactical EW Division of the US Naval Research Laboratory and now leads the Systems Automation Section. He is the principal author of a number of algorithms used to automate ESM systems. He holds BS and PhD degrees in physics from The Pennsylvania State University and is a member of the American Physical Society, American Association of Physics Teachers, and Association of Old Crows.

Andrew Borden is a retired Air Force officer and a mathematician with a long background and many publications in the area of decision making algorithms. He was Principal Scientist for Electronic Warfare at the NATO C3 Agency. His last active duty assignment was as Deputy Chief of Staff for Intelligence, USAF Air Intelligence Agency (then Electronic Security Command). He is the Vice President and Chief Scientist of INFERLOGIX, L.L.C., an emerging, woman-owned company with headquarters in Galveston, Texas.

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David C. Jenn received the Ph.D. degree in electrical engineering from the University of Southern California in 1987. From 1976 to 1978 he was with McDonnell Douglas Astronautics Co. and from 1978 to 1990 with Hughes Aircraft Co. In 1990 he joined the Department of Electrical and Computer Engineering at the Naval Postgraduate School as a Professor. His research has focussed on the design and analysis of high-performance phased array antennas for radar and communication systems, electromagnetic wave propagation, and radar cross section analysis. Dr. Jenn is author of the book Radar and Laser Cross Section Engineering.

Lock Wai Lek (Willy) received the B. Eng. degree from the National Defense Academy, Japan, in 1998. In December 2003 he completed a program in Engineering Science at the Naval Postgraduate School. Currently he is a Major in the Singapore Armed Forces (Army).

Richard W. (Bill) Bambrick was born in Toronto. He was an instructor on jet engines in the Royal Canadian Air Force (RCAF). He studied Electrical Engineering at the University of Toronto and undertook extensive courses in Aeronautical Engineering and Radar and Communications Engineering with the RCAF. Bill was with Boeing Aerospace Company where he worked on several major military programs, including the B-52 and B-1 avionics systems, the Short Range Attack Missile, and AWACS. He also had temporary assignments at Northrop Aircraft, Rockwell International, ITT Avionics, ITT Gilfillan and AIL Systems.

Ming-Chiang Li received a BS degree in Physics from Beijing University in 1958 and Ph.D. in Physics from the University of Maryland in 1965. He was a member of the Institute for Advanced Study in Princeton, New Jersey from 1965 to 1967, a professor in the Physics Department at the Virginia Polytechnic Institute and State University from 1967 to 1983, a senior technical staff member at MITRE from 1983 to 1984, and a senior scientist at Naval Research Laboratory from 1984 to 1994. His research involves scattering theory, high precision measurements, laser optics, medical imaging, wafer inspections, advanced radars, smart and generic ECM.

David C. Moody is a Captain with the USAF 333rd Training Squadron, Keesler Air Force Base, MS. He is the primary instructor for Deployable Communications Supplemental Course and also teaches Multiplexing and Modulation Theory as well as Satellite Communications. He earned a BSEE degree (with honors) from The University of South Alabama in 2000 and an MSEE degree from The Air Force Institute of Technology in 2004.

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Major Richard G. Cobb is Assistant Professor of Aerospace Engineering in the Department of Aeronautical and Astronautical Engineering, Air Force Institute of Technology, WPAFB, Ohio. He received his BS in aerospace engineering from the Pennsylvania State University in 1988, an MS in astronautical engineering from the Air Force Institute of Technology in 1992, and a Ph.D. from the Air Force Institute of Technology in 1996. He has been assigned to Kirtland AFB and served as a launch operations officer at Cape Canaveral AFS on the Global Positioning System program.

Anthony N. Palazotto, Professor of Aerospace Engineering, Air Force Institute of Technology, WPAFB OH, received his Ph.D. from New York University in 1968 with a specialty in the area of solid mechanics including a minor in applied mathematics. Dr Palazotto has over 348 presentations and publications, 168 of which are in archival journals. He is the co-author of a text entitled” The Nonlinear Analysis of Shell Structures” published by AIAA in 1992 and “Buckling and Postbuckling of Composite Plates” published by Chapman and Hall in 1995. He is a Fellow in ASCE, an Associate Fellow in AIAA, and a registered Professional Engineer.

Dr. Richard A. Raines is Director of the Center for Information Security Education and Research and Associate Professor of Electrical Engineering at the Air Force Institute of Technology (AFIT), Wright Patterson Air Force Base, Ohio. He earned a BSEE (with honors) from The Florida State University in 1985, a MS degree in Computer Engineering from AFIT in 1987 and a Ph.D. in Electrical Engineering from Virginia Polytechnic Institute and State University in 1994. He has more than 50 technical publications and is a Senior Member of the IEEE.

Thomas W. Küsel, BEng (Electronic) (Cum Laude), MSc (Electrical Engineering) is the manager of the Technical Visioneering Team at the Council for Scientific and Industrial Research in the Defence Electronics Programme. His experience and interest is in research and development of radar and radar countermeasures, with an emphasis on the development of radar test and evaluation facilities.

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Michael R. Inggs, PrEng, BSc (Hons) Rhodes, PhD London, MIEEE is professor at the Department of Electrical Engineering at the University of Cape Town in the division of radar and remote sensing. His interests are in the fields of radar remote sensing and parallel processing.

Erlank J. Pienaar, BEng (Electronic), BEng (Hons)(Electronic) is the manager of the radar and electronic warfare unit at the Council for Scientific and Industrial Research in the Defence Electronics Programme. His experience and interest is in research and development of radar and radar countermeasures, with an emphasis on the development of radar test and evaluation facilities.

Dr. Arthur G. Self was born in the United Kingdom. He earned a Ph. D. in Nuclear Physics and spent eleven years at the UK Ministry of Defence(MOD) leading the development of EW systems and research into advanced EW technologies for tri-Service applications. He represented the UK on NATO and TTCP EW panels. He emigrated to Canada to join an EW company. He has led R&D into novel, new receiver, processing, and frequency extension technologies including the development of naval ESM and ECM products. His career has spanned Government R&D, and senior positions in the Defence Industry. He

continues to publish widely and he is a technical reviewer for the IEEE and AOC. He is a registered Professional Engineer of Ontario and a member of the AOC, IEEE, and AFCEA.

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The Farey Series in Synchronisation andIntercept-Time Analysis for Electronic Support

I. Vaughan L. ClarksonSchool of Information Technology & Electrical Engineering

The University of QueenslandQueensland, 4072

Australiav.clarkson@itee.uq.edu.au

Invited Paper

Abstract

In Electronic Support, periodic search strategies for swept-frequency superheterodyne receivers(SHRs) can cause synchronisation with the radar it seeks to detect. Synchronisation occurs when theperiods governing the search strategies of the SHR and radar are commensurate. As a result, theradar may never be detected. In this paper, we find that, under certain conditions, the number ofratios of periods that can cause synchronisation is finite. We develop theory that can enumerate allof the ratios and determine the intercept time.

Index Terms

Electronic support, superheterodyne receiver, emitter intercept, synchronisation, Farey series,radar warning receiver, scan-on-scan.

I. Introduction

The superheterodyne receiver (SHR) has long been a primary tool for Electronic Support (ES).The swept-frequency SHR has the advantage of being able to cover a wide bandwidth and,by virtue of its narrow instantaneous bandwidth, it is selective and sensitive. However, a keyelement to the effectiveness of the swept-frequency SHR in operational environments is itssearch strategy.

The simplest strategy, and traditionally the most widely used, wholly or partly, is a simpleperiodic strategy, whereby the SHR repeatedly sweeps through the entire band of interest ata constant rate [1]. Thus, the times at which the SHR is tuned to any particular frequency areseparated by a fixed period, the sweep period.

When a swept-frequency SHR is searching for a radar which is also employing a periodicsearch strategy, such as a circularly scanned or raster scanned radar, it is well known thatsynchronisation can be a problem [1], [2]. Synchronisation can occur when the sweep period ofthe SHR and the scan period of the radar are commensurate, which is to say that the ratio of theperiods is rational. The effect of synchronisation is that energy from the radar is intercepted bythe SHR either very regularly or not at all. The latter possibility is ordinarily regarded as highlyundesirable. If the SHR and the radar are not precisely synchronised but nearly so (in a sensethat can be made precise) then it is possible that the time to intercept could be arbitrarily long.

This paper results from work that was performed under Contracts 4500100103 and 4500302428 for the Defence Scienceand Technology Organisation (DSTO), Department of Defence, Australia.

Preliminary versions of this paper have been published as a DSTO Research Report and presented at the IEEE Aerospace2003 conference.

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To the operator of an SHR, it is usually important to intercept any radars of interest in theminimum possible time. Therefore, synchronisation or near synchronisation is to be avoided.

In this paper, we consider a swept-frequency SHR employing a simple periodic search strategywhose sweep period is adjustable. We consider this question: for a given circularly scannedradar whose scan period and beamwidth are known, which sweep periods cause synchronisat-ion?

We find that there is usually a finite number of ratios between scan and sweep period thatcause synchronisation. We show that these ratios are the elements of a novel generalised Fareyseries and develop an algorithm for enumerating them. We also show that these ratios have anumber of interesting geometric interpretations.

A. The Synchronisation Problem in Electronic Support

In ES, a key operational requirement is the ability to detect or intercept users of the elec-tromagnetic spectrum in the shortest possible time. The ideal instrument for maintainingsurveillance would be a receiver that is able to monitor and resolve all of the spectrum at once.The instant any activity commenced, or came within range, it could be separately detected fromother users. Unfortunately, such a receiver is too large, heavy and expensive to be practicalwith today’s technology. Even to limit the bandwidth to that in which most radars operate —a bandwidth of many gigahertz — is beyond the reach of current technology, both in terms ofthe antenna and front-end technology required and the computational power needed to keepup with the data. Instead, a compromise must be sought.

A very commonly employed tool in ES is the swept-frequency superheterodyne receiver (SHR)[1]. This type of SHR aims to maintain surveillance over a wide search bandwidth by tuning andre-tuning a receiver of smaller bandwidth to different frequencies within the search bandwidth.We assume that the antenna on the SHR is omni-directional. This is typical of a radar warningreceiver.

In detecting radars, the SHR may encounter problems with synchronisation when this searchstrategy is used. Synchronisation is defined as a situation in which two or more recurrent eventsoccur in such a way that the pattern of their coincidences is periodic. In this case, it meansthat the SHR receives energy from the radar very regularly or not at all. The latter possibilityis considered to be highly undesirable in operational scenarios. The problem arises because ofthe periodic nature of the search strategy of the SHR and of the radar it is trying to detect.

The periodicity in an SHR employing a simple periodic search strategy exists because of thefixed number of dwells and the fixed dwell period. The times at which the SHR is dwellingon any particular frequency is therefore periodic. The period is called the sweep period of theSHR.

The chief source of periodicity in radars is the scanning pattern of its main beam, eitherthrough mechanical movement of the antenna or, in more modern and sophisticated radars,through electronic ‘beam steering’. For instance, a very common configuration for a radar isto have a mechanically rotated antenna which rotates continually at a constant angular ratethrough 360◦. The times at which the main beam of the radar is pointed towards the SHR isperiodic and the period is known as the scan period or seconds per revolution (SPR) of theradar. Thus, interception of the radar by the SHR is a so-called scan-on-scan problem.

Synchronisation of the type that results in a failure to detect occurs if, each time the SHRvisits the frequency on which the sought radar operates, the radar’s main beam is directedelsewhere. More precisely, it occurs when two conditions are satisfied. The first condition isthat the sweep period of the SHR and the scan period of the sought radar be commensurate,which is to say that the ratio of the sweep period to the scan period is rational, such as 4 : 3or 7 : 5. The second condition is that the integers which make up this ratio be not ‘too large’

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(in a sense which can be made precise). For instance, a ratio of 4 : 3 between scan and sweepperiod might produce synchronisation, whereas a ratio of 49 : 47 might not. These conditionsappear to have been discovered by Richards [3], who was the first to study synchronisationrigorously, in connection with an (unstated) problem in theoretical physics. His original workhas been extended, refined and applied to the SHR problem by others [4]–[6].

Even if the ratio between the scan and sweep period does not exactly satisfy the conditionsfor synchronisation, but instead is ‘close’ to satisfying them, then the two events required fordetection — namely, that the SHR is visiting the radar’s operating frequency and that the SHRis being illuminated by the radar’s main beam — may remain out of step for some considerabletime. For instance, although a ratio of 4 : 3 might be required for synchronisation, a ratio of4.001 : 3 may produce long periods in which the two events do not coincide. Clearly, for theoperator of a SHR, it is not desirable to be synchronised with a sought radar or to be evennearly so. Therefore, if a SHR is to employ a simple periodic search strategy, it is advantageousfor the operator to set it to a sweep period which is as far as possible from those that causesynchronisation.

B. Organisation of This Paper

The paper begins in Section II with some mathematical preliminaries. Here, the notation tobe used throughout the rest of the paper is introduced. The theory of intercept time, from theearliest work of [3] to the more recent work of [6], is briefly reviewed, as is its relationship tothe Farey series. It is the Farey series — which are particular sequences of rational numbers— that embody the conditions for synchronisation.

The hitherto established theory allows only for the analysis of intercept time for pulse trainsfor which the sum of pulse widths is assumed constant, although the periods or pulse repetitionintervals (PRIs) may be varied. As we shall see, for the determination of a sweep period for aSHR, it is more appropriate to assume that the duty cycles are constant, rather than the sumof the pulse widths. It is therefore necessary to extend previous results.

In Section III, we discover the conditions for synchronisation between pulse trains withconstant duty cycles. We will find that, usually, there are only a finite number of synchronisationratios in this case. We present three geometrical interpretations of the ratios and present ageneralised Farey series which contains them and an algorithm for generating them.

In Section IV, we show that the generalised Farey series provides all the information necessaryto compute the intercept time for any combination of PRIs. A procedure is presented whichallows variations in intercept time to be quickly computed in response to variations in theratio of PRIs.

Finally, in Section V, we briefly discuss the beginnings of a generalisation of the synchron-isation theory to scan-on-scan-on-scan problems. In these problems, three scanning processesmust coincide for detection to take place, for instance, where the SHR, in addition to sweepingin frequency, is also scanning a directional antenna in angle. We are able to give preliminaryresults establishing some conditions under which synchronisation can and cannot occur.

II. Mathematical Preliminaries

In this section, the intercept-time problem is interpreted mathematically as a problem con-cerning the coincidence of window functions or pulse trains. The notation used throughout theremainder of the paper is introduced, and the key results from the literature are reviewed. Theresults presented here are distilled from [3]–[6].

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A. Coincidence of Multiple Pulse Trains

Consider the situation where a SHR with an omni-directional antenna is employing a simpleperiodic search strategy and (amongst other emitters in a threat emitter list) seeks a particularcircularly scanned radar. The SHR visits the band in which the radar operates every sweepperiod. Each visit lasts a period of time equal to the dwell period. This can be representedmathematically as a function whose value is 1 when the SHR is visiting the radar’s band and 0otherwise. This function is periodic with the sweep period and can be interpreted as a periodicwindow function or pulse train. The width of each window or pulse is the dwell period.

Similarly, assume the SHR can only receive energy from the radar when the radar’s mainbeam is directed towards it. This occurs once every scan period and lasts for an amount oftime which is proportional to the beamwidth. Again, this can be expressed mathematically as apulse train with period equal to the scan period and pulse width determined by the beamwidth.

The radar is detected by the SHR only when both functions simultaneously take the value1. More generally, a situation can be considered where multiple pulse trains exist and it is ofinterest to determine when all of them simultaneously take the value 1. Given Nt pulse trains,we assign to each a PRI of Ti, a pulse width τi and a phase φi, where i = 1, . . . ,Nt. Throughoutthis paper, the PRI of any pulse train is assumed to be greater than zero and less than infinity.The phase is taken between the time origin and the centre of a pulse. A pulse from pulse traini occurs at all times t where ∣∣t − kiTi −φi∣∣ 6 1

2τi (1)

for some integer ki (ki ∈ Z) which we call the pulse index for the ith pulse train. Coincidenceof all Nt pulse trains occurs when (1) is satisfied for all i. This is illustrated for the case Nt = 3in Figure 1.

�������������������

t = 0 //t

oo �//φ1

oo �//φ2

oo �//φ3

�oo �//T1

�oo �//T2

�oo �//T3

oo //τ1

oo //τ2

oo //τ3

Pulse train 1

Pulse train 2

Pulse train 3

Coincidences

Fig. 1. Coincidence (intercept) of three pulse trains.

It is not difficult to show that a necessary and sufficient condition for coincidence is that∣∣(kiTi +φi)− (kjTj +φj)∣∣ 6 12(τi + τj) (2)

for all i, j = 1, . . . ,Nt. If this condition is satisfied for some set of pulse indices k1, . . . , kNt , thenthis combination of pulse indices produces a coincidence.

If we are only interested in coincidences that are of a minimum duration, say d, then it canbe shown that this is easily accommodated within (2) by replacing the ‘true’ or ‘natural’ values

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of the pulse widths by new values that are reduced by d. We observe that it is impossible fora coincidence of duration d to occur unless each of the pulse widths is not less than d.

As a special case, when Nt = 2, instead of reducing both pulse widths by d, we can reduceeither of them by any amount, so long as the sum of the reductions is 2d. For instance, wecould choose to reduce τ1 by 2d and leave τ2 unchanged. We reiterate that this reduction isonly for the purposes of finding coincidences through (2) — the true pulse widths must eachbe not less than d for coincidences of duration d to occur.

For the SHR problem, we have Nt = 2. We can assign the sweep period to T2, the dwell timeto τ2, the scan period to T1 and we can assign beamwidth (in degrees) to τ1 according to theformula

τ1 =beamwidth

360× T1. (3)

This is often called the illumination time.Recall that most radars emit periodic pulses with a certain PRI. Let this PRI be T3. It is

generally a requirement that the SHR should dwell in a band long enough to receive a specifiednumber, Nc, of consecutive emitted pulses from the sought radar. In some receiver processingarchitectures, these consecutive pulses are necessary to measure the PRI in order to declare adetection. Some authors then draw a distinction between detection and interception, the latterbeing said to have occurred if any energy (even a single pulse) from the radar is registered at thereceiver. We will use the terms ‘interception’ and ‘detection’ interchangeably, while continuingto recognise that a distinction may be made between the minimum duration of overlap required.

Hence, for detection, we require an overlap between the ‘SHR sweep’ pulse train and the‘radar scan’ pulse train that is of a duration at least equal to NcT3. We can assign this valueto d and reduce either τ1 or τ2 by 2d in (2) to find coincidences that result in the receptionof at least Nc consecutive emitted pulses from the radar.

B. Intercept Time for Two Pulse Trains

We now review the established results for intercept time between two periodic pulse trains.Here, we define the (maximum) intercept time as the maximum number of consecutive PRIsrequired from pulse train 2 until a coincidence occurs with a pulse from pulse train 1, regard-less of their phases. The intercept time is then defined as an integer multiple of T2. In termsof the SHR problem, the intercept time is therefore defined as the number of sweep periodsrequired to detect a given radar.

To examine intercept time more closely, it is useful to define the ratio of PRIs,

α = T2

T1, (4)

the normalised sum of pulse widths,1

ε = τ1 + τ2

T1, (5)

and the normalised phase difference,

β = φ2 −φ1

T1. (6)

Hence, we will examine the problem of intercept time relative to the PRI of pulse train 1, T1.

1Although the presentation of maximum intercept time here is essentially an abridgment of Section 3.1 of Chapter 5 of[6], the definition of the normalised sum of pulse widths, ε, and normalised phase difference, β, are different. The valuefor ε defined here is twice its value in [6] and β has opposite sign. These changes simplify the discussion of the newmaterial which appears later.

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With (4), (5) and (6), we can rewrite (2) so that we see that a coincidence occurs between thepth pulse of pulse train 1 and the qth pulse of pulse train 2 whenever∣∣qα− p + β∣∣ 6 1

2ε.

Clearly, for any p,q ∈ Z, there exists a range of normalised phase differences β for whichcoincidence will occur between these two pulses. Let Ip,q be this interval on R, which can beformally defined as

Ip,q ={x ∈ R |

∣∣qα− p + x∣∣ 6 12ε}

=[p − qα− 1

2ε,p − qα+12ε]. (7)

Thus, a coincidence with a pulse from pulse train 1 occurs with the 0th, 1st, . . . , or (n − 1)thpulse from pulse train 2 if

β ∈⋃

p,q∈Z;06q<n

Ip,q.

Let Cn(β) be the characteristic function of this union. That is, Cn(β) = 1 if there exists somep,q ∈ Z with 0 6 q < n such that β ∈ Ip,q and Cn(β) = 0 otherwise. Now, Cn(β) is periodicwith period 1. Therefore, a coincidence must have occurred with one of these n consecutivepulses if Cn(β) = 1 over any interval of length 1, regardless of the phases of the two pulsetrains. Thus, the intercept time is nT2, where n is the least value of n such that this conditionis true.

I1,0I1,1I1,2I1,3I1,4I0,0

1

_0

C5(β)

I1,0I2,5I1,1I2,6I1,2I2,7I1,3I2,8I1,4I0,0

1

_0

C9(β)

I1,0I2,5I1,1I2,6I1,2I2,7I1,3I2,8I1,4I0,0

I3,10

LLI3,11

LLI3,12

LLI3,13

LLI2,9

LL

1

_0

C14(β)

�������������������������������

β = 1

�������������������������������

β = 0

Fig. 2. The value of the characteristic function Cn(β) for n = 5, n = 9 and n = 14.

Figure 2 illustrates the value of the characteristic function Cn(β) over the unit interval [0,1]for n = 5, n = 9 and n = 14 where α = 0.217 and ε = 0.1. Note that it is not until the union

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with I3,13 in C14(β) that this function becomes uniformly equal to 1 across the entire unitinterval. Hence, the intercept time in this example 14T2.

Finally, we observe that it is possible to derive a simple lower bound on intercept time.Notice that, as n increases, the contribution of each new interval to the union eventuallylessens because of overlap. The union would grow most quickly to cover the unit interval ifoverlap did not occur. Since each Ip,q is of width ε, we can conclude that the intercept timetherefore cannot be less than 1/ε PRIs of pulse train 2. Thus,

intercept time >T2

ε= T1T2

τ1 + τ2. (8)

C. Intercept Time and Diophantine Approximation

We have seen how intercept time is related to a characteristic function Cn(β). When, as nincreases, this function becomes equal to 1 for all β, the value of n yields the intercept time.Now, we will describe how intercept time is related to Diophantine approximation.

Diophantine approximation is the study and practice of finding integers p and q, not bothzero, that make the expression

∣∣qα− p∣∣, or similar expressions, small for some real numberα. Of importance here is the definition of a best approximation. We define a best approximationp/q to α as one where q > 0 and, for all p′/q′ with q′ > 0, it is true that

q′ 6 q ⇒∣∣q′α− p′∣∣ >

∣∣qα− p∣∣and ∣∣q′α− p′∣∣ 6

∣∣qα− p∣∣ ⇒ q′ > q.

Note that we are abusing notation here. Strictly speaking, p/q may not properly be a fractionsince we allow q to be 0. Really, we should think of p and q as integer coordinates, but wewill persist with the fractional notation because, as we shall see, these numbers will usuallyrepresent ratios.

For any real number α, there is a series of best approximations to it, which we may write

p1

q1,p2

q2, . . . ,

pnqn, . . . ,

such that the absolute approximation error |ηn|,

ηn = qnα− pn,

is non-increasing (and, apart from a single exception when α = k + 12 for some integer k,

strictly decreasing) from one element of the series to the next. If α is rational, the seriesof best approximations is finite, the last element of the series being the expression of α as afraction in lowest terms, and the absolute approximation error of which being zero. Otherwise,if α is irrational, the series is infinite. The series can be found using Euclid’s algorithm, andthe elements correspond to the convergents of the simple continued fraction expansion of α.

We say that p/q is a best approximation of α to within ε if p/q is a best approximation andit is the first in the series of best approximations to α with absolute approximation error notgreater than ε.

The series of best approximations can be ordered in such a way that they exhibit the followingproperties [6], [7].• The approximation errors of successive elements of the series have opposite sign, i.e.,ηnηn+1 < 0, unless ηn+1 = 0, in which case the (n+ 1)th element is the last in the series.

13

• Successive elements obey a unimodularity property such that

pn+1qn − pnqn+1 ={

1 if ηn > 0,

−1 otherwise.(9)

The intercept time of any particular pair of pulse trains with PRI ratio α and normalisedsum of pulse widths ε can be determined by the following procedure.

1) Determine the best approximation of α to within ε, which we denote pn(ε)/qn(ε).2) If the approximation error of this best approximation is zero then α is rational and

corresponds to a synchronisation ratio. The intercept time in this case is infinite.3) Otherwise, determine the next element in the series, pn(ε)+1/qn(ε)+1.4) Calculate the value k according to the equation

k =⌊ε−

∣∣ηn(ε)+1∣∣∣∣ηn(ε)∣∣⌋,

where b·c is the floor function, i.e., that function which returns the greatest integer notgreater than its argument.

5) The intercept time is T2[qn(ε)+1 + qn(ε) − kqn(ε)

]. That is, a coincidence with pulse train 1

is guaranteed after qn(ε)+1+qn(ε)−kqn(ε) consecutive pulses from pulse train 2, regardlessof the phases.

We also observe that, instead of finding the succeeding best approximation in Step 3, wecould instead find any pair of integers (r , s) such that

rqn(ε) − spn(ε) = ±1,

where the sign on the right-hand side is positive if ηn(ε) > 0 or negative otherwise, as in (9). Ifwe replace k in step 4 with κ where

κ =⌊

ε− |sα− r |∣∣qn(ε)α− pn(ε)∣∣⌋

then the intercept-time expression in step 5 can be replaced with T2[s + qn(ε) − κqn(ε)

].2

D. Synchronisation, Intercept Time and the Farey Series

One method to enumerate the synchronisation ratios and to determine the intercept timeis to examine the Farey series of appropriate order. The Farey series of order n, Fn, is theseries of fractions in lowest terms in ascending order, such that the denominators of each arepositive and less than or equal to n [7]. Table I lists the Farey series between 0 and 1 for ordersone to five.

Consider two adjacent elements of Fn, h/k < h′/k′. The mediant of the elements is definedas (h+ h′)/(k+ k′). The adjacent elements h/k and h′/k′ remain adjacent in higher orders ofthe Farey series until the order reaches k+ k′, at which point they become separated by theirmediant. Adjacent elements also obey a unimodularity property, in that

h′k− hk′ = 1.

The determination of intercept time from the Farey series can be made by the procedure setout below.

2This observation, not made in either [5] or [6], significantly simplifies the procedure for determining intercept timefrom the Farey series. This manifests itself in (12) by obviating the need to define and keep track of so-called ‘left parents’and ‘right parents’ in the Farey series, as is prescribed in those earlier works.

14

TABLE I

The Farey series up to order five between 0 and 1.

01

11

01

12

11

01

13

12

23

11

01

14

13

12

23

34

11

01

15

14

13

25

12

35

23

34

45

11

1) We calculate the ratio of PRIs, α, and the normalised sum of pulse widths, ε, accordingto (4) and (5).

2) We calculate the appropriate order of the Farey series, which is d1/εe − 1.3) From this Farey series, we locate the adjacent pair of elements h/k and h′/k′ which

surround α. If α is in fact precisely equal to one of the elements of the series then theintercept time is infinite and no further steps need to be taken.

4) We calculate the value x1, where

x1 =

h+ εk

if k < k′,

h′ − εk′

otherwise,(10)

the values(p,q, P,Q

), where

(p,q, P,Q

)=

(h, k,h′, k′) if α < x1 or both k < k′

and α = x1,

(h′, k′, h, k) otherwise,

(11)

and the value of κ, where

κ =⌊ε− |Qα− P |∣∣qα− p∣∣

⌋. (12)

5) The intercept time is T2[Q+ q − κq

].

If we hold the sum of the pulse widths constant, so that ε is held constant in (5), then itis easy to vary T1 or T2 and observe the effect on the intercept time. By either increasing ordecreasing T1 or T2, the ratio α moves through the Farey series. As α approaches one of theelements of the series, the intercept time approaches infinity. As α moves between adjacentelements, the intercept time reaches a minimum around x1, as defined in (10).

When intercept time becomes infinite, the cause is synchronisation. Thus, the elements ofthe Farey series of order d1/εe − 1 are the complete series of ratios between T2 and T1 forwhich synchronisation occurs when the sum of pulse widths is held constant.

The value of κ in (12) is piecewise constant in sub-intervals between elements of the Fareyseries, and so the intercept time is piecewise linear over these same sub-intervals. Specifically,κ is constant in the sub-intervals

d1(κ) 6 α < d1(κ − 1) when α < x1,

15

or

d2(κ − 1) < α 6 d2(κ) when α > x1,

where

d1(j)= h

′ + jh− εk′ + jk (13)

and

d2(j)= h+ jh

′ + εk+ jk′ . (14)

These equations can be verified by direct derivation from (12), bearing in mind the additionalfacts that the sign of Qα− P is always opposite to that of qα−p and that qα−p is positiveor negative depending on whether α is less than or greater than x1, respectively.3

III. Synchronisation with Constant Duty Cycles

The discussion of Section II-E outlined a procedure to examine the variation of intercept timewith variation of the ratio of PRIs, α. The procedure demands that the pulse width of bothpulse trains are held constant (or, at least, that their sum is held constant) in order to fix anorder for the Farey series.

Instead, suppose that it is not the pulse widths that are held constant, but the duty cycles.The duty cycle of a pulse train is the ratio of the pulse width to the PRI.

To motivate this investigation, we return to the SHR problem. We should like to examinethe variation of intercept time with a given radar as we vary the sweep period, which we haveassigned to T2. However, usually, the number of frequency bands on which the SHR dwells isnot dependent on the sweep period, so the dwell period is a fixed proportion of the sweepperiod. Thus, τ2 is a fixed proportion of T2, and so the duty cycle of pulse train 2 is constant.

In general, let us define λ1 and λ2 as the duty cycles of pulse train 1 and pulse train 2,respectively. We restrict both λ1 and λ2 to be strictly greater than 0 and strictly less than1. We do this to simplify the following discussion by restricting the multiplication of specialcases which would otherwise result. In the case where one of the duty cycles is 0, the problemcan be formulated in such a way that the normalised sum of pulse widths is constant, and thiscase has already been dealt with in Section II. In the case where one of the duty cycles is 1,intercept is always immediate and the problem is trivial.

Now, in terms of the PRI ratio, α, the normalised sum of pulse widths, ε, from (5), can berewritten as

ε(α) = λ1 + λ2α.

Because ε is now a function of α in this new régime, it is not possible to directly use themethod given in Section II-E to find the synchronisation ratios or to evaluate the variation inintercept time with α from the Farey series, because the required order of the Farey series isnot necessarily constant.

In this section, we discover several interesting geometric interpretations of the intercept-timeproblem between pulse trains with constant duty cycles. Mathematically, these interpretationsarise from what appears to be a novel generalisation of the Farey series.

3The expressions for d1(·) and d2(·) in (13) and (14) should be able to be reconciled with the expressions for d1, d2,f1 and f2 in (30)–(33) of [5] and those which appear again in revised form on p. 176 of [6]. That they cannot is due to thepropagation of error on the author’s part which has only been discovered in the course of preparing this paper.

16

A. First Geometric Interpretation

For any particular value of α and ε, we know from our discussion of the Farey series inSection II-E that synchronisation can occur if and only if there exists some integers p and qsuch that ∣∣qα− p∣∣ = 0 and 0 < p and 0 < q <

1ε.

Additionally, in order to represent a proper ratio, we require that p and q be co-prime, i.e., thatthey have no common factors. Geometrically, the condition qα − p = 0 means that a line orray drawn from the origin with slope α passes directly through the point

(q,p

). Furthermore,

q <1ε= 1λ1 + λ2α

= 1λ1 + λ2

(p/q

) .This can be rewritten simply as

λ1q + λ2p < 1. (15)

Together with the conditions that p > 0 and q > 0, geometrically these conditions describe atriangle within which all possible synchronisation ratios p : q must lie. Thus, when both pulsetrains have a non-zero duty cycle, there can be only a finite number of synchronisation ratios.

//q

OOp

0 1 2 3 4 5 6 7 8

1

2

3

4

0.26p + 0.13q < 1

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

• • • • •

••

•

Fig. 3. First geometric interpretation of synchronisation ratios.

In Figure 3, we present a graphical depiction of the situation for λ1 = 0.13 and λ2 = 0.26.The line representing the boundary of the inequality (15) is drawn as a dotted line. The pointsof Z2 which correspond to synchronisation ratios are indicated with a bullet (‘•’), all othersare marked with a plus (‘+’). All of the points within the triangle delimited by (15) and theconstraints p > 0 and q > 0 correspond to synchronisation ratios, except the point (2,2),since the coordinates are not co-prime. Thus, reading from the graph, there are 7 possiblesynchronisation ratios for two pulse trains when the duty cycle of the first pulse train is 0.13and that of the second is 0.26. As a ratio of the PRI of the second pulse train to the first, theyare 1 : 5, 1 : 4, 1 : 3, 1 : 2, 2 : 3, 1 : 1, 2 : 1 and 3 : 1.

17

B. Second Geometric Interpretation

To develop a second geometrical interpretation for synchronisation between pulse trains withconstant duty cycles, we return to the construction we used in Section II-C for determining theintercept time between two pulse trains. In particular, we recall the definition of the indicatorfunction Cn(β). Recall that β is the normalised phase difference between the pulse trains andthat Cn(β) is the indicator function of the union of of intervals Ip,q with 0 6 q < n. Where thefunction Cn(β) is equal to 1, this means that, for this value of β, intercept between the pulsetrains is assured after n consecutive pulses from pulse train 2.

In the case of pulse trains with constant duty cycles, we can employ the following geometricconstruction to arrive at Cn(β). Consider an arrangement of identical rectangles centred overthe elements of Z2 in R2. The sides of the rectangles are aligned with the horizontal and verticalaxes. The lengths of the sides are λ2 along the horizontal axis, which we will call the q axis,and λ1 along the vertical axis, which we will call the p axis.

Let Rp,q represent the rectangle centred on the point with the specified p and q coordinates.The projection of Rp,q along the line of slope α onto the p axis is the interval Ip,q of (7),which can be deduced from simple geometrical considerations. The function Cn(β) is then theindicator function of these projections for the first n columns of rectangles from q = 0 toq = n− 1.

//q

OOp

0

1

1 2 3 4R0,0

XX+ +R1,0 +

R2,0 +R3,0 +

R4,0

+ R0,1 +R1,1 +

R2,1 +R3,1 +

R4,1

I4,1

I3,1

I2,1

I1,1

I0,0

LL

I0,1 ��

Fig. 4. Geometric construction of C5(β) for pulse trains with constant duty cycles.

In Figure 4, the geometric construction of Cn(β) for n = 5 is illustrated. As for Figure 2, thevalue of α is 0.217 and ε = 0.05. However, we have now assigned duty cycles of 0.0566 forpulse train 1 and 0.2 for pulse train 2. It can be verified that these duty cycles are consistentwith ε = 0.1 for α = 0.217. The rectangles Rp,q are drawn and their centres marked with a‘+’. The intervals Ip,q are indicated by heavy lines along the p axis. They are formed by theprojections of the corresponding Rp,q along lines of slope α = 0.217, indicated by the dottedlines. The indicator function of the union of the Ip,q gives the same function as depicted inFigure 2 for C5(β).

To complete the second geometric interpretation, and as an aid at many points in thediscussion thereafter, we find the following theorem useful. The proof of this theorem andall other theorems, propositions and lemmas in this paper are to be found in the Appendix.

Theorem 1: Suppose qα− p = 0 for some integers p and q > 0, and p and q are co-prime.Then there exist no solutions in integers r and s to the inequality |sα− r | < 1/q, unless r isan integer multiple of p and s is the same multiple of q. However, solutions do exist to theequation sα− r = ±1/q. If α > 0, solutions exist with 0 6 r 6 p and 0 6 s 6 q.

Consider a fraction p/q which corresponds to a synchronisation ratio. First, Theorem 1 tells

18

us thatI0,0 = Ip,q = I2p,2q = . . .

Furthermore, apart from multiples of(p,q

), since q < 1/ε, any other interval Ir ,s must lie a

distance greater than ε from the origin along the p axis. Thus any point sufficiently close to I0,0does not belong to any interval Ir ,s whatsoever. Now, I0,0 is the projection onto the p axis ofthe rectangle R0,0. Therefore, if we project a ray with slope α = p/q from a point sufficientlyclose to (but above and to the left of) the top left corner of R0,0, it will not pass through anyother rectangle Rr ,s .

On the other hand, if a fraction p/q does not correspond to a synchronisation ratio becauseq > 1/ε, then we can find intervals Ir ,s that partially overlap I0,0. Therefore, from a point nearthe top left corner of R0,0, a ray projected with slope α will always intersect some rectangleRr ,s .

These arguments lead us to our second geometric interpretation of synchronisation betweenpulse trains of constant duty cycle. From the upper left corner of the rectangle R0,0, projecta ray with slope 0 < α < ∞. If the ray continues to infinity without being obstructed by anyother rectangle Rr ,s then that value of α yields a synchronisation ratio. Figure 5 depicts theserays for the case where the duty cycles are 0.13 and 0.26, as for Figure 3.

//q

OOp

0 1 2 3 4 5 6 7 8

1

2

3

4

JJ

α=

3

GG

α=

2

??

α= 1

::

α =2/3

77

α =1/2

44

α = 1/3

33α = 1/4

22

α = 1/5

Fig. 5. Second geometric interpretation of synchronisation ratios.

C. Third Geometric Interpretation

Consider rectangles Sp,q which, like the Rp,q, are each centred on points in Z2. However, theSp,q have twice the width and twice the breadth of the Rp,q. Moreover, suppose there is norectangle S0,0. When projected along lines of slope α onto the p axis, they form intervals Jp,qcentred at qα− p of width 2ε.

If Sp,q corresponds to a synchronisation ratio then a line segment from the origin to thecentre of Sp,q is not obstructed by any other rectangle Sr ,s . To see this, consider the projectionsonto the p axis. From Theorem 1, all other intervals Jr ,s must be centred at least a distance1/q > ε from the origin. Therefore, none of these other intervals contain the origin.

19

Notice that if Sp,q corresponds to a synchronisation ratio then the rectangles Skp,kq areinvisible from the origin for all positive integer multiples k. By invisible from the origin, wemean that the line segment from any point in one of these rectangles to the origin must passthrough another rectangle — in this case, it must pass through Sp,q.

Conversely, if Sp,q, with p and q co-prime and positive, does not correspond to a synchron-isation ratio then a line extending from the origin to the centre of Sp,q must be obstructedby another rectangle Sr ,s . This is because Theorem 1 guarantees that there is some pair ofintegers r and s with 0 6 r 6 p and 0 6 s 6 q such that sα− r = 1/q 6 ε. Thus, Jr ,s containsthe origin.

Furthermore, in this case, Sp,q is wholly invisible from the origin. To understand this, considera point X within Sp,q and suppose, without loss of generality, that its projection onto the paxis is on the positive p axis at, say, x. Now, x < ε. Thus, any point on the line segment OX,when projected onto the p axis lies in the interval [0, x]. Again, Theorem 1 guarantees that wecan find integers r and s with 0 6 r 6 p and 0 6 s 6 q such that sα−p = 1/q. The projectionJr ,s contains [0, x]. Therefore, at some point, the line segment must pass through Sr ,s .

Together, these arguments yield our third geometric interpretation of synchronisation be-tween pulse trains of constant duty cycle. Any rectangle Sp,q that is (even partly) visible fromthe origin corresponds to a synchronisation ratio if p and q are both positive.

//q

OOp

0 1 2 3 4 5 6 7 8

1

2

3

4

Fig. 6. Third geometric interpretation of synchronisation ratios.

Figure 6 illustrates this third geometric interpretation for the now familiar case where thetwo pulse trains have duty cycles of 0.13 and 0.26, such as in Figure 3 and Figure 5. Thelimits of visibility from the origin are indicated by dotted lines. Those portions of the sides ofthe Sp,q that are visible are indicated by heavy lines. As expected, only those rectangles thatcorrespond to synchronisation ratios are even partly visible.

This geometric interpretation of synchronisation is remarkably similar to that discovered byAllen [8] in relation to phase locking between coupled neurons. The only substantial differenceis that, in Allen’s construction, the rectangles are not centred on the points of Z2, but offsetfrom them. Therefore, our interpretation might be considered a ‘homogeneous’ or ‘central’version of a class of similar constructions of which Allen’s is one other (inhomogeneous ornon-central) example.

20

D. A Generalised Farey Series

In Section II-E, we saw that the synchronisation ratios are given by the Farey series ofappropriate order in the case where the pulse widths of the two pulse trains are held constant.This allows us to vary the ratio between the PRIs and quickly recompute the intercept time. Wehave now shown that, for the case of constant duty cycles, the ratios could be found from theintersection of Z2 with a triangle. We now describe these ratios as a generalised Farey series.We set out an algorithm for computing the complete series in order.

Let us define a generalised Farey series G(λ1, λ2) as the series of fractions p/q in lowest termsin ascending order such that each fraction corresponds to a synchronisation ratio between twopulse trains of duty cycles λ1 and λ2, respectively. For the case λ1 = 0.13 and λ2 = 0.26, aswas used in Figures 3, 5 and 6, the series G(0.13,0.26) is

15,14,13,12,23,11,21,31.

We now present an algorithm which recursively outputs the generalised Farey series.Algorithm 1:

1 proc genfarey(h, k,h′, k′, λ1, λ2) ≡2 if λ1(k+ k′)+ λ2(h+ h′) < 1 then3 genfarey(h, k,h+ h′, k+ k′, λ1, λ2);4 output((h+ h′)/(k+ k′));5 genfarey(h+ h′, k+ k′, h′, k′, λ1, λ2);6 fi.

To output the entire generalised Farey series, we execute genfarey(0,1,1,0, λ1, λ2). We makethe following observations about the outputs of this algorithm when it is executed like this.

1) The outputs are in ascending order.2) Any particular output p/q satisfies the inequalities p > 0, q > 0 and (15).3) When genfarey is first executed, and on subsequent calls to itself, the parameters h, k,h′ and k′ satisfy the unimodularity property:

h′k− hk′ = 1. (16)

4) As a consequence of this unimodularity, the outputs are always in lowest terms. Forsuppose, on the contrary, that the procedure were to produce an output that was not inlowest terms. That is, at line 4, suppose h+h′ and k+k′ have some common factor c > 1.Then (h+ h′)/c and (k+ k′)/c must be integers. Hence, any integer linear combinationof these integers should also yield an integer. But

kh+ h′c

− hk+ k′

c= h

′k− hk′c

= 1c,

as a result of unimodularity, and clearly 1/c is not an integer, contradicting our suppo-sition.

5) If 1/0 is taken here to equate to ∞ then we can say that, when genfarey is first executed,and on subsequent calls to itself, it is always true of the parameters that h′/k′ > h/kand all outputs (if any) lie strictly between these two fractions.

6) The procedure completes in a finite amount of time. For if this were not so, it could onlybe because genfarey kept calling itself forever. But this cannot occur, because either k+k′or h+ h′ must increase by at least one each time genfarey calls itself and, since λ1 andλ2 are both positive, the test at line 2 must eventually fail.

From observations 1–4 above, we can conclude that the following proposition is true.Proposition 1: The output from the execution of the algorithm genfarey(0,1,1,0, λ1, λ2) is a

properly ordered sub-series of G(λ1, λ2). 21

The converse, which we now state, is also true. Its proof can be found in the Appendix.Proposition 2: All of the elements of G(λ1, λ2) are contained in the output from the execution

of the algorithm genfarey(0,1,1,0, λ1, λ2).Consider adjacent elements in G(λ1, λ2). Since we have now shown that these are equivalent

to the outputs of genfarey , consider what must occur in the execution of this procedurebetween two adjacent outputs. It is clear that if, at some stage in the execution, genfareycalls itself at line 3 and, upon return, no output has been produced, then the output of(h+ h′)/(k+ k′) on line 4 will be adjacent in the output sequence to h/k, unless h/k = 0/1.Similarly, if, at some stage in the execution, genfarey calls itself at line 5 and, upon return, nooutput has been produced, then the output of (h+ h′)/(k+ k′) on line 4 will be adjacent inthe output sequence to h′/k′, unless h′/k′ = 1/0. Indeed, for every pair of adjacent elements inthe series, one of these two situations must have arisen. This leads to the following theorem.

Theorem 2: Every pair of adjacent elements p/q < r/s in G(λ1, λ2) satisfies the unimodu-larity property

rq − ps = 1.

IV. Intercept Time for Constant Duty Cycles

So far, we have determined that, for two pulse trains with constant, positive duty cycles, thereare a finite number of PRI ratios that can give rise to synchronisation. We have discovered threegeometric interpretations for these ratios, proposed a generalisation of the Farey series whichdescribes them and an algorithm for producing them.

In this section, we describe how this generalised Farey series can be used to compute theintercept time for two pulse trains with constant duty cycles. This allows us to observe theeffect on intercept time as one or both of the PRIs are varied. The resulting procedure is verysimilar to that presented in Section II-E for pulse trains with constant pulse width.

As discussed in Section II-D, in order to compute intercept time for any particular value ofα and ε, it is sufficient to find the best approximation of α to within ε and the subsequentbest approximation.

First, let us define the augmented, generalised Farey series G∗(λ1, λ2) as the series whichconsists of the elements of G(λ1, λ2) in their original order, but with the addition of 0/1 as thefirst element and 1/0 as the last element. We observe that adjacent elements in the augmentedseries still obey the unimodularity property of Theorem 2.

The following lemma and theorems establish that, for any given α, one of the two adjacentelements of G∗(λ1, λ2) must be a best approximation.

Lemma 1: If h/k < h′/k′ are adjacent elements of G∗(λ1, λ2) then

h+ λ1

k− λ2>h+ h′k+ k′ >

h′ − λ1

k′ + λ2. (17)

Theorem 3: Suppose h/k < h′/k′ are adjacent in G∗(λ1, λ2). If h/k 6 α 6 h′/k′ then either

|kα− h| 6 ε(α) or∣∣k′α− h′∣∣ 6 ε(α).

Theorem 3 tells us that, for any given α, one of the adjacent elements in the augmented,generalised Farey series must be a ‘good’ approximation, but it remains to prove that one ofthem is a best approximation.

Theorem 4: Suppose h/k < h′/k′ are adjacent in G∗(λ1, λ2) and suppose h/k 6 α 6 h′/k′.Either h/k or h′/k′ is a best approximation for α to within ε(α). Specifically, if k < k′ thenh/k is a best approximation to within ε(α) when

α 6h+ λ1

k− λ2, (18)

22

otherwise h′/k′ is a best approximation. On the other hand, if k > k′ then h′/k′ is a bestapproximation when

α >h′ − λ1

k′ + λ2,

otherwise h/k is a best approximation.We have now shown that, for any PRI ratio α, the best approximation of α to within ε(α)

can be found by a simple procedure from the two surrounding elements in the augmented,generalised Farey series, G∗(λ1, λ2). From the procedure for calculating intercept times frombest approximations in Section II-D, we can then verify that the following procedure can beused to calculate intercept time between pulse trains with constant duty cycles (and this is astraightforward adaptation of the procedure from Section II-E for calculating intercept timesfrom the Farey series).

1) Calculate the ratio of PRIs, α, according to (4).2) Calculate the augmented, generalised Farey series, G∗(λ1, λ2).3) From G∗(λ1, λ2), locate the adjacent pair of elements h/k < h′/k′ that surround α. Ifα = h/k or α = h′/k′ then the intercept time is infinite.

4) Otherwise, we calculate the value x1 where

x1 =

h+ λ1

k− λ2if k < k′,

h− λ1

k+ λ2otherwise,

and the values(p,q, P,Q

)and κ from (11) and (12), respectively.

5) The intercept time is T2[Q+ q − κq

].

As with the procedure for calculating intercept times from the Farey series given in Section II-E, we can see that the value of κ in (12) is constant in sub-intervals between elements of theaugmented, generalised Farey series. Therefore, the intercept time is piecewise linear overthese same sub-intervals. However, because ε is now a function of α, the intervals on which κis constant are those for which

d∗1 (κ) 6 α < d∗1 (κ − 1) when α < x1,

or

d∗2 (κ − 1) < α 6 d∗2 (κ) when α > x1,

where

d∗1(j)= h

′ + jh− λ1

k′ + jk+ λ2

and

d∗2(j)= h+ jh

′ + λ1

k+ jk′ − λ2.

In Figure 7, we present a plot of the intercept time between two pulse trains. Again, we usea duty cycle of 0.13 for pulse train 1 and a duty cycle of 0.26 for pulse train 2. The PRI ofpulse train 1, T1, is held constant with T1 = 1, and T2 is allowed to vary between 0.1 and 4.The intercept time is plotted as a solid line. From the plot, we can see that the intercept timegoes to infinity at all the points of G(λ1, λ2) — the synchronisation ratios.

23

0 1 2 3 4

5

10

15

20

Pulse repetition interval of pulse train 2, T2

Inte

rcep

tti

me

Fig. 7. Intercept time for two pulse trains with constant duty cycles. (Key: intercept time, __ theoretical lower bound.)

The theoretical lower bound on intercept time from (8) is also plotted in Figure 7 in a dashedline. We see that the intercept time approaches this lower bound at several points.

For the SHR problem, it is now apparent that the procedure we have described in this sectioncould be used to generate plots such as that depicted in Figure 7. From a plot of this type, itis possible to deduce a sweep period within operating limits that minimises the intercept timewith a radar of known scan period.

V. Synchronisation in Scan-On-Scan-On-Scan Problems

We conclude the paper with a brief discussion of the conditions which give rise to synchron-isation in so-called ‘scan-on-scan-on-scan’ problems. In these scenarios, we have three periodicwindow functions representing different ‘scanning’ processes. For instance, we might have ascenario in which the SHR does not have an omni-directional antenna but a directional one, andso must scan in angle in the same was as we have assumed the radar does. For interceptionto occur now, the receiver antenna must be pointing at the radar, the radar antenna must bepointing at the receiver and the receiver must be tuned to the radar’s band. Three pulse trainsare now at work: two which, as before, represent whether the radar antenna is pointing towardsthe receiver and whether the receiver is in the right band, as well as one new one, whetherthe receiver antenna is pointing towards the radar. Our question here is then this: given theduty cycles of the three pulse trains (corresponding to beamwidths and dwell time), are thereconditions in which interception may never occur and, if so, what are they?

We are not yet able to give a complete characterisation of three-way synchronisation, butwe are able to make some important observations, expressed as theorems. The first one isobvious.

Theorem 5: Synchronisation between three periodic pulse trains can occur if it can occurbetween any two of them.

24

To prove the second theorem, we need to appeal to Kronecker’s Theorem, for which weneed the notion of linear independence of numbers [7]. We say that the numbers ξ1, . . . , ξn arelinearly independent if no linear relation

a1ξ1 + · · · + anξn = 0

with integral coefficients, not all zero, also known as an integer relation, holds between them.Lemma 2 (Kronecker’s Simultaneous Approximation Theorem): If θ1, . . . , θk,1 are linearly in-

dependent, α1, . . . , αk are arbitrary real numbers and ε is a positive real number then thereexist integers n,p1, . . . , pk such that∣∣nθm − pm −αm∣∣ < εfor m = 1, . . . , k.

We can now make our second observation about three-way synchronisation. Here, we definethe pulse repetition frequency (PRF) of a pulse train as the inverse of its PRI.

Theorem 6: If the PRFs of three pulse trains are linearly independent and their pulsewidthsare positive then synchronisation cannot exist between them.

Proof: Let θ1 = T1/T2 and θ2 = T1/T3. Clearly, θ1, θ2 and 1 are linearly independent.Consider any combination of phases φ1, φ2 and φ3 and any positive pulsewidths τ1, τ2 andτ3. Set

α1 = (φ2 −φ1)/T2,α2 = (φ3 −φ1)/T3

and

ε = 12 min {τ2/T2, τ3/T3}

By Kronecker’s Theorem, there exist integers k1, k2 and k3 such that

|k1θ1 − k2 −α1| < ε|k1θ2 − k3 −α2| < ε

Multiplying by T2 and T3, respectively, these inequalities become∣∣(k1T1 +φ1)− (k2T2 +φ2)∣∣ < 1

2τ2 6 12(τ1 + τ2)∣∣(k1T1 +φ1)− (k3T3 +φ3)

∣∣ < 12τ3 6 1

2(τ1 + τ3)

By the triangle inequality, we further have that∣∣(k2T2 +φ2)− (k3T3 +φ3)∣∣ < 1

2(τ2 + τ3).

But these inequalities are just the conditions for three-way interception. Since the phases werearbitrary, we conclude that three-way synchronisation is not possible.

We note that these theorems are easily generalised to higher-order intercepts. In particular,a simple extension of Theorem 6 is that Nt pulse trains cannot be synchronised if theirpulsewidths are positive and their PRFs are linearly independent. On the other hand, for twopulse trains, this extension of Theorem 6 is equivalent to stating that synchronisation cannotoccur when the PRFs (and therefore the PRIs) are not commensurate: a fact already well knownto us.

Clearly, a full characterisation of the conditions under which synchronisation occurs inscan-on-scan-on-scan problems is not provided by Theorems 5 and 6. It is the author’s beliefthat a more thorough consideration of the implications of Kronecker’s theorem will yield thenecessary insights.

25

Appendix

This Appendix sets out the proofs of the theorems, propositions and lemmas stated in themain body of the text.

Proof of Theorem 1: Rewrite the inequality |sα− r | < 1/q as∣∣sp − rq∣∣ < 1 by multiplying

throughout by q. The expression sp − rq involves only integer variables so it must take aninteger value. If its absolute value is less than 1, then it must be zero, which implies, exceptwhere r = s = 0, that r/s = p/q. Since p and q are co-prime, r must be an integer multiple ofp and s the same multiple of q.

If sα− r = ±1/q then, by again multiplying throughout by q, we have sp − rq = ±1. It is abasic result from the moduli of integers that solutions to this equation exist [7].

If any one solution (r , s) is found then all integer pairs of the form(r + kp, s + kq

)are

solutions, for k ∈ Z. Therefore, it is clear that we can find a solution for which 0 6 r 6 p or0 6 s 6 q. It remains to show that both conditions can be satisfied simultaneously when α > 0.

If α > 0 then p > 0. Suppose we choose a solution with 0 6 s < q. Then, rq = sp±1 impliesthat rq > −1 and rq 6 pq. Since q > 0, this implies that −1 6 r 6 p. If r = −1, it can only bebecause s = 0. Thus, apart from (r , s) = (−1,0), we have shown that, when α > 0, the equationsα− r = ±1/q has solutions with 0 6 r 6 p and 0 6 s 6 q. But if (r , s) = (−1,0) is a solution,then (r , s) =

(p − 1, q

)must also be a solution, and the theorem is proved.

Proof of Proposition 2: If an element p/q of G(λ1, λ2) is missed, it must be because at somestage genfarey is called, or calls itself, with parameters h, k, h′ and k′ with h/k < p/q < h′/k′and the test at line 2 fails.

Now, to complete the proof, we show that

pq= ah+ bh

′

ak+ bk′ (19)

where a and b are positive integers. The numerator and denominator of (19) can be viewed astwo simultaneous linear equations. We can quickly verify that the solutions for a and b are

a = kp − hqh′k− hk′ = kp − hq,

b = h′q − k′ph′k− hk′ = h

′q − k′p.

Hence, both a and b are integers. Now a and b are positive because, for a, p/q−h/k > 0, whichimplies that kp − hq > 0, and, for b, h′/k′ − p/q > 0, which implies that h′q − k′p > 0. (Thespecial case where h′/k′ = 1/0 requires a trivially different treatment but does not invalidatethis conclusion.)

Finally, since p/q ∈ G(λ1, λ2), we have

λ1q + λ2p < 1

and we have supposed that the test on line 2 fails, so that

λ1(k+ k′)+ λ2(h+ h′) > 1.

Subtracting these two inequalities and substituting (19), we have

λ1[(a− 1)k+ (b − 1)k′]+ λ2[(a− 1)h+ (b − 1)h′] < 0.

But this is impossible because the terms on the left-hand side — λ1, λ2, h, h′, k, k′, a− 1 andb − 1 — are all non-negative.

Proof of Theorem 2: From the discussion preceding the theorem statement, we knowthat, for each pair of adjacent elements, there was a point in the execution of genfarey where

26

(h+ h′)/(k+ k′) was output and it was adjacent to and preceded by h/k or (h+ h′)/(k+ k′)was output and it was adjacent to and succeeded by h′/k′. In both cases, the unimodularityproperty (16) furnishes the required result.

Proof of Lemma 1: Consider the left-hand inequality of (17). We have

h+ λ1

k− λ2− h+ h

′

k+ k′ =(h+ λ1)(k+ k′)− (k− λ2)(h+ h′)

(k− λ2)(k+ k′).

Using the unimodularity property, this simplifies so that

h+ λ1

k− λ2− h+ h

′

k+ k′ =λ1(k+ k′)+ λ2(h+ h′)− 1

(k− λ2)(k+ k′). (20)

Now, the numerator of the right-hand side is non-negative since, if it were negative, it wouldmean that (h+ h′)/(k+ k′) satisfies (15) and therefore that this fraction was an element ofG∗(λ1, λ2), separating h/k and h′/k′, contrary to our assumption. Similarly, the denominator ispositive since k > 1, k′ > 0 and λ2 < 1, ensuring that the right-hand side of (20) is non-negative.Thus, we have confirmed the left-hand inequality of (17).

The right-hand inequality of (17) can be confirmed using the same method.Proof of Theorem 3: Suppose that

hk

6 α 6h+ h′k+ k′ .

From the left-hand inequality of (17), it is then true that

hk

6 α 6h+ λ1

k− λ2.

The left-hand inequality implies that

kα− h > 0

and the right-hand inequality implies that

kα− h 6 λ1 + λ2α = ε(α).

On the other hand, ifh+ h′k+ k′ 6 α 6

h′

k′

then, using the right-hand inequality of (17), we find that

−ε(α) 6 k′α− h′ 6 0.

Proof of Theorem 4: It has already been shown that the theorem is true if α = h/k orα = h′/k′. Let us therefore assume that α lies strictly between these two values. We will provethe theorem only for the case where k < k′. The same technique can be used when k > k′,although care must be taken in the special case where h′/k′ = 1/0 (and this case can be avoidedby simply swapping the two pulse trains).

Suppose (18) is satisfied. Then, to show that h/k is a best approximation for α to withinε(α), we show that |sα− r | > ε(α) for any non-zero integer pair (r , s) ≠ (h, k) with 0 6 s 6 k.If sα− r > 0 then write

sα− r = A+ B (21)

27

where

A = shk− r ,

B = s(α− h

k

).

Consider the value of B. We have

0 < B < s(h′

k′− hk

)= sh

′k− hk′kk′

= skk′

61k′. (22)

From Theorem 1, we know that |A| > 1/k. If A < 0, this would mean that sα− r = A+ B < 0,contrary to our assumption. On the other hand, if A > 0 then sα− r > 1/k > ε(α).

If sα− r < 0 then writesα− r = C +D (23)

where

C = sh′

k′− r ,

D = s(α− h

′

k′

).

Similar to (22), we have

0 > D > s(hk− h

′

k′

)= − s

kk′> − 1

k′.

Again, from Theorem 1, |C| > 1/k′. If C > 0, this would mean that sα − r = C + D > 0,contrary to our assumption, so instead we have sα−r < −1/k′ < −ε(α). Therefore, regardlessof whether sα− r is positive or negative, we have |sα− r | > ε(α). Thus, h/k must be a bestapproximation of α to within ε(α).

Now, suppose (18) is not satisfied. The argument is almost the same, with one small dif-ference. Let (r , s) ≠ (h′, k′) be any integer pair with 0 6 s 6 k′. We want to show that|sα− r | > ε(α). If (r , s) is an integer multiple of (h, k) then, because of (18) not being satisfied,sα−r > ε(α). Apart from this distinction, the remainder of the argument that r/s cannot be abest approximation of α to within ε(α) is the same. That is, suppose (r , s) ≠ (h′, k′) and (r , s)is not an integer multiple of (h, k) with 0 6 s 6 k′. If sα − r > 0 then we decompose sα − rinto the sum of A and B as in (21). As before, it then follows that sα− r > ε(α). Similarly, ifsα − r < 0 then we decompose sα − r into the sum of C an D as in (23). As before, it thenfollows that sα−r < −ε(α). Thus, h′/k′ must be a best approximation of α to within ε(α).

Acknowledgement

The author would like to express his sincere gratitude to Dr. Greg Noone of the Defence Sci-ence & Technology Organisation, Australia, for many helpful discussions that have contributeddirectly to the improvement of the material presented in this paper.

References

[1] R. G. Wiley, Electronic Intelligence: The Interception of Radar Signals. Norwood, Massachusetts: Artech House, 1985.[2] A. G. Self and B. G. Smith, “Intercept time and its prediction,” IEE Proc., vol. 132F, no. 4, pp. 215–222, July 1985.[3] P. I. Richards, “Probability of coincidence for two periodically recurring events,” Ann. Math. Stat., vol. 19, no. 1, pp.

16–29, Mar. 1948.[4] S. W. Kelly, G. P. Noone, and J. E. Perkins, “The effects of synchronisation on the probability of pulse train interception,”

IEEE Trans. Aerospace Elec. Systems, vol. 32, no. 1, pp. 213–220, Jan. 1996.[5] I. V. L. Clarkson, J. E. Perkins, and I. M. Y. Mareels, “Number theoretic solutions to intercept time problems,” IEEE Trans.

Inform. Theory, vol. 42, no. 3, pp. 959–971, May 1996.[6] I. V. L. Clarkson, “Approximation of linear forms by lattice points with applications to signal processing,” Ph.D.

dissertation, The Australian National University, 1997.[7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed. Oxford University Press, 1979.[8] T. Allen, “On the arithmetic of phase locking: Coupled neurons as a lattice on R2,” Physica, vol. 6D, pp. 305–320, 1983.

28

Higher Order Shift Register Sequences

John Fielding

CACI Technologies, Inc. Abstract Binary shift registers often are used to generate pseudo-random sequences. The maximum length sequence generated by such a shift register is 2n – 1, where n is the number of shift register stages. When more than two states need to be represented, the contents of multiple stages (k) in the shift register can be used as binary values to represent additional states ranging from “1” to “2k − 1.” An alternative to forming a binary word is to change the base from “2” to a larger prime number (p). Then, the number of states represented is “p” and the maximum length sequence generated from the shift register is pn – 1. This paper illustrates how higher order (> Base 2) shift register sequences can be generated. Introduction Pseudo-random sequences are useful for a variety of applications – radio frequency hopping, pulse interval modulation, intrapulse phase modulation, scan sequencing, etc. By far, the most common method of generating pseudo-random sequences is with a binary linear shift register configured to produce a maximal length recursive sequence. However, the commonly used binary shift register generator can be extended to operate using larger prime bases. Extending the number base provides additional flexibility in the generation of pseudo-random parameter sequences. Discussion The traditional representation of a shift register within the ELINT/EW community is given in Figure 1. The shift operation is to the right (rather than to the left as in communications and mathematical treatments of this subject). The contents of the selected shift register stages are added Modulo-2 and the result is used as the next input to the first stage of the shift register. As the shift register is clocked to the next cycle, the contents of the current stages are shifted to the right and the result of Modulo-2 addition of selected stages is moved into Stage 1 of the shift register. The process continues until terminated, which usually happens at the point where the sequence repeats.

29

Figure 1. Conventional Shift Register Representation In the example shown in Figure 1, the first and third stages of a three-stage shift register are added Modulo-2 and the result becomes the input to Stage 1. Any sequence of “1’s” and “0’s” except all “0’s” can be used for the initial fill of the shift register. In Figure 1, the initial fill is chosen to be 1-1-1. Modulo-2 addition of the initial contents of Stages 1 and 3 results in a value of “0” [1 +1 (Mod 2) = 0]. After the first shift, the contents of the shift register are 0-1-1. The next Modulo-2 addition of the contents of Stages 1 and 3 results in a value of “1” [0 +1 (Mod 2) = 1]. Thus, the contents of the shift register after the second shift are 1-0-1. The process continues until the sequence repeats. Since there are three stages, the maximal sequence length is 23 – 1 = 7. When more than two states must be represented, the stages of a binary register can be used to represent a larger numerical value. For example, the three registers shown in Figure 1 can be treated as a number to represent seven distinct values. Figure 2 repeats the shift register generator and shows the mapping of shift register stages to numerical values.

Figure 2. Using a Binary Shift Register to Represent More than Two Values

30

The shift register diagrams given in Figures 1 and 2 show the contents of two stages being sent to an adder and the contents of another stage being ignored. These diagrams can be re-drawn to show all stages being sent to the adder and weights of either “1” or “0” attached to each path. Such a reconstruction of the shift register generator is shown in Figure 3. The weights in Figure 3 that are equivalent to the configuration shown in Figures 1 and 2 are W1 = 1, W2 = 0, and W3 = 1. In the case of Base-2 operation, a weight of “1” usually is represented by connecting the appropriate shift register stage to the Modulo-2 adder and a weight of “0” usually is represented by no connection. The representation given in Figure 3 explicitly assigns the weight values.

Figure 3. Explicitly Assigning Weights to Shift Register Stages The representation given in Figure 3 is not particularly helpful when only two states are possible, but it becomes very useful when more than two states (and more than two weights) are to be represented. Take, for example, a Base-5 shift register that can represent five states (integers 0, 1, 2, 3, and 4). The weights possible with such a shift register generator coincide with the five states (i.e., integers 0, 1, 2, 3, and 4). By properly choosing the weights for a three-stage shift register using base-5 arithmetic, it is possible to generate a maximal length sequence of 124 values (53 – 1 = 124). One such allowable configuration is shown in Figure 4. The initial fill of the shift register stages is arbitrarily set to 1-1-1, and the results of only the first seven shift register states are shown in Figure 4.

31

Figure 4. Three-Stage Shift Register Using Base-5 Operations

The full sequence resulting from the configuration shown in Figure 4 is given below.

11144214224300404244012102324103332213211240020212 20313014123044411341331200101311043403231402223342 344310030343302420414320

These numerals (0 through 4) can be used as symbols to represent any parameter that has five states (e.g., five radio frequencies, five pulse intervals, five polarization states, five phase states in a pulse compression waveform, etc.). An Application – Intrapulse Modulation One application of the 124-value sequence just presented is for intrapulse phase shift keying for the purpose of pulse compression. In this application, there are five phase states, each of which is an integer multiple of 360°/5 = 72°. The value “0” represents 0°, the value “1” represents 72°, the value 2 represents 144°, etc. The magnitude and phase of the resulting pulse is given in Figure 5. With five phase states, the intrapulse phase characteristics appear more similar to noise than to predictable signal structure.

32

Figure 5. Intrapulse Modulation with Five Phase States The intrapulse phase modulation resulting from this maximal length shift register sequence has desirable autocorrelation properties, and therefore provides good pulse compression performance. The autocorrelation plot for the five-phase maximal length sequence is given in Figure 6.

Figure 6. Autocorrelation of the Five Phase State Maximal Length Sequence Conclusions Shift register sequences derived using prime-number bases greater than Base 2 provides additional flexibility in generating pseudo-random sequences with desirable properties. The sequences can be used to randomize radar parameters such as radio frequency or pulse interval, randomize jammer waveform sequences, generate intrapulse polyphase modulation sequences with desirable autocorrelation properties, etc. Sometimes it is useful to add a tool to the bag of tricks. Specific uses of higher order shift register generators are left to the imagination of the reader. Reference [1] Carlson, E. J., “Low Probability of Intercept (LPI) Techniques and Implementations for Radar Systems,” IEEE National Radar Conference, Ann Arbor, MI, Apr. 20, 1988, Proceedings

33

Implementation and Testing of Fuzzy Adaptive Resonance Theory

Algorithm to Analyze Complex Radar Data Sets

Michael J. Thompson and John C. Sciortino Jr. Naval Research Laboratory, 4555 Overlook Ave., Washington, DC. 20375

ABSTRACT This paper evaluates one promising method used to solve one of the main problems in electronic warfare. This problem is the sorting of radar signals in a tactical environment. The sorting process requires clustering of collected radar pulse descriptor words and the comparison of known emitters. The method described here, fuzzy adaptive resonance theory (fuzzy ART), is a self-organizing neural network algorithm. This paper discusses the theory behind the fuzzy ART and also shows results of the processing of four “real” radar pulse data sets. It also shows the implementation and design of the test setup that was developed for a fuzzy ART application specific integrated circuit (ASIC), which was developed by Defence Research and Development Canada (DRDC) Ottawa. Keywords: artificial intelligence, electronic warfare, self-organizing systems, neural networks, fuzzy logic

INTRODUCTION

The fuzzy ART algorithm is one of the adaptive resonance theory (ART) algorithms developed by the Department of Cognitive and Neural Systems at Boston University.1-3 The basic algorithms in this group are ART, ARTMAP, fuzzy ART and fuzzy ARTMAP. They are based on the same mathematical model. The ART and fuzzy ART are unsupervised algorithms and require no a priori knowledge of the environment to perform the clustering of data. The ARTMAP and fuzzy ARTMAP are supervised algorithms and require a priori knowledge of the environment to perform both the clustering and classification of data. The fuzzy part of both algorithms allows the analysis of analog data whereas the ART and ARTMAP require binary data to perform their analysis. The difference between analog data and binary data is as shown in Figure 1.

34

Figure 1 Binary data and analog data

The fuzzy ART algorithm requires user defined initial parameters, which are the vigilance parameter, choice parameter and learning rate parameter. The vigilance parameter, ρ, determines the closeness of the match between the presented input pattern and the previously formed clusters. If there is a match within the vigilance parameter the input pattern is assigned to that cluster. If the match between the input pattern and any of the clusters is not within the vigilance parameter then a new cluster is formed and the input pattern is assigned to that cluster. This means that the lower the vigilance parameter, the more clusters will be formed. The choice parameter, α, is set to a value greater then zero. It is usually not much greater then zero. The value in most cases is on the order of 0.001. This parameter is a factor in the decision in the category choice function defined in step 4. The learning rate parameter, β, is the parameter that has an effect on the convergence rate to the answer. This means that it is the rate at which input patterns are matched with clusters. In most cases this parameter is set to one (fast learning) as opposed to zero (slow learning).

Fuzzy ART Algorithm and Architecture

The fuzzy ART algorithm has five basic steps as shown in the flow chart shown in figure 2, which are as described below. 1. Weight and parameter initialization Initially, all neuron weights, wj, are set equal to 1, i.e. wij =1, for j=1…N (number of input patterns presented ) and i=1…M (number of elements in each pattern) the vigilance parameter, ρ, is chosen between 0 and 1, i.e. ρ ∈ [0,1], and the choice parameter, α, is chosen greater than 0, i.e. α > 0. For the sake of simplicity, the learning rate parameter, β, is assumed to be equal to one, i.e. β=1. This corresponds to fast learning.

Binary Data

0 1

Analog Data

0 1 Inputs can be either 0 or 1 or between 0 and 1 Inputs are either 0 or 1

35

2. Input vector coding A new input pattern a= (a1, a2, ..., aM) of M elements, where each element ai is a real number in the interval [0,1], undergoes complement coding; aci=1- ai. This yields an input vector I = [a;ac] of 2M elements. The compliment values ac are added to the input pattern structure to prevent unnecessary cluster formation. The reason for this is that with more features in the input pattern structure it is more likely that the pattern will be associated with an existing cluster rather than cause the forming of a new cluster. 3. Vigilance test “For every committed neuron j, if the vigilance test, M?j ⋅≥∧ wI , is passed, then the

neuron is considered to be a potential winner. The set of all potential winners is denoted

by C. The fuzzy logic AND operator is defined by:

( ) ( )( )Mj2M2j11j ,wI,...,,wI minmin=∧ wI , whereas x is the norm of x defined by

∑ == iN

1i xx . Steps 4 and 5 are performed for every neuron j that has passed the

vigilance test.” 4

4. Category choice

“The activation Tj of each neuron j ∈ C, is calculated using the choice function, which is

( )jjj aIT ww +∧= . The winner J is selected as the one having the highest activation,

}: max{ CjTT jJ ∈= .”4 This basically means that the input pattern is placed in the

category, for which TJ is the largest.

5. Prototype vector update

“If the subset test, which is JJ wIw| ∧=| , fails, then the winner neuron is updated, i.e.

JJ wIw ∧=′ otherwise no update is required.”4 The equation JJ wIw ∧=′ is for when

β=1. Note if β does not equal 1 then the equation

)(w)1()(wI)(w oldoldnew JJJ ββ −+∧=′ applies.

36

j ≤ N

j = j + 1

3 Vigilance test

MwI ⋅≥∧ ρj

j

jjT

w

wI

+

∧=

α

Subset Test

= I ∧

5 Prototype vector update

JJ wIw ∧=′

Computation of and wjI ∧

4 Category choice

( ) N1jjJ TmaxT ==

j = 1

Input a

yes

no

noyes

yes

no

no

yes

Newinput?

Tj=0

2 Input vector codingI=(a,ac)

1 Weights and parametersInitialization: wij, β, α, ρ

(i=1..2M; j=1..N)

wj

wJwJ

Activation

Figure 2 Fuzzy ART flowchart4

DATA

Four datasets have been evaluated using this algorithm. Each dataset consists of several pulse descriptor words (PDWs) and is described as follows: � Dataset1

1. 5 day collection 112 tests (10 maximum emitters per test)) � Dataset2 and dataset3

2. 10000 pulses all together, 5242 pulses labeled 31 emitters � Dataset4 ( One day 1 million pulses; 5 days collection)

3. 1 million pulses; over 5 days, 7 emitters total

The PDWs that were used by the algorithm for these sets of data were radio frequency (RF) and pulse width (PW). The statistical evaluation of the data sets using ground truth to distinguish emitters is as shown in Figure 3 (dataset 2 and 3) and Figure 4 (dataset1), respectively. These diagrams depict elliptical regions with the mean given by the center and the width corresponding to the lengths of the semi-major and semi-minor axes. As shown in the diagrams, there are several regions of overlap between the elliptical regions for the parameters RF and PW. This is an indication of the possibility of clustering pulses from the one emitter with the pulses from another emitter.

37

Figure 3 Statistical representation of dataset 2 and 3

Figure 4 Statistical representation of dataset1

38

RESULTS

The fuzzy ART algorithm has been tested along with four other algorithms that are also unsupervised and therefore require no a priori knowledge of the environment to perform the clustering operation. The other four algorithms include; � Supervised Piriform Hierarchical Clusterer5 (Super PHC) � K-Means � Fuzzy Min-Max Clustering 6 (FMMC) � Integrated Adaptive Fuzzy Clustering8 (IAFC)

“Super PHC is a biologically inspired neural network algorithm that is derived from simulations of the rat olfactory bulb and olfactory cortex. It builds a tree to hierarchically partition the input vector space according to the statistical structure of the training set.”7 It then utilizes the partitions to form the clusters, which it then uses to place the input patterns in the formed clusters. K-Means is a statistical method of clustering and can be considered a baseline method. This means that the results from K-Means show how clustering using the statistics, which are used to form the ellipsoids in Figure 3 and Figure 4, would perform. FMMC and IAFC are other methods of performing fuzzy clustering and are also along with fuzzy ART in the family of self-organizing neural networks. The results of the analysis of all four datasets using these algorithms are given in Table 1 and Table 2. Table 1 shows each algorithm’s percentage of correct correlation for each dataset and Table 2 provides the related convergence time for each algorithm and dataset.

Table 1. Correct correlation

Table 2. Time required to converge to the correct answer (seconds)

63 % 55 % 75 % - - Dataset4

- 92 % 90 % - - Dataset3

58 % 65 % 78% 56 % 48 % Dataset2

37 % 49 % 50 % 46 % 48 % Dataset1

IAFC FMMC FA K-Means SuperPHC Data Set

IAFC FMMC FA K-Means SuperPHC Data Set

1431 22365 9482 - Dataset4

- - - - 2 Dataset3

58 314 39 - 2 Dataset2

39,611 12,308 334 - 30 Dataset1

39

The figures labeled as Figure 5 (Correct Correlation) and Figure 6 (Processing Time) show the results of the analysis of dataset 1, which because of its size, 107850 pulse descriptor words from 112 different emitters, needed to be split into 10 separate files each with 10785 pulse descriptor words for processing purposes. The algorithms compared in Figure 5 and 6 are fuzzy adaptive resonance theory MAP (FAM), fuzzy adaptive resonance theory (FA), fuzzy min max clustering (FMMC) and integrated adaptive fuzzy clustering (IAFC). FA is the clustering part of the fuzzy ARTMAP algorithm. FMMC and IAFC are other types of self-organizing neural network algorithms that were used for comparison.

Figure 5. Correct correlations for the segments of dataset1

Figure 6. Processing time (seconds) for each segment of dataset2

1

100

10000

FAM FA FMMC IAFC

12345678910

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FAM FA FMMC IAFC

12345678910

40

HARDWARE IMPLEMENTATION AND TEST SETUP

The fuzzy ART algorithm has been implemented in the form of an application specific integrated circuit (ASIC) which has been developed and provided to us for testing by Defence Research and Development Canada (DRDC). The test setup for this device is as shown in Figure 7. This test setup has been constructed and utilizes a field programmable gate array (FPGA), to perform all of the First in First Out (FIFO) and register operations as shown in the diagram. Figure 8 is a photo of the test hardware.

Input FIFO

FPGA

Fuzzy ART Device

Input FIFO

FPGA

Rclk1

A_in1

clkb B0-11

Entre_acc AE

Rclk2

A_in2

clkb

B0-11

Output FIFO

FPGA

TJ

J

subset

A0-16

clka Write_tj

Reg

iste

rs Looser

Winner Modification

Control

Dip

Sw

itche

s Enable_chip Cascade

AF 1

11

5

log2G

1

11

11

ID

17+log2G

1

1

Figure 7. Device test setup diagram4

Figure 8. Hardware implementation of device test setup

The board shown in Figure 8 will be connected to a PC using a RS232 interface on the board. A text file in the proper format from the PC is loaded to random access memory

41

(RAM) and then sent to a FIFO for processing through the device. The output from the device will go to another FIFO and then to RAM and finally out to a file to be processed and displayed by a graphical user interface. This work is currently ongoing and will be discussed in future publications.

CONCLUSION The results of the processing of all of the datasets with the fuzzy ART algorithm provided a better value of the correct correlation measure with actual ID, and exceeded the results of processing for the majority of these datasets with other algorithms such as K-Means and other self-organizing neural networks as shown in Table 1. The processing times shown in Table 2 also show that fuzzy ART was also the quickest for the majority of the datasets considered. The results of this analysis also show that while it does look promising, further research will need to be completed to support the conclusions presented in this paper using other datasets. Also further research needs to be done to show how this algorithm does in the areas of convergence time, which is how long it takes to arrive at an answer and computational complexity, which means how many resources from a computational stand point does this algorithm require for operation.

REFERENCES 1. Teresa Serrano Gotarredona, Bernabé Linares-Barranco, G. Andreou, “Adaptive

Resonance Theory Microchips Circuit Design Techniques” 1998 Kluwer Academic Publishers

2. Eric Granger and Mark A. Rubin and Stephen Grossberg and Pierre Lavoie, "A What-and-Where fusion neural network for recognition and tracking of multiple radar emitters", Neural Networks, Volume 14, Number 3, pages 325-344, year 2001

3. Gail A. Carpenter, Stephen Grossberg, Natalya Markuzon, John H. Reynolds and David B. Rosen, “Fuzzy ARTMAP: A Neural Network Architecture for Incremental Supervised Learning of Analog Multidimensional Maps”, IEEE Transactions on Neural Networks, Vol. 3, No. 5, September 1992

4. Marc Andre-Cantin, “Integration Support for ICDPMFA3 fuzzy ART Neural Network ASIC(v.3)” (NRL internal manual for ASIC(not published))

5. Kowtha, V., Satyanarayana, P., Granger, R., and Stenger, D. (1994), “Learning and classification in a noisy environment by a simulated cortical network.”, Proc. Third Ann. Comp. Neural Systems Conference, Boston: Kluwer, pp. 245-250

6. P. K. Simpson, “ Fuzzy min-max neural networks”, IEEE Transactions on Neural Networks, vol.3, p.776-786 (1992)

7. Yang, S., V. C. Kowtha, G. L. Barrows, J. C. Sciortino, Jr., D. A. Stenger (1999). “Enhanced Emitter Identification Using Scaled Conventional Parameters and Intrapulse Parameters”, NRL Formal Report NRL/FR/572099-9912, 8 September 1999. Naval Research Laboratory. (winner of 1999 NRL ARPAD award).

8. Kim, Y.S., Mitra, S. “An Adaptive Integrated Fuzzy Clustering Model for Pattern Recognition”, Fuzzy Sets and Systems, No. 65, pp 297-310, 1994

42

THE SHANNON-HARTLEY THEOREM AS A UNIFYING PRINCIPLE IN ELECTRONIC WARFARE AND INFORMATION WARFARE

ANDREW BORDEN INFERLOGIX, L.L.C.

Abstract Claude Shannon developed the quantitative definition of uncertainty and information. He also proved the Shannon-Hartley theorem which determines how much information can be sent through a noisy communications channel with a specified bandwidth. This theorem can be used as a unifying principle in Electronic Warfare. All EW attack and defend measures are attempts to affect the variables in the Shannon-Hartley Theorem. An analog of Shannon Hartley serves the same purpose in Information Warfare. Keywords Conditional Entropy, Electronic Warfare, Information Warfare, Entropy, Information, Mutual Information, the Shafer-Dempster Mathematical Theory of Information, the Shannon-Hartley Theorem THE SHANNON-HARTLEY THEOREM AS A UNIFYING PRINCIPLE IN ELECTRONIC WARFARE AND INFORMATION WARFARE The Shannon Hartley Theorem The Shannon-Hartley Theorem (SHT) is one of the most elegant mathematical theorems of the twentieth century. Its beauty is derived partly from the fact that the proof uses results from geometry and other mathematical disciplines to link two different, but related, properties of electromagnetic transmission and produces the stunning result that we can determine the limiting potential of the information that can be carried by the signal…measured in bits. The quantification of information in terms of bits was also developed by Shannon…arguably the Newton of the Information Age. The SHT says that, if the noise has certain regular properties (Gaussian and white), then the limit of information in bits per second that can be carried by a band-limited, communications channel (the channel capacity) is: C = S * log2(1 + S./N) 1 S is the bandwidth of the signal and the Signal to Noise ratio (S/N) is an absolute ratio, not expressed in dB. (LaFrance) The relevance of SHT to Electronic Warfare is that it is a unifying principle that can be used to describe the use of electromagnetic energy to obtain or transmit information and, the use of energy to attack this use (Offensive countermeasures) and to defend it (Defensive countermeasures). Moreover, an analog of SHT can be used to describe and

43

unify Information Operations and the corresponding offensive and defensive measures taken in this world. The Shannon-Hartley Theorem and Electronic Warfare For an electromagnetic signal in a noisy environment, the Signal to Noise ratio is computed as follows: S/N = (σ/(SPG)/4Π)(Pr/Pj)(G2rt/GjGrj)(Bj/Br)(R2j/R4t) 2 Where:

σ = Target cross section in meters SPG = Signal Processing Gain Pr = Transmitted power of the radar in Watts Pj = Transmitted power of the jammer in Watts Grt = Radar antenna gain in the direction of the target (dB) Gj = Jammer antenna gain in the direction of the radare (dB) Grj = Radar antenna gain in the direction of the jammer (dB) Bj = Bandwidth of the jammer signal Br = Bandwidth of the radar Rj = Distance between the radar and the jammer in meters Rt - Distance between the radar receiver and the target in meters Note: If Br < Bj, then (Bj/Br) is taken to be 1, that is, all the jammer energy is in the bandpass of the radar receiver. (Eaves and Reedy)

ENCODE DECODETRANSMIT(SIGNAL)MESSAGE MESSAGE

BANDWIDTH

NOISE

FIGURE 1: SENDING INFORMATION THROUGH A NOISY CHANNEL

Figure 1 represents the transmission described. The SHT guarantees that there exists a coding of the message such that the information from the message can be transmitted at a rate arbitrarily close to the Channel capacity.

All Offensive Countermeasures in Electronic Warfare are an attempt to reduce the bandwidth or the Signal to Noise Ratio the signal being used by an adversary to receive information, thereby reducing the rate at which uncertainty can be reduced. All

44

Defensive countermeasures are an attempt to increase or protect the Bandwidth or Signal to Noise ratio of the signal being used. For example, noise jamming against a radar signal is an attempt to reduce the adversary’s S/N. If S/N is low enough, the rate at which information can be received is reduced so that the adversary cannot make decisions on time. As S/N approaches zero, the channel capacity also approaches zero. For another example, a Low Probability of Intercept (LPI) radar may exploit the tradeoff between Bandwidth and S/N by using a much lower S/N to minimize detection, increasing bandwidth to achieve the needed channel capacity. A frequency hopping signal uses an instantaneous bandwidth sufficient to support the required channel capacity according to the SHT, but uses a greater total bandwidth to gain an advantage over the jammer. Strictly speaking, the total bandwidth is not the W in the Shannon-Hartley formula, but it is still an indication of the Protect margin over the jammer and the capability of the system to obtain information. The Shannon-Hartley Theorem and Information Warfare For a probability distribution: {pi} i = 1,2,3,…,n,

H = ∑pi*log2pi 3

is the formula for Shannon’s Entropy (uncertainty). Shannon’s Entropy is maximal when all the pi’s are equal, that is, when all outcomes are equally probable. Hmax = log2n 4 For a given probability distribution, the Percent of Max Entropy (PME) is: PME = H/Hmax 5

PME is a measure of the “Don’t Know” factor for a given probability distribution. In any decision making task, there is initial Entropy, H(Problem), associated with the a priori probability distribution. This is a measure of how difficult the problem is and how much information (in bits) must be generated to solve it. For each data source which contributes to Entropy reduction, the expected value of the Conditional Entropy (after exploiting the data source) can be computed. (LaFrance). The expected value of the information contributed by the data source (conditional information) is the following:

I(ProblemData) = H(Problem) – H(ProblemData) 6 H(Problem) is the initial uncertainty (in bits) associated with a Situation Assessment task.

45

H(problemdata) is the average amount of uncertainty (in bits) remaining after data is processed. I(ProblemData) is the average reduction in uncertainty (in bits) contributed by the data. If the data source contributes no information to the solution of the problem, H(Problem) is equal to H(ProblemData) and the conditional information is zero. The data source is useless. If the conditional Entropy after consulting the data source is zero, then all the uncertainty has been removed and the problem is solved. When reasoning in the presence of uncertainty, it is almost always necessary to consult several data sources. It is valuable to know however, how much conditional information the data source will contribute to the solution of the problem. Moreover, data sources contain redundant information, so after the most lucrative data course has been consulted, the remaining ones must be re-evaluated in terms of their new conditional information potential. This process is computationally expensive, but the benefit is that the most direct path to the solution can be found and the data exists to provide great explanatory power. It would be desirable to ensure efficiency by determining and selecting the most productive data source at each node in the decision process and this is done by the use of Shannon’s formulae for uncertainty and information. . In the Information Operations analog to the SHT, the number of data sources is the Bandwidth and I(Problem|Data) is the Signal to noise ratio. If, in performing situation assessment, I(Problem |Data) is too small…that is, too much uncertainty remains after processing the available data, then additional data sources are needed to increase both bandwidth and S/N.

OBSERVATIONS/MEASUREMENTS

PROCESSING(MUTUAL

INFORMATION)

CLASSIFICATION/IDENTIFICATION

OBJECT/SITUATION

BANDWIDTH KNOWLEDGE

CONDITIONALENTROPY(NOISE)

FIGURE 2: SITUATION ASSESSMENT IN THE PRESENCE OF UNCERTAINTY Figure 2 is the Information Warfare analog to electromagnetic transmission. All IW Attack measures are designed either to reduce bandwidth by denying observations/measurements or to increase conditional entropy by delaying, degrading or corrupting measurements.

46

For example, Concealment or Camouflage might be used to deny the use of an optical or EO sensor, thereby reducing the Information Bandwidth available to do Situation Assessment. Deception or other corruption of data could be used to increase the Conditional Entropy and reduce the Mutual Information between the problem and the data to be used to reduce uncertainty. The computation and use of mutual information suggest the requirement for a canonical method for processing data and evaluating all the conditional probabilities needed. This method exists and uses Bayes theorem and Shannon’s definitions of information and uncertainty in a computationally intensive, but efficient and practical way. Given that a standard and efficient method exists for reducing uncertainty, the nest step is to identify the mathematical signature of deception, if it exists. Unfortunately, the use of probabilities alone gives too many false positives, that is, reports anomalies whenever unlikely events occur. However, unlikely events do occur and they do not always indicate anomalous behavior or deception. In trying to develop a deception detector, we attempted to compare the probability distribution based on the current perception of the problem with the probability distribution computed from processing a new data point when it arrived. If the new data suggests conflict with the existing picture, the possibility of deception, corrupted or otherwise anomalous data is present. In the search for the elusive coefficient of alienation however, we found that, when the “Don’t Know” factor (PME) was large, small asymmetries in probability were exaggerated and too many false positives occurred. The “Don’t Know” factor was taken as conflict between the current perception of the situation and a specific data element. This suggested the use of the Shafer-Dempster Mathematical Theory of Evidence and Belief. In S-D, there are belief assignments instead of probabilities. The belief assignments are computed from weights of evidence using an exponential function. It is very difficult to arrive at weights of evidence, accounting for one of the reasons S-D is not widely used. The other reason is that Dempster’s Rule of Combinations (of beliefs), analogous to the compounding of probabilities, is computationally expensive. We have addressed both difficulties and have found that S-D is very suitable for analyzing the presence of deception. The reason for this is that S-D explicitly acknowledges the “Don’t Know” factor based on insufficient evidence. It is called “Uncommitted Belief” in S-D and can be excluded from the analysis of the conflict of beliefs. So the approach is to map the PME associated with a probability distribution to the “Uncommitted Belief” in S-D. PME and “Uncommitted Belief” are different concepts, but the way the SAET selects data sources makes it reasonable to associate the two. Moreover, many iterations of a simulation of the process of mapping probabilities to S-D has convinced us that the S-D approach gives more reasonable values and fewer false positives when the “Don’t Know” factor is high. (Shafer) A probability distribution: (p1, p2, p3,…,pn, PME) is mapped to

47

(((1-PME)*p1), ((1-PME)*p2), ((1-PME)*p3),…((1-PME)*pn), PME)) which is then interpreted as: (b1, b2, b3, …, bn, (Uncommitted Belief)) So, Shannon’s Entropy is mapped to Uncommitted Belief or “Don’t know” in S-D. where the bi are beliefs in the possible outcomes. This mapping of probabilities directly to beliefs eliminates the need to determine the exponential values for evidence reauired by S-D theory. Moreover, we have built a very efficient Dempster engine that can run S-D decision making in parallel with the SAET’s Bayesian approach. When we have two belief assignments {b1,I,… Uncommitted} and {b2,i,… Uncommitted}, the coefficient of alienation is computed as follows: CofA = (∑i,(b1,i – b2,j))/2 7 If all the beliefs are the same for both assignments, the CofA is zero. If the belief assignments are orthogonal, the CofA is 1. In experiments, we have found that a value exceeding 0.5 suggests that the data should be re-evaluated for validity and the possible presence of deception. The Situation Assessment Evaluation Tool (SAET) The SAET is a Bayesian classifier that uses Shannon’s definitions of uncertainty and information to ensure efficiency. Many conditional probabilities are computed, but this has the corresponding advantage that statistics are available to evaluate the reliability and efficiency of decision making and to determine when increased bandwidth and/or S/N are needed. It is possible to perform an instant online simulation to determine how statistically representative inputs are handled by the system. The SAET has been used successfully in projects to classify the intentions of airborne targets and to improve maintenance diagnostics in an advanced SIGINT system. A considerable amount of internal R&D has been done pertaining to the design of ESM classification algorithms, Indications and Warning and medical diagnosis. The deception (anomaly) detector is embedded into the classification process and an alarm is reported whenever data appears to be in conflict with the current perception of the situation. The deception detector seems to be robust and reliable. SUMMARY The Shannon-Hartley Theorem and its information theoretic analog can be unifying principles for EW/IW respectively. The computational tools exist to apply the information theoretic theory to Information Warfare. Using these computational tools a Deception or Anomaly Detector can be developed and used in parallel with the Situation Assessment algorithm.

48

Bibliography

Borden, Andrew and Elliot, Linda: “Intuition and Decision-Making Systems”, Information and Security, an International Journal. Volume 1, Number 2. March 1999. Borden, Andrew: “Intuition and Decision-Making Systems - II”, Information and Security, an International Journal, Volume 1, Number 3, Fall 2000. (Also presented at Technet International, Prague, Czech Republic, October 2000). Borden, Andrew. “Intuition and Decision-Making III”, Proceedings of Technet International, a symposium sponsored by the Armed Forces Communications and Electronics Association, Budapest, Hungary, October 2002. Eaves, Jerry L. and Reedy, Edward K. (Editors), Principles of Modern Radar, Van Nostrand Reinhold Company, New York, 1987. LaFrance, Pierre: Fundamental Concepts in Communication, Prentice-Hall International Editions, Prentice-Hall International, Inc. A division of Simon and Schuster, Englewood Cliffs, N.J. 1983.

Shafer, Glenn: “The Mathematical Theory of Evidence, Princeton University Press, 1965.

Shannon, Claude E. and Weaver, Warren; The Mathematical Theory of Communication, Urbana, The University of Illinois Press, 1949.

Curriculim Vitae

Andrew Borden is a retired Air Force officer and a mathematician with a long background and many publications in the area of decision making algorithms. He was Principal Scientist for Electronic Warfare at the NATO C3 Agency. His last active duty assignment was as Deputy Chief of Staff for Intelligence, USAF Air Intelligence Agency (then Electronic Security Command). He is the Vice President and Chief Scientist of INFERLOGIX, L.L.C.

INFERLOGIX L.L.C. is an emerging, woman-owned company with headquarters in Galveston, Texas.

President: Beth Cloyd, 12420 E. Ventura Dr., Galveston, TX 78554.

409 939-5009

49

EFFECTS OF RADIO WAVE PROPAGATION IN URBANIZED AREAS ON UAV-GCS COMMAND AND CONTROL

Lock Wai Lek

Singapore Armed Forces wlock@starnet.gov.sg

David C. Jenn

Naval Postgraduate School Department of Electrical & Computer Engineering

833 Dyer Road Room 437

Monterey, CA 93943 jenn@nps.navy.mil

831 656 2254

ABSTRACT

In an urban environment, the linkage between UAVs and ground control stations are subjected to multipath interference due to reflection, diffraction, and scattering between the transmitter and receiver. Severe multipath can result in a nearly complete loss of command signals, which can limit the UAV’s operational area or even cause a loss of the vehicle. This paper examines the propagation of RF signals through an urbanized area using a ray-tracing computational electromagnetics software package. Several scenarios were developed to approximate actual operational situations. Given the UAV’s transmitter power and other system parameters, the signal levels are computed on a grid of specified observation points. Variations in the simulation included observation point locations, building material composition, building density, UAV operating altitude and frequency, number of deployed UAVs and their locations, and theoretical ray bounce considerations.

Based on a large number of simulations several guidelines for operating a UAV in a dense urban environment are suggested, such as how to select an optimum altitude and the potential use of “urban canyons” for communications.

50

I. INTRODUCTION

Many armed forces around the world have recognized the tremendous potential of unmanned aerial vehicles (UAVs) for battlefield surveillance and reconnaissance. UAVs are remotely piloted or self-piloted aircraft that can carry cameras, sensors, communications equipment or other payloads. They have been used in reconnaissance and intelligence gathering roles since the 1950s and more challenging roles like combat missions are envisioned [1]. A study conducted by United States Office of the Secretary of Defence/C3I regarding the use of a UAV as an Airborne Communication Node (ACN) [2] concluded that tactical communication needs can be met much more responsively and effectively with ACNs than with satellites.

Because of the growing trend towards urbanization, future wars may no longer be fought on large expanses of open terrain like pastures and deserts, but in built-up areas. At the same time, the trend towards network centric warfare puts a high demand on battlefield comprehensive awareness for commanders and troops linked by communication and sensor nodes. Existing command and data links for UAVs are point-to-point communication links between the UAV and a ground control station (GCS). However, future concept of operations (CONOPS) would involve UAV or payload control from soldiers in units other than the controlling units. These operations require future systems to evolve from control center to network centric application.

UAV-GCS linkage in an urbanized area is subjected to multipath interference due to reflection, diffraction and scattering. Opaque or absorbing materials, corners of buildings, and window openings can cause large fluctuations in the signal. Shadow regions are formed when the line of sight (LOS) is blocked from radio-frequency (RF) signals. Severe multipath can result in a nearly complete loss of command signals, which can limit the operational area or even cause a loss of the vehicle. On the other hand, diffraction at edges and corners causes illumination behind obstructions, and spreading through small apertures, which can actually help in extending propagation in some cases.

This paper examines the propagation of RF signals through an urbanized area using a commercially available ray-tracing software package. Several scenarios were developed to approximate actual operational situations. Given the UAV’s transmitter power and other system parameters, the signal levels are computed on a grid of specified observation points. Variations in the simulation included observation point locations, building material composition, building density, UAV operating altitude and frequency, number of deployed UAVs and their locations, and theoretical ray bounce considerations.

The signal strength level necessary to establish a link depends on the receiver sensitivity as well as interference and noise at the GCS. In order to make the contours system independent, the signal power levels are computed in decibels relative to a milliwatt (dBm). Once the specific parameters of the GCS are known, the contours that will satisfy the minimum signal strength requirements can be easily converted into operational range contours. Two common frequency bands used by existing UAVs were examined. They are 5 GHz (C band) and 15 GHz (Ku band) for line-of-sight (LOS) command, control, and data links.

II. PROPAGATION MODELING

The mechanisms that govern radio propagation in urban areas are complicated, but they are generally attributed to three basic propagation methods: (1) reflection, (2) diffraction, and (3)

51

scattering [3]. UAV-GCS command links in urban environments are subjected to severe degradation due to the superposition of these components. As a result, the received signal strength at the GCS can be roughly characterized by three nearly independent phenomena of large-scale path loss, large-scale shadowing, and multipath fading.

For most data and communication links, a narrow band sinusoidal case can be assumed

where the time dependence is of the form j te ω ( 2 ,fω π= f is frequency, and 1j = − ) so that phasor quantities can be used. Furthermore, because the reflecting and diffracting objects are large compared to the wavelength, high-frequency ray-tracing approximations can be applied, as shown in Figure 1. The received power rP at the GCS due to a UAV transmitting a power tP with antenna gain tG at range R is given by the Friis equation [4]

22

2 2| |

(4 )t t r p

rP G G LG

P FR

λ

π= . (1)

The wavelength is /c fλ = , where 83 10c = × m/s is the phase velocity in free space. The GCS antenna gain in the direction of the UAV is rG . The miscellaneous loss and processing gain factors, L and pG , respectively, are system dependent.

UAVTRANSMITTER

GCS 1

GCS 2

GCS 3

METAL

GLASS

DIFFRACTION

REFLECTION

TRANSMISSION

REFLECTION

Figure 1: Illustration of some possible ray paths for the simple case of a glass slab and metal wall, for the case of the GCS transmitting.

The path-gain factor (PGF) F gives the total electric field intensity at the GCS relative to the direct free-space electric field intensity. In addition to the direct path signal, for ray-based propagation modeling, the contributors to the total electric field intensity are the reflected and diffracted signals that occur in the environment. They arise from the ground and foliage, or buildings and other manmade objects on the ground or in the air. Given an observation point in

52

space, the total field will be the sum of all of the direct, reflected, and diffracted fields arriving at that point.

Formulas are available for the reflected and diffracted fields based on geometrical optics (GO) and the geometrical theory of diffraction (GTD) [5]. They incorporate reflection (diffraction) coefficients that linearly relate the reflected (diffracted) fields to the incident fields at the reflection (diffraction) points. In the case of reflection, the traditional Fresnel coefficients for planar boundaries can be used [5]. Specular (mirror-like) reflections must satisfy Snell’s law. The diffracted fields lie on a cone whose axis is coincident with the diffracting edge and half angle is determined by the angle of the incident ray with the edge. The number of terms needed for a converged value of F in equation (1) must be determined from the minimum signal level that is to be reliably computed, and in an urban environment this is difficult to predict in advance.

The formulas for the reflection and diffraction coefficients depend on the electrical properties of the materials. The materials are defined by their complex relative permittivity ( r r rjε ε ε′ ′′= − ), relative permeability ( r r rjµ µ µ′ ′′= − ), and surface resistivity (Rs) [6]. Materials

with infinitely large Rs are transparent to waves while those with small Rs ( ~< 50) are highly reflective. Combinations of these parameters can be used to achieve the electrical characteristics of most common building materials.

While there are several engineering tools to predict antenna radiation and wave propagation, Urbana [7] was selected for this research. The propagation model is essentially a three-dimensional ray-tracing process that in principle predicts the local mean power received at any given point. For each observation point, reflection and diffraction points are determined, the ray paths between the transmitter and receiver are traced, and then the vector sum of multipath signals computed.

III. SCENARIOS AND SIMULATION RESULTS

A. UAV System Parameters

For the purpose of defining the simulation scenarios, the UAV is assumed to be similar to the multipurpose security and surveillance mission platform (MSSMP) [8]. It is capable of hovering, and designed to provide a rapidly deployable, extended-range surveillance capability for a variety of operations and missions.

A vertically polarized dipole antenna transmitting at 5 GHz with a power of 1 W was used to simulate a data signal from the MSSMP. The signal contours for other power levels can be determined by simple scaling, since all of the scattering phenomena and the media are linear. The receiving antenna pattern is isotropic, and the signal levels are plotted in dBm. A combination of GO and GTD were used in computing the signal contours. Single diffraction and up to seven reflections were included in the ray tracing. Mixed modes (e.g., reflected-diffracted rays) were not included.

B. Central City Area

A number of urban environments were examined during the course of the research [9], however, only two general cases will be discussed here. The first, shown in Figure 2, is based on a sample file that is included with the Urbana software. It represents a central city area with a

53

mix of high and low buildings. The tallest building (357 m) is in the category of a skyscraper, such as the Sears building in Chicago. The size of the observation plane (x−y plane) is 1620 m by 800 m. The observation cell size is a square with edge lengths of 1 m. Therefore the “pixel” size for the resulting contour images is 1 m by 1 m. Since this is much greater than the wavelength of the frequencies under consideration, the field strength at other points in the cell will fluctuate about the center value. Small-scale variations in the field are not captured, but large-scale path loss and shadowing are. More resolution can be achieved at the expense of increased computation time.

For buildings comprised of several materials, an average value can be estimated based on the constituent materials. When both the transmission and reception points are outdoors, the details of the building walls (i.e., the locations of doors and window openings) are not as crucial to the computation as they are for outdoor-to-indoor and indoor-to-outdoor propagation. Therefore homogeneous exterior walls with averaged material properties are sufficient in most cases. Roads were simulated with a thin layer of concrete over a semi-infinite plane.

To illustrate the effect of different building materials, Figures 3 through 5 show the outdoor signal levels for buildings comprised predominately of concrete, glass and wood, respectively. The UAV is modeled at the open area near the right portion of the city coordinates (x, y, z) = ( − 353 m, − 69 m, 187 m).

Figure 2. Model of a dense urban environment. (From reference [7].)

54

800,400

800,−400 −820, −400

-820, 400

x

y

Figure 3. City with predominately concrete buildings (distance unit is meters).

800,400

800,−400 −820, −400

-820, 400

x

y

Figure 4. City with predominately glass buildings (distance unit is meters).

55

800,400

800,−400 −820, −400

-820, 400

x

y

Figure 5. City with predominately wood buildings (distance unit is meters).

The results demonstrate that building materials have a significant effect on the signal contours. In the concrete model shown in Figure 3, propagation is limited to the open areas with large regions being shadowed. However, in Figures 4 and 5, due to a higher transparency of the building materials, there is more transmission through the buildings.

C. Ducted Waves in an Urban Canyon

A main feature of an urbanized area is the presence of long straight avenues of roads and pavements with high buildings on both sides, thus forming a so-called “urban canyon.” Analogous to surface wave ducting in the atmosphere [10], waves are trapped in the canyon and propagate extended distances with little attenuation. The effect of urban canyons is clearly visible in Figure 6 as a corridor for strong radio-wave propagation. For this calculation metal buildings were used and the UAV was located at (752 m, 203 m, 150 m). The buildings are acting as a waveguide, effectively preventing the signal from dispersing.

These data indicate that in certain parts of the city, despite having no LOS, troops are still able to receive and transmit with higher headquarters (HHQ) through ACNs. Tactically, identification of urban canyons becomes important for total communications coverage throughout the military operation with the minimal logistics tail.

D. Flying a UAV Over a City

In this scenario, the UAV is flown over the city along the x-axis in support of troops advancing from the right. The altitude of the UAV is kept constant at 358 m, which is just above the tallest building in the city. The y-coordinate is maintained constant at − 7 m. Figures 7 to 9 represent a series of time snapshots that shows the signal contours in the city as the UAV moves along its flight path.

The figures clearly show the shifting of shadow regions. In order to achieve the required signal strength level for an effective UAV-GCS link, it is necessary to anticipate the changes of signal contours and to locate, if possible, the ideal position for the GCS. The figures suggest that, for low transmission powers, the location of the GCS will need to displace with the advance of

56

the UAV. This scenario highlights the needs of a UAV deployed in support of MOUT should have the inherent capability to hover in order to capitalize on communications opportunities.

800,400

800,−400 −820, −400

-820, 400

x

y

URBAN CANYON EFFECT

Figure 6. Example of strong propagation along streets of a city with predominately metal buildings (distance unit is meters).

800,400

800, − 400 − 820, − 400

− 820, 400

x

y

Figure 7. UAV flying over a city (currently at x = − 385 m).

57

800,400

800,−400 −820, −400

−820, 400

x

y

Figure 8. UAV flying over a city (currently at x = 0 m).

800,400

800,−400 −820, −400

−820, 400

x

y

Figure 9. UAV flying over a city (currently at x = 1200 m).

E. Operating Altitude

The UAV signal was examined for various flight altitudes. The location of the UAV is near the tallest building in the city (x, y) = ( − 385 m, − 10 m). The z-coordinate was set at 1,000 m and 10,000 m with the resulting signal contours shown in Figures 10 and 11, respectively. At three times the height of the tallest building, coverage within the city using a single UAV provided better coverage contours than using two UAVs operating at lower altitudes or perched at rooftops. A larger grazing angle creates smaller shadow areas around buildings. However, a hovering UAV at high altitude will be consuming energy during flight as compared to one that is perched at rooftops on standby mode and will require considerable planning.

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800,400

800,−400 −600, −400

−600, 400

x

y

Figure 10. Varying UAV altitude (z = 1,000 m).

800,400

800,−400 −600, −400

−600, 400

x

y

Figure 11. Varying UAV altitude (z = 10,000 m).

Figure 11 shows that, at a high altitude, the area directly under the UAV antenna (in its null) will experience low signal levels, while locations further away are in a region of higher antenna gain. Path loss increases substantially at high altitudes, which must be compensated for by transmitting at a higher power. The advantage of operating at high altitude would be a larger area of uniform illumination, because shadows are smaller. Requirements for adaptive positioning of the GCS would subsequently diminish.

Note that no strong urban canyon effect was created at these altitudes. In order to effectively use an urban canyon for propagation, the transmitter must couple into the guiding structure. The coupling is most efficient when the height of the UAV is below the buildings

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forming the walls of the canyon. Thus positioning of the UAV is pivotal in utilizing the urban canyon effect.

F. Frequency Dependence

Finally, the operating frequency of the UAV was varied to observe its effects on coverage. The location of the UAV was fixed at the tallest building ( − 385 m, − 7 m, 358 m) and the frequency was set at 5 GHz and 15 GHz. The signal contours are shown in Figures 12 and 13, respectively.

Comparing both figures, it is observed that there is an insignificant change in signal contours, as long as the wavelength is small compared to the size of the buildings. This is not surprising because the low signal areas coincide with shadows. The nature of shadows is geometric, and therefore independent of frequency. However, the small-scale fluctuations will change with frequency, but they cannot be observed with the large cell size of 1 m. Calculations with a smaller cell size confirmed that the signal fluctuations across the width of a 1-m cell were typically in the range of 1 to 3 dB.

800,400

800,−400 −820, −400

-820, 400

x

y

Figure 12. Signal distribution for a frequency of f = 5 GHz.

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800,400

800,−400 −820, −400

−820, 400

x

y

Figure 13. Signal distribution for a frequency of f = 15 GHz.

G. Indoor Reception For GCS

The second set of scenarios consisted of a series of single story buildings that might be used to house a portable GCS. The research focused on indoor reception of RF signals, as would be the case when a soldier operates from the safety of a building. One configuration is shown in Figure 14. The footprint of the base building is 800 inches by 800 inches with a wall height of 132 inches. The observation cell size is 12 inches by 12 inches and the observation plane for all of the simulations was set at mid-window level. Since outdoor-to-indoor propagation is being modeled, the window openings are included, and various window materials were considered. The barrier wall is of the same thickness as the walls of the buildings.

Figure 14. Two buildings with a barrier wall.

The signal contours are shown in Figure 15 for a UAV located at (504 inches, 354 inches, 250 inches). The strong transmission through the windows is evident, and it is these areas that offer the highest potential for good reception. Diffraction from the edges of windows, which propagates into the shadow regions, may also permit reception if the receiver sensitivity is

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sufficient. When the UAV is moved to (− 50 inches, 450 inches, 400 inches) the signal contours shown in Figure 16 result. Instead of a pattern with stronger regions at windows, multipath inside of the building created a cross-like pattern with a significant signal level at the intersection. In this configuration, troops are able to operate a UAV away from windows, out of enemy fire.

504, −506

−1206, 1354

x

y

−1206, −506

504, 1354 Figure 15. Two single level buildings with barrier wall with the UAV at (504 inches, 354 inches, 250 inches).

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504, −506

−1206, 1354

x

y

−1206, −506

504, 1354 Figure 16. Two single level buildings with barrier wall with a UAV at (− 50 inches, 450 inches, 400 inches).

IV. SUMMARY AND CONCLUSIONS

UAV-GCS linkage in an urbanized area is subjected to multipath interference due to reflection, diffraction, and scattering. Shadow regions are formed when LOS areas are blocked. However, diffraction at corners causes illumination behind walls, below towers, and spreading through small apertures, that can actually help extend propagation in some cases.

This research focused on the signal contours generated when UAVs transmit towards an urbanized target area. Specific examples were shown to illustrate the effects of varying operating frequency, operating altitude, material composition of the building structures, and number of deployed UAVs on the signal contours. The examples are extracted from a larger set of data that can be found in references [9] and [11].

The signal strength level necessary for establishing a link will depend on the receiver sensitivity of the GCS. Once the GCS antenna and receiver characteristics are specified, the observation cells that satisfy the minimum signal strength requirements are easily identified by the color contours.

The simulation results indicate that in order to adapt to the dynamic propagation environment, an UAV deployed for MOUT must have the inherent capability to hover or fly at low speeds. Upon arriving at the pre-determined ideal location, the UAV will subsequently remain in situ for maximum coverage.

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Operating at high frequency has merits of higher data rate transfer, which is crucial to support the large quantity of voice and live video feeds to be transmitted via UAV-GCS linkage. However, high frequencies are attenuated more rapidly in lossy materials like cement and glass commonly found in urbanized areas. At the same time, higher frequencies are more susceptible to attenuation due to weather such as rain, snow, and fog.

This research indicates that there exists an optimal operating altitude of UAV for signal coverage. Perching at rooftops to minimize power consumption may not be ideal, as most of the signals will simply be reflected upwards. If the UAV is positioned too high above buildings, the path loss becomes high and the areas beneath the UAV experience low signal levels when using a vertical monopole antenna (due to a downward pointing null in its radiation pattern).

Urbanized areas are made up of mostly straight roads lined with buildings. Through simulation, the locations of urban canyons can be identified and exploited. Meanwhile, identified shadow regions can either be avoided or illuminated by deploying ACNs. When limited to the use of a single UAV, it was found that operating a three times the height of the tallest building in the central city provides concentric, uniform coverage.

This research has established the process that can be used to predict the signal levels in an urban environment. By understanding the unique propagation modes possible in an urban environment, planners for MOUT will be able to provide continuous, uninterrupted and constant signal linkage between all the nodes (troops, artillery, planes, ships, sensors, etc). Using UAVs as ACNs will allow the edge in information dominance.

V. REFERENCES

[1] United States Office of the Secretary of Defense, “Unmanned aerial vehicles roadmap 2002 − 2027,” pp. 33, December 2002.

[2] United States Office of the Secretary of Defence/C3I, “Unmanned aerial vehicles as communications platforms,” pp. 52, November 1997.

[3] J. B. Andersen, T. S. Rappaport, and S. Yoshida, “Propagation measurements and models for wireless communications channels,” IEEE Commun. Mag., Vol. 33, No. 1, pp. 42 − 29, January 1995.

[4] C. A. Balanis, Antenna Theory, Analysis and Design, second edition, Wiley, 1997.

[5] C. A. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.

[6] D. C. Jenn, Radar and Laser Cross Section Engineering, AIAA Education Series, 1985

[7] Urbana Wireless Tooslset, Training Manual, SAIC (DEMACO)

[8] http://www.spawar.navy.mil/robots/air/amgsss/mssmp.html accessed on 11 November 2003.

[9] Lock Wai Lek, “Effects of Radio-Wave Propagation in Urbanized Areas on UAV-GCS Command and Control,” Masters Thesis, Naval Postgraduate School, Monterey, CA, December 2003.

[10] J. Griffiths, Radio Wave propagation and Antennas, Prentice-Hall International (UK), 1987.

[11] F. Pala, “Frequency and Polarization Diversity Simulations for Urban UAV Communication and Data Links,” Masters Thesis, Naval Postgraduate School, Monterey, CA, September 2004.

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IMPACT OF LOW ALTITUDE PROPAGATION LOSSES ON AIRCRAFT SURVIVAL

AND SENSOR DESIGN

By

Richard W. (Bill) Bambrick

ABSTRACT

This paper presents a brief explanation of the nature of propagation effects at low grazing angles, followed by a discussion of the model used to evaluate them. The paper focuses on evaluations of the impacts of these effects on three important aspects of penetrating bomber effectiveness: enemy radar performance in detecting penetrating bombers, required passive threat warning receiver sensitivity levels to detect enemy emitters before exposure, and the effectiveness of the bomber’s defensive jammers. The goal is to provide an appreciation of the sensitivity of these important analytical tasks to real-world propagation conditions, and thus to stress the importance of including them in engagement simulations. PART I: INTRODUCTION

Radar operators are well acquainted with the difficulties in attempting to detect and track low-flying targets, where propagation effects such as multipath lobing, diffraction, atmospheric fading, etc. produce frequent losses of the reflected signal. Similar effects disrupt the operation of ESM and other radio receivers and radar jammers when used at low grazing angles. From the standpoint of the penetrating bomber or other tactical aircraft, these effects may be good news or bad, depending upon the situation. Denial of detection by enemy radars is advantageous, but loss of the radar signal to the bomber’s ESM receiver can be very serious, and of course anything that weakens the effectiveness of its ECM system is equally debilitating.

Low-grazing-angle RF propagation is a complex, non-linear function of geometry,

frequency, polarization, earth characteristics, and several other parameters. In the case of radar operation, for example, detection and tracking performance is severely penalized; but at the same time attenuation of signals beyond the horizon is far from infinite, which may be advantageous to ESM receivers attempting to detect enemy radar emissions before being exposed. When these effects are ignored in simulation models, the results may be overly pessimistic from the bomber’s standpoint, and the opposite for the enemy radar. Similar errors plague evaluations of ESM and ECM performance.

Admittedly, including these highly nonlinear effects in a simulation is not easy. Either one

must utilize empirical data, which is not that easy to come by, especially when dealing with

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enemy terrain characteristics, or he must incorporate models of these effects in the simulation, which can result in huge, slow-running programs that are costly to construct and operate. However, for many practical cases it is not necessary to have such precision in the simulation. The approach taken by the present author in past analyses was to run a proven propagation simulation to generate a comprehensive multi-dimensional table of propagation losses, which was then loaded into engagement simulations to evaluate the end-game results, using table look-up methods that are very fast.

This paper presents a brief review of propagation effects at low grazing angles, followed by

evaluations of the impacts of these effects on three important aspects of penetrating bomber effectiveness: enemy radar performance in detecting penetrating bombers, required ESM receiver sensitivity levels to detect enemy emitters before exposure, and the effectiveness of the bomber’s radar jammers. The goal was to provide an appreciation of the sensitivity of these important analytical tasks to real-world propagation conditions, and thus to stress the importance of including them in engagement simulations.

PART II: THE PROPAGATION LOSS MODEL

The transmission loss experienced by radio energy along a tropospheric path depends on several factors, including the antenna heights and separations, the nature of the propagation medium, and the electrical characteristics of the earth. The model used to generate the propagations losses in this paper follows the general method developed by the Environmental Science Services Administration (ESSA). Three propagation regions are used. In the line-of-sight (LOS) region, a two-ray model is utilized in a routine based primarily on work done by the Institute of Telecommunication Sciences (ITS), a branch of ESSA. This program considers the modified system geometry as determined by the effective earth radius, as well as terrain, climate, and phenomenological effects, which require an analysis of both the non-statistical and statistical aspects of the problem. Non-statistically, free space path loss, atmospheric absorption, and signal facing due to a coherently reflected multipath signal are calculated. Since the propagation medium affects the transmitted signal in a random way, a statistical evaluation must be made of the lang- and short-term signal fades which tresult from random reflections off the earth’s irregular surface, scattering from atmospheric turbulence, or propagation through regions of extreme irregular radio refractivity. For certain systems, such as high-reliability communication links operating at maximum separation, it also becomes necessary to treat the radio horizon distance as a statistical parameter. This can be achieved by using available radio refractivity data and a ray-tracing program, to provide effective earth radius data for use in the tropospheric system algorithms. As the radio horizon is approached, diffraction processes dominate the signal propagation. Ray theory cannot describe propagation at distances beyond the horizon, and wave theory must be invoked. The calculation of mean signal levels in the diffraction region consists of solving, for each point in space, the differential equation involving the radiation wave function, subject to appropriate boundary conditions.

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For paths which extend well beyond the radio horizon, forward scatter becomes the dominant propagation mode. This depends upon irregularities in the atmosphere. The methods used to predict these losses are based on earlier ITS work, and are semi-empirical, depending on the propagation path characteristics, the radio refractivity, the amount of energy reflected from the ground for low antennas, and the decrease of scattering efficiency with height in the upper atmosphere. Long-term facing due to climatic variability is also considered. At frequencies above about 1 GHz, atmospheric absorption along the transmission path becomes significant. In this model absorption due to water vapor and oxygen are considered, using a theoretical model, which has been modified to agree with empirical data. PART III: TYPICAL PROPAGATION LOSSES The table in Figure 2 lists the ranges of parametric variations for the key parameters used to generate the database for the present analyses. The database consists of numerous curves of one-way path loss versus range. Each set contains three curves representing the median and the upper and lower decile loss values, for a given set of propagation parameters. Twelve frequencies were used: 50, 100, 150, 400, 900 MHz; 1.2, 2.0, 5.0, 10.0, and 20.0 GHz, with both horizontal and vertical polarizations. Other variables included six transmitter antenna heights and five receiver antenna (or target) heights. Isotropic antennas were assumed throughout the analyses. For radar cases, the receiver antenna was replaced by a target with a given reflectivity. Data were computed for range increments of two nautical miles, from zero to 60 nmi. For the present case terrain typical of maritime coastal regions was modeled. Figure 1 includes some statistical parameters that need definition. The analyses were performed for average seasonal conditions of temperature, humidity, etc., and the climate modeled was typical of maritime coastal regions. Service Probability refers to the probability that a specified grade of service will be met or exceeded for the specified time availability. Grade of Service refers to the degree of reliability over a short period of time during which the statistics of the signal-to-noise ratio may be considered stationary. It might be expressed as the percentage of binary errors allowable for a digital system, or the percentage of word intelligibility for voice systems, for example. When the system is assumed to be receiver noise limited, the grade of service is equivalently expressed as a required receiver power input level. Time Availability is the percentage of time during which the specified grade of service, or better, is achieved. The time period involved should include all of the expected variations, and could extend over a full year, a season, a month, or certain hours of the day during a specified longer period. Figure 3 presents a plot of the predicted transmission losses for a typical case. The transmitting antenna was at 100 feet elevation, and the receiving antenna at 200 feet. The transmitter was vertically polarized, operating at 900 MHz. The three loss curves are compared

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to the free space prediction, which follows the inverse-square-law form out to the radio horizon, then breaks abruptly beyond that value, which means that losses become infinite beyond the horizon.

The effect of varying either of the antenna heights is illustrated in Figure 4, which represents a VHF transmission.

Figure 5 illustrates the effect of varying the transmitted frequency. In this case the upper

decile loss curves are presented for a 50-foot transmitter antenna and a 200-foot receiver, for path lengths of 16, 24, and 32 nmi. At the 16 nmi separation the transmission is within the line-of-sight region. The 24 nmi curve represents propagation in the trans-horizon region, where diffractions effects predominate. The 32-nmi case represents propagation beyond the RF horizon of 25.5 nmi (assuming a four-thirds earth radius model). It will be noted that all three curves predict losses greater than the free-space predictions, emphasizing the importance of including these nonlinear effects.

It is a simple matter to convert the one-way losses to the two-way losses appropriate to the

radar case using the relation

L2 = 2L1 – 10Log10(K/Lambda2) (dB) (1) where K = 4Pi, and Lambda is the wavelength.

The correction factor in this expression is necessary since the one-way losses are calculated

on the basis of isotropic transmitting and receiving antennas. In the radar (2-way) case, one end of the path contains a reflecting target. Figure 6 illustrates the results for a one-square-meter target. instead of an isotropic receiving antenna at one end of the path, as indicated by the 2-way loss equation. PART IV: IMPACT ON RADAR PERFORMANCE The ability of radars to detect low-flying targets was evaluated parametrically for a family of radars. For simplicity each radar was assumed to have vertical polarization, with an antenna 25 feet above the ground, and was assumed to require a 13 dB signal-to-noise ratio for reliable detection on a single scan. Clutter and masking effects were ignored, to focus on propagation effects. The radar’s performance was represented by its free-space normalized range, R0, which is the range at which a one-square-meter target will produce a received S/N level of zero dB. It is obtained simply by solving the well-known free-space radar range equation for a one-square-meter target and a S/N of unity. An iterative procedure was used to determine detection performance (S/N = 13 dB) for each member of the family of radars, using the pre-computed table of propagation losses. The procedure is repeated for decreasing ranges until a value of 13 dB is obtained. The equations used were:

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S/N = (R0/R)4 SIGT(LFS2/LP2) = 13.0 dB (3) Where R0 is the normalized range for a one-square-meter target, R0 = RD[(S/N)D]0.25 (4) and R = actual target range, SIGT = target cross section (square meters), LFS2 = free space path loss (2-way), LP2 = actual path loss (2-way), RD = free-space detection range versus 1 M2 target, and (S/N)D = required S/N for detection, assumed = 13 dB.

Thus, R0 = RD[(10)1.3]0.25 (4)

The results are shown in Figure 7, which pertains to a one-square-meter target flying at 200 feet altitude. Two observations are apparent. First, at all frequencies, the radars’ detection ranges were severely reduced below the free-space value. Secondly, there is some frequency above which the radar is able to detect the target beyond the horizon, which cannot be done with the conventional “cookie-cutter” model that uses the radar range equation within the LOS, and infinite attenuation beyond it.

The lower frequency bands are frequently used by long-range Early Warning (EW) and Ground Control Intercept (GCI) radars. The upper bands are typically reserved for fire control and tracking radars used by Surface-to-Air Missile (SAM) sites and Anti-Aircraft Artillery (AAA) batteries. Intermediate frequencies in the L- to C-bands are populated by search, height-finding, and acquisition radars often co-located with weapon sites. Figure 8 illustrates a somewhat surprising result when the target cross-sectional area is varied. A family of VHF radars with normalized ranges varying from 50 to 400 nmi was evaluated for a broad range of RCS values, from very stealthy targets to those typical of large transport aircraft. The broken-line curves represent free-space detection, whereas the solid lines illustrate the effects of real low-altitude losses. The relative insensitivity of detection range to target size is evident and noteworthy. These effects are even more pronounced at the higher frequencies evaluated in Figure 9 for a radar with a normalized range of 100 nmi. At frequencies above 900 MHz the detection curves cross over at a target size of about two square meters. Detection occurs beyond the horizon at UHF and higher frequencies for typical aircraft cross sections. In general, the higher the frequency, the less sensitive the detection performance is to variations in target cross-section at low altitudes.

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It should be noted that all of the preceding radar evaluations were made using the lower decile transmission losses. This was done to make the results pessimistic from the penetrating bomber’s standpoint. The significance is that the predicted radar performance would be worse than the presented estimates 90% of the time. PART V: IMPACT ON ESM RECEIVER SENSITIVITY Many military aircraft employ ESM receivers to detect, locate, identify, and evaluate threat-associated emitters in their path. Emitters of interest may be communications transmitters used to support GCI links, or radars over a broad band of frequencies used for detection, tracking, and fire control functions. It is highly desirable that ESM receivers have sufficient sensitivity to be capable of performing their design goals at ranges sufficient to provide an adequate safety margin. A set of received signal predictions was made using the real propagation loss data to evaluate these requirements. Figure 10 presents the first set of results for emitters typical of communications transmitters operating in the VHF and UHF bands, with radiated power levels of one kilowatt from 50-foot antennas. ESM receiver antennas were assumed to be isotropic, flying at 200 feet. The curves of received signal level vs. range were computed for the upper decile (90%) case, which is penetrator-pessimistic. The major factor of interest is the sensitivity required to detect the emissions far enough beyond the horizon to react safely, which is a subjective criterion dependent on penetrator speed and processing delay. When used to detect and assess EW/GCI, acquisition, and height finding radars, ESM receivers must function over a broader range of frequencies. Figure 11 illustrates the received signal levels, again at the 90% confidence level, for typical radars in this category. The curves, labeled A, C, and D pertain to actual threat radars, the identity of which cannot be revealed in an unclassified paper.

Figure 12 presents similar results for the higher frequencies used by three typical SAM and AAA fire control radars. Again, the identities of the radars labeled E, F, and G cannot be given, for security reasons. PART VI: IMPACT ON ECM EFFECTIVENESS Traditional methods of evaluating the performance of jammers against enemy radars utilize an equation that describes the jam-to-signal ratio at the victim radar’s receiver. Separate analyses are performed to determine how large this ratio should be to achieve the desired effect, e.g. range denial, range-gate stealing, false-target generation, etc. Once this is obtained, the procedure becomes simply an assessment of J/S versus encounter geometry. The conventional approach assumes that J/S varies directly with the square of the range and inversely with the jamming aircraft’s radar cross section, which is the free-space form.

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As should now be evident, this idealized result is seldom realized in the real world of low-altitude penetration. The preceding sections have shown that there are very significant departures from the inverse-fourth-power law for radars and the inverse-square-law for the one-way transmissions applicable to many jammers. A comparison was made of effectiveness estimates for a jammer using real losses and for conventional free-space losses. For the free-space assumption, the jam-to-signal ratio is given by (J/S)FS = K(PJGJ/PTGT)(BR/BJ)(R2/SIGT). (5) In this equation the J subscript refers to the jammer, T to the transmitter, R to the receiver, P is power, G is antenna gain, and B is bandwidth. All quantities are either self-evident or have already been defined. Neglecting the bandwidths for the moment, the equations representing the radar and jamming signals are

SREAL = PTGTGRSIGT/LP2 (6) and

JREAL = PJGJGR/LP1, (7) where LP1 is the one-way loss for the jammer. Thus,

J/S)REAL = (PJGJ/PTGT)(LP2/LP1)(BR/BJ)(1/SIGT). (8)

The bandwidth ratio has been inserted to account for mismatches. LP1 and LP2 are the one-way and two-way real losses at range R. The two approaches can be compared by taking the ratio of (8) to (5):

(J/S)REAL/(J/S)FS = (LP2/LP1)/(KR2). (9) But

LP2 = (LAMBDA2/K)LP12, (10)

so we have finally

(J/S)REAL/(J/S)FS = (LAMBDA2/K)LP1/R2. (11)

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When this ratio exceeds unity, or zero dB, the implication is that real low-altitude losses will benefit the jammer by producing a higher J/S than would be the case using free-space methods. This ratio was evaluated as a function of the jammer-radar separation for the same radars evaluated in the last section. The results are shown in Figures 13 and 14. The increase in predicted J/S performance against the VHF-band radar in Figure 13 is large at all ranges, due to the higher ratio of real to free-space losses at VHF for short ranges. (See Figure 5.)

The remaining radars all operate at frequencies at or above S-band, so the onset of improvement in the performance predictions does not begin until longer ranges. At ranges less than 10 miles the curves would go through rapid oscillations due to multipath effects. These were omitted for clarity.

The significance of these curves is that actual transmission losses tend to enhance jammer

effectiveness at ranges in excess of about 10 miles, where such enhancement is indeed welcome, and it gets better farther out. Alternatively, the jammer output power can be reduced at these ranges and still provide the same effectiveness. This would be advantageous in reducing the enemy’s capability to perform passive tracking on jamming signals, or to deny him the capability of home-on-jam missile guidance. PART VII: CONCLUSIONS This paper has examined the impact of real RF transmission losses at low grazing angles in three key areas of concern for advanced penetrating bombers and other air vehicles: the detectability of low-flying targets by enemy surface radars; the ability of passive ESM receivers to detect and respond to potentially hostile surface threat emitters; and the effectiveness of active electronic jammers against surface radars. In all cases the conclusions were clear: there is a serious danger in using simpler, closed-form methods of analysis to evaluate these important effects, for the answers can be orders of magnitude in error, and most often the errors are in the wrong direction. The predictions made in this paper were penetrator-conservative. Radar detection ranges were calculated using the lower decile (10%) confidence level, so that there is only a 10% chance that the predicted ranges would be exceeded. Threat warning receiver sensitivity calculations were based on upper decile (90%) losses, which is a worst case for the ESM receiver. Received signal levels would only be less than those predicted 10% of the time. The ECM calculations used the upper decile losses, rendering them jammer-conservative. Clearly, it is most important to recognize these effects in constructing meaningful evaluations of bomber-threat engagements. Many analysts have already recognized this and have incorporated the effects into their engagement models. For those who have not as yet done so, or those questioning the method they have used, the approach used in the present paper may provide some ideas. Incorporating a realistic propagation model into a fast-running engagement simulation would be a daunting task, one which has consumed countless hours of labor. Using

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the table look-up method sacrifices a little accuracy, but gains immeasurably in speed and ease of programming. And one must always be cognizant of the old programmer’s axiom: garbage in, garbage out. The loss data are only as good as the assumptions used to generate them, which adds further endorsement of the table-look-up method. One final note is necessary. Apparently there has been a trend to change the names given to the two major EW functions discussed in this paper. ESM is now called ES, which stands for Electronic Support; and ECM is now called EA, for Electronic Attack. ES is somewhat understandable, although an unnecessary shortening of a well-known term. But attempting to suggest that electronic countermeasures represent some sort of attack is not only illogical, it is downright wrong. Traditionally, ECM is a “black art” that has more of stealth than blazing saddles. The real objectives are to hide one’s position and intentions from the enemy, or to confuse or saturate or deceive him. None of these functions imply an attack. That is reserved for airborne platforms equipped for defense suppression, such as the Wild Weasels of Air Force and Navy fame.

So please, in honor of the generations of “Crows” since the early forties, let us hear no more of ES and EA. The terms ESM and ECM are as familiar as household names to Old Crows, and should be learned by “Young Crows” as well. And they fit within a whole family that includes Expendable Countermeasures (EXCM), IR Countermeasures (IRCM), EO Countermeasures (EOCM), and so on. All part of the EW tradition.

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81

New Channelized Receivers

Ming-Chiang Li

11415 Bayard Drive, Mitchellville, MD 20721, U.S.A.

(Email: liceimer@yahoo.com)

Abstract

A theory is described in support of new types of channelized receivers utilizing

optical fiber recirculation loops. These loops are capable of reproducing thousands of

replicas from a single RF signal. By analyzing the replicas, the new channelized

receivers are able to resolve simultaneous pulses in a RF signal. At the current

technology, the new channelized receivers can have a band width of 40 0. GHz and a

power of resolving these pulses to their theoretical limits. Hence the adoption of

recirculation loops will advance electronic warfare technology.

Introduction

The objective of a channelized receiver is to investigate transient and non-

cooperative RF signals over a wide range of frequencies and to identify their

characteristics. It is an important and indispensable tool in intelligence gathering of

electronic warfare to defeat hostile military operations. The conventional channelized

receiver requires a bank of filters to process hostile RF signals instantaneously. The use

of the filter bank leads to bulky and expensive channelized receivers.

An alternative is the Bragg cell receiver. It can perform as a channelized receiver

without hundreds of filters. The attractive feature of such a receiver is its potentially

small size and low cost. Although its feasibility has been demonstrated, research and

development are still needed to realize its full capability.

There is also a digital receiver, which is often referred to as the sampling

oscilloscope and is capable of analyzing very short transient pulses. But during a

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measurement, these transient pulses are required to be produced identically and

repeatedly. It means that these transient pulses must be cooperative. Hence a sampling

oscilloscope cannot function as a channelized receiver. To overcome the requirement on

the repeated production, a simple and straightforward approach is to develop an ultra fast

digitizer, which has the ability to capture all information from a single transient RF pulse.

However such a digitizer is still not within our reach.

The advancement of fiber optic technology provides us with an alternative. An

optical fiber recirculation loop can keep a single transient RF signal of interest for a

second and reproduce ten thousand identical replicas. Hence, we have a repeated

production mechanism for any transient RF signal. This reproduction mechanism

provides a new technical base. It is the objective of this paper to lay the theoretical

foundation for new channelized receivers and to stimulate their experimental efforts.

Basic elements of a recirculation loop will be presented first. The loop was

originally conceived to investigate the low attenuation of optical fibers and the versatility

of optical amplifiers1. These fibers and amplifiers are essential components in optical

fiber based communication, computer, radar2, and deception3 networks. Although the

potential of using recirculation loops to advance the RF signal processing was recognized

quite early4, the experimental response is slow. However, recently an experiment5 has

demonstrated such potential.

The theory presented here is a frequency domain theory for RF measurements.

First the theory is applied to instantaneous frequency measurements of a single RF pulse

by utilizing delay lines. The existence of multiple simultaneous pulses will lead to errors

in instantaneous frequency measurements. In the section on multiple pulses, recirculation

loops are introduced to replace the delay lines. These loops are able to generate replicas

from a single RF signal. The replacement removes the errors mentioned above and leads

to a new channelized receiver.

The recirculation loops open doors for other new channelized receivers as well.

Another example will be described in the section on the digital receiver. Finally, we

discuss the advantage of the new channelized receivers.

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Recirculation Loop

An optical fiber recirculation loop is depicted in Fig. 1. OA is an optical

amplifier for maintaining strength of the circulating RF signal. Isolator is to keep

circulation in one direction. When the optical amplifier is directional, the isolator is not

needed. Switchable coupler is to switch the signal in and out of the loop. After a

circulation, a replica is produced. Loop switch is to clean the loop in preparation for the

arrival of a new signal. For simplicity, optics and RF converters have been omitted. As

depicted in Fig. 1, many identical replicas are created from a single RF signal.

LOOP SWITCH

ISOLATOR

O A

SWITCHABLE COUPLER

Fig. 1 Optical fiber recirculation loop stores an input RF signal and regenerates its replicas after circulations. OA is an optical amplifier to maintain signal strength. Isolator is to keep circulation in one direction. Switchable coupler is to switch a signal into and out off the loop. Loop switch is to clean the loop for a new signal.

One experiment5 has demonstrated the creation of 1000 replicas in 8.0

microseconds from a single pulse with pulse length of 1 nanosecond. The degradation on

these replicas was insignificant. At the present state of optical fiber technology, it is

feasible for a 25-picosecond RF or optical pulse to circulate in a recirculation loop for

84

one second and generate hundred thousands of replicas. Recirculation loops have

provided us with a mechanism to store transient RF signals for repeated analysis.

Theory

A channelized receiver uses a large number of filters to detect and sort its

received RF signals. The main character of a filter is its frequency response, thus it is

natural to construct a frequency domain theory to describe channelized receivers. The

theory is described in a general form for the purpose of extending channelized receivers

with recirculation loops.

In the frequency domain, a transient RF signal is represented by frequency spread

function f ( )ω . The receiver needs a reference signal f ' ( )ω to process the received RF

signal. The reference signal is usually generated by a local oscillator, but subjected to

change according to the problem on hand. The receiver compares the received RF signal

with a reference signal and yields its measured complex quantity of

d f Exp i t f Exp i tω ω ω ω ω[ ' ( ) ( ' )] ( ) ( )*∫ − −0 0 (1)

where t0 is the time when the received signal arrives at the receiver, and t0 ' is the time

when the reference signal is utilized by the receiver to compare with the received signal

and to achieve the measurement. The ability for the measurement of complex quantity is

often referred to as quadrature detection, which is well known and will not be repeated

here. Furthermore, it is simple to verify that Eq. (1) denotes a correlation measurement

between the receiving and reference pulses in time domain.

Instantaneous Frequency Measurement

The transient RF signal in Eq. (1) can have many pulse components. The

measured quantity in Eq. (1) is only a complex number. It is not enough to determine all

85

these components. In this section, we will discuss a simple case. There is only one pulse

in the RF signal.

f Exp A i( ) { ( ) }ω ω ω α= − − +02 (2)

where ω0 is the center frequency of the pulse. Positive constant A is related to the pulse

width, and constant α is a phase associated with the pulse. The phase can depend on

many factors, but in the present case it is an unknown constant. We assume here that the

frequency spread function of Eq. (2) has a Gaussian shape. The integration in Eq. (1)

then can be carried out directly and the theoretical analysis becomes simple.

An instantaneous frequency measurement (IFM) receiver6 uses a delay line to

measure the center frequency ω0 (carrier frequency) of Eq. (2). In measurement, IFM

receiver divides the received pulse of Eq. (2) into two equal parts. One of them is

delayed by the delay line and the other is utilized as the reference signal. Eq. (1) now can

be written as

d Exp A i Exp i t

Exp A i Exp i t

d Exp A Exp i td

ω ω ω α ω

ω ω α ω

ω ω ω ω

[ { ( ) } ( ' )]

{ ( ) } ( )

{ ( ) } [ ]

*∫

∫

− − + −

− − + − =

− − −

02

0

02

0

022

(3)

where t t td = −0 0 ' is the time delay of the delay line. The integration in Eq. (3) can be

carried out. We arrive at

π

ω2 80

2

AExp i t

tAdd[ ]− − (4)

The above complex quantity means that we can measure these two functions

π

ω2 8

2

0AExp

tA

Cos tdd[ ] ( )− (5)

and

π

ω2 8

2

0AExp

tA

Sin tdd[ ] ( )− (6)

through quadrature detection. Eqs. (5) and (6) leads to the experimentally measured

phase angle

86

θ ω= 0td (7)

Since the time delay td of the delay line is a known quantity. From Eq. (7), we obtain

the center frequencyω0 . In order to keep the frequency ω0 ambiguous, the phase angle

θ must be less than 2π . This limits the range of time delay td for an IFM receiver to

cover a specific frequency band. The delay range depends on the frequency band of

interest. The above result is known to the radar community. We have derived such a

result based on our theory.

Multiple Pulses

The IFM receiver measures only one complex quantity of Eq. (1). It is only able

to handle one pulse signal at a time. If multiple pulses arrive simultaneously, then Eq. (7)

will lead to erroneous results. The situation can be improved by using recirculation

loops outlined above. We will first consider two RF pulses arriving at an IFM receiver

simultaneously. The RF signal of these two pulses has the form

f Exp A i Exp A i( ) { ( ) } { ( ) }ω ω ω α ω ω α= − − + + − − +1 012

1 2 022

2 (8)

where Ai and α i for i = 1 2, denote respective constants in describing their pulse widths

and unknown phase constants. ω01 and ω02 are the center frequencies of these pulses.

OPT ICAL F IBERRECIRCULAT ION

LOO P

OPT ICAL F IBERRECIRCULAT ION

L O O PQUADRATURE

R FRECEIVER

Fig. 2 A channelized receiver uses two optical fiber recirculation loops to regenerate replicas of an RF

signal for analysis. The lengths of these loops are slightly different. An RF receiver through quadrature

detection measures the interference between replicas from two loops.

87

For simplicity, signal strengths between these two pulses are assumed to be the same.

The difference in signal strengths will not alter our discussions, but lead to unnecessary

complexity.

The new channelized receiver in Fig. 2 is an interferoceiver4. After the arrival,

the RF signal of Eq. (8) is divided into two equal parts. Each part, after alignment, is sent

to a respective recirculation loop of the interferoceiver. These two recirculation loops

have different lengths and yield two trains of replicas. The length difference between

two loops is much less than either length of the loops. Each replica in one train is paired

with its corresponding replica in the other train. An RF receiver with the ability in

quadrature detection measures the complex quantity of each replica pair

d Exp A i Exp A i Exp i t nt

Exp A i Exp A i Exp i t nt

ω ω ω α ω ω α ω

ω ω α ω ω α ω

[ { ( ) } { ( ) }] { ( ' )}

[ { ( ) } { ( ) }] { ( )}

* *∫ − − + + − − + − +

× − − + + − − + − +

1 012

1 2 022

2 0

1 012

1 2 022

2 0

l

l

for n = 1 2 3, , ,.... (9)

where t0 is the arrival time of these two pulses. tl and tl ' are loop delays after a

circulation in these two loops. n is the number of circulations. Carrying out the

integration, we have

πω

πω

πα α

ω ω ω ω

2 8 2 8

4

101

2 2

1 202

2 2

2

1 21 2

1 01 2 02

1 2

2 2

1 2

1 2 01 022

1 2

AExp in t

n tA A

Exp in tn t

A

A AExp i i

Exp inA A

A At

n tA A

A AA A

c c

dd

dd

dd

[ ] [ ]

[ ]

[( )

( )] . .

− − + − − +

+− +

× −++

−+

−−

++

ll

ll

ll

for n = 1 2 3, , ,.... (10)

where

t t tdl l l= − ' (11)

88

is the time delay difference of recirculation loops and is much smaller than loop delays.

The relative delay difference between members in each pair, according to Eq. (10),

increase with the number n of circulations. It is clear from Eq. (10) that recirculation

loops lead to thousands of measurable quantities by a quadrature RF receiver. These

quantities, with a help of fast Fourier transformation, provide us an opportunity to

remove erroneous results of conventional IFM receivers.

In arriving Eq. (9), we had assumed that signals before entering recirculation

loops are perfectly aligned with respect to each other. We could delay one of the signals

with time Mtdl where M is an integer, then the ntdl dependence in Eq. (10) should be

replaced by

nt n M t n td d dl l l→ − =( ) ' (12)

where n' is another integer and can be negative. To count for a possible time delay, we

simply take

n = − −..., , , , , , ....2 1 0 1 2 (13)

in Eq. (10).

The fast Fourier transformation of Eq. (10) will yield three distinct frequency

components ω01 ,ω02 , and

( ) / ( )A A A A1 01 2 02 1 2ω ω+ + (14)

First two of them are the center frequencies of two pulses in Eq. (8). The last one comes

from the interference of these two pulses. The magnitude of the cross term in Eq. (10)

depends on the difference of frequencies ω01 andω02 . For a large difference, due to the

exponential damping,

ExpA A

A A[

( )]−

−+

1 2 01 022

1 2

ω ω (15)

it is quite small and can be neglected. Hence, through the help of recirculation loops,

each individual frequency of two pulses can be uniquely determined, when they arrive at

the optical fiber recirculation loops simultaneously. This is quite different from the

conventional IFM receiver.

89

The above discussion can be extended to the case of several simultaneous pulses.

Let us assume that there are six simultaneous pulses of equal magnitude with center

frequencies 9 950. GHz , 9 900. GHz , 9 800. GHz , 9 600. GHz , 9 400. GHz , and 9 200. GHz respectively. Furthermore, all of these pulses have the same pulse width A A A= = =1 2 ...

= × −0 25 10 12 2. sec . The pulse length of these simultaneous pulses is about 16. secµ and

the width of pulse envelop is about 0 6. MHz . Similar to Eq. (10), there will be six center

frequency terms and fifteen cross terms. Cross terms come from interference.

Interferences among pulses with center frequency differences in the order of 0 200. GHz

or 0 050. GHz , due to the exponential damping of Eq. (14), are still small. Cross terms

associated with these small interferences can also be neglected.

We have written a program7 to simulate the ability of recirculation loops in

resolving these six simultaneous pulses. The simulation is based on Eq. (10) and fast

Fourier transformation. The time delay difference of these two loops is assumed to be

tdl = −10 10 sec , which corresponds to the loop length difference of about 3cm . The total

number of circulation is 2 1638414 = which corresponds to the number of points in fast

Fourier transformation. The simulated result is given in Fig. 3. We observe distinct

1 0 G H z0.0GHz 9 .0GHz 1 0 G H z

Fig. 3 Magnitudes of six simultaneous pulses with frequencies 9 950. GHz , 9 900. GHz , 9 800. GHz , 9 600. GHz , 9 400. GHz , and 9 200. GHz . The unit for the vertical axis is arbitrary. The horizontal axis is from 0 0. to 10 0. GHz .

Fig. 4 is the same as in Fig. 3 except the horizontal

axis which is from 9 0. to 10 0. GHz .

90

pulse peaks, but these peaks are not resolved. Upon close examination, six distinct peaks

appear in Fig, 4. They are expected and correspond to peaks of frequencies 9 950. GHz ,

9 900. GHz , 9 800. GHz , 9 600. GHz , 9 400. GHz , and 9 200. GHz respectively. Although

these RF pulses haves the same signal strength, their peaks in Fig. 4 have slightly

different heights. The difference comes from the carrier frequency dependence in the

pulse correlation measurement of Eq. (1).

The horizontal coordinate in Fig. 3 is from 0 to 10 000. GHz . The alias free range

of frequency is related to the time delay difference of recirculation loops. The choice of

the time delay difference tdl = −10 10 sec in the simulation determines the alias free range

as from 0 to 10 000. GHz . In the principle, we should be capable of resolving

simultaneous pulses from DC to 10 000. GHz with a help of above recirculation loops.

But there is a limitation. We have to use a quadrature receiver in resolving these peaks.

The quadrature phase shifter is band dependent. If we are able to find a frequency

independent phase shifter, then we would have the capability stated above. However, the

band independent phase shifter is not available at the present. But the limitation still can

be overcome even in the absence of such a phase shifter. We will return to this topic later.

It will be quite interesting to find such a phase shifter. In principle, we should be

able to find one. A straight forward approach is to upper converting the RF signal to an

optical signal through a single side modulation. Then the optical signal is phase shifted

by a quadrature. The shifted optical signal is down converted back to the RF signal.

Since the frequency of the RF signal is only a small fraction of the optical signal. The

above scheme will lead to a band independent quadrature phase shifter.

In the next simulation, we select the frequencies as 9 610. GHz , 9 609. GHz ,

9 606. GHz , 9 603. GHz , 9 601. GHz , and 9 600. GHz . Furthermore, we keep all other

parameters the same as that in Fig. 4 above. The simulated result is depicted in Fig. 5.

However we only observe five peaks instead of six from the inspection of Fig. 5.

Frequency differences among adjacent pulses above are 10. MHz , 2 0. MHz , and

3 0. MHz . The difference of 10. MHz is comparable to the pulse width of 0 6. MHz .

91

The interference is not negligible between pulses of 9 601. GHz , and 9 600. GHz . Peaks

from these two pulses emerge together to form a single peak. Hence one peak is missing

in Fig. 5.

9 .597 GHz 9 .622 GHz 9.597 GHz 9.622 GHz

Fig. 5 Magnitudes of six simultaneous pulses with frequencies 9 950. GHz , 9 900. GHz , 9 800. GHz , 9 600. GHz , 9 400. GHz , and 9 200. GHz . The unit for the vertical axis is arbitrary. The horizontal axis is from 9 597. to 9 622. GHz . The simulated

measurement has 2 1638414 = circulations. Six peaks are not resolved

Fig. 6 is the same as in Fig. 5 except that the

simulated measurement has 2 6553816 = circulations. Six peaks are resolved. More circulations and replicas lead to better resolving powers.

We had chosen 2 1638414 = points in fast Fourier transformation. These points

run from -8192, -8191… 0, 1, …, 8190, and 8191. At the point n = 0 , the RF replica

from one recirculation loop is perfectly aligned with the replica from the other loop. It

means that two members of the replica pair at n = 0 are overlapped completely. Before

and after the point 0, members in replica pairs are partially overlapped as indicated by

Eqs. (10) and (15).

It is easy to observe that, for replica pairs at initial point n = −8192 and final

point n = 8191, some portions of their members are still overlapped. By taking 16384

points, we truncated the series in Eq. (10) too early and discarded valuable information.

To assure the contribution of all overlapped replica pairs are included in evaluating the

correlation result of Eq. (10). We had to increase more measurement points in

performing fast Fourier transformation.

92

We recalculate the result in Fig. 5 again, by taking 2 6553816 = points. The

recalculated result is depicted in Fig. 6. We observe six peaks. The emerged peak

between frequencies 9 601. GHz , and 9 600. GHz is now split into two. All peaks in Fig. 6

are shaper than those in Fig. 5. Hence complete correlation evaluation leads to better

resolution. However, Nyquist imposes a theoretical limit on best resolution based on the

pulse width. It is about 0 6. MHz in simulations presented above. The degradation on the

replicas is negligible as observed in experiments. It is feasible to reach the theoretical

limit by increasing number of the replicas.

When multiple pulses arrive simultaneously, an IFM receiver with single delay

line would lead to erroneous results. Efforts have been made to remove them by using

multiple delay lines, but with limited success. The approach of multiple delay lines is

parallel in nature. A received signal has to be amplified first, and then evenly divided

into these lines. Meanwhile the divided signals should not be degraded. It is impossible

to carry out the above tasks when the number of multiple delay lines increases to

hundreds.

The approach of recirculation loops can be considered as an approach of multiple

delay lines. The recirculation loops convert a parallel operation to a series operation.

Not only hundreds, but thousands of replica pairs in a series can be generated by these

loops without degradation.

The instrument described in Fig. 2 is a new IFM receiver with recirculation loops.

Such an instrument is able to resolve multiple simultaneous pulses without ambiguity.

Hence it is a new channelized receiver in a general sense. Furthermore, erroneous

results of conventional IFM receivers, in view of the new IFM receiver, arise from

insufficient delay lines. The insufficiency leads to errors as induced by irresolvable

ambiguities.

Digital Receiver

93

We have indicated in the introduction that it is feasible to have a new channelized

receiver based on the digital receiver of sampling oscilloscope. The digital RF receiver

emits own short pulse for reference. For simplicity, the reference signal f ' ( )ω is taken

as

f Exp A' ( ) { ' }ω ω= − 2 (16)

where A' is a constant and denotes the width of the short pulse. If the pulse is extremely

short, then constant A' is zero and Eq. (16) becomes

f ' ( )ω = 1 (17)

Eq. (17) simplifies Eq. (1) of measurement as

d Exp i t Exp i t f t tω ω ω[ ( ' )] ( ) ~( ' )*∫ = −0 0 0 0 (18)

where t '0 denotes the instant when the RF signal is digitized by a sampler of the digitized

receiver. Function ~( ' )f t t0 0− denotes the shape of the RF signal in time domain. Other

notations in Eq. (18) are the same as that in Eq. (1). In writing down Eq. (18), we have

implicitly used a quadrature detection method to digitize the RF signal.

OPTICAL FIBERRECIRCULATION

LOOP

SAMPLING

OSCILLOSCOPE

SAMPLING

OSCILLOSCOPE

2p

Fig. 7 A channelized receiver has a recirculation loop and two sampling oscilloscopes. Replicas in one path have to pass a quadrature phase shifter. Both oscilloscopes are coordinated and have a same shift size. An interferoceiver as the new channelized receiver of another type is depicted in

Fig. 7. An RF signal with simultaneous pulses is sent to a recirculation loop. Every

replica from the loop is split into a replica pair. Members in pairs are sent to two

94

separated paths. There is a quadrature phase shifter in one path. The receiver comprises

two sampling oscilloscopes to digitize these members from two paths respectively. Two

oscilloscopes are coordinated in time shifts to sample sequential replicas. Their shift

sizes are the same.

The digital receiver through quadrature detection measures the complex quantity

of each replica in the train

d Exp i t nt f Exp i t f t t ntd dω ω ω ω[ ( ( ' ))] ( ) ( ) ~( ' )*∫ + = − −0 0 0 0

n = − −...., , , , , ,....2 1 0 1 2 (19)

where td is the time shift between subsequent measurements. The circulation number n

in the above equation has the same meaning as that in Eq. (13) to account for an initial

delay.

Now let us consider the RF signal with two simultaneous pulses of Eq. (8). We

get the measured complex quantity from Eq. (19)

π

ω απ

ω αA

Exp in tn t

Ai

AExp in t

n tA

idd

dd

101

2 2

11

202

2 2

224 4

[ ] [ ]− − + + − − + (20)

where we have chosen t t0 0 0− =' . In comparison with Eq. (10), cross terms do not exist,

but two terms in the above equation will interfere with each other.

The above discussion can be extended to the case of several simultaneous pulses

as well. Let us return to the two cases above. For the case of six simultaneous pulses

with center frequencies 9 950. GHz , 9 900. GHz , 9 800. GHz , 9 600. GHz , 9 400. GHz , and

9 200. GHz , the separation between these frequencies is large. Eq. (20) leads to a similar

result as that of Eq. (10), which is depicted in Figs. 3 and 4. For the case with center

frequencies of 9 610. GHz , 9 609. GHz , 9 606. GHz , 9 603. GHz , 9 601. GHz ,

and 9 600. GHz , there is a slight difference between the results from Eqs. (10) and (20).

The result of Eq. (20) is depicted in Fig. 8 after taking 2 6553816 = Fourier points and

that of Eq. (10) was in Fig. 6. Pulse peaks in Fig. 6 and Fig. 8 are similar. The difference

95

between these two is that pulse peaks in Fig. 6 are slightly sharper than that in Fig. 8. It

means that sampling oscilloscopes lead to a slightly larger interference between adjacent

9.622 GHz9.597 GHz

Fig. 8 A simulation from channelized receiver with sampling oscilloscopes. Simultaneous pulses and number of circulations in Figs. 6 and 8 are the same. The simulated results in both figures are very similar, but peaks are slightly sharper in Fig. 6 than in Fig. 8.

pulses than that from the quadrature receiver. This is because pulse widths as indicated

by Eq. (10) are narrower than that by Eq. (20). The narrowing factor is equal to the

square root of two from inspection. Hence the instrument with sampling oscilloscopes

and recirculation loops as described in Fig, (7) is able to resolve multiple simultaneous

pulses with ambiguity. Such an instrument is also a new channelized receiver in a general

sense.

We have assumed that the digitizer of the sampling oscilloscopes only samples

each replica once. For the given examples, these replicas have a pulses width

about 16. secµ . Within such a width, many existing digitizers can sample several times.

A use of these digitizers will reduce the number of needed replicas to achieve resolutions

of interest.

The gate width of sampling oscilloscopes has been assumed to be infinitesimally

small. If the gate width is not small, then there is gate dependence in Eq. (20). From Eq.

(16), Eq. (18) should be replaced by

d Exp i t Exp A f Exp i tω ω ω ω ω[ ( ' ) ( ' )] ( ) ( )*∫ −02

0 (21)

Let us consider the RF signal with two simultaneous pulses of Eq. (8) again. After the

integration, the digital receiver through quadrature detection leads to the measured

complex quantity

96

π

αω ω

πα

ω ω

A AExp i in

AA A

tn tA A

A AA A

A AExp i in

AA A

tn tA A

A AA A

dd

dd

11

1 01

1

2 2

1

1 012

1

22

2 02

2

2 2

2

2 022

2

4

4

++ −

+−

+−

++

++ −

+−

+−

+

'[

' ( ' )'

']

'[

' ( ' )'

']

ll

ll

(22)

It is clear from the above equation that a finite width of sampling gate will shift measured

frequencies as well as damping signal strengths. The width of sampling gate should be as

small as possible.

Removal of Contamination

In the preceding sections, we had assumed that a broad band quadrature phase

shifter exists. The assumption leads the measurements of complex quantities as

expressed in Eqs. (10) and (20). In the absence of the quadrature phase shifter, we are

only able to measure real parts of the above complex quantities

πω

πω

πα α

ω ω ω ω

2 8 2 8

2

4

101

2 2

1 202

2 2

2

1 21 2

1 01 2 02

1 2

2 2

1 2

1 2 01 022

1 2

ACos n t Exp

n tA A

Cos n t Expn t

A

A ACos

Cos nA A

A At Exp

n tA A

A AA A

dd

dd

dd

[ ] [ ] [ ] [ ]

{ [ ]}

[ ] [( )

( )]

ll

ll

ll

− + − +

+−

×++

−+

−−

+

(23)

and

π

ω απ

ω αA

Cos n t Expn t

A AExp n t Exp

n tAd

dd

d

101 1

2 2

1 202 2

2 2

24 4[ ] [ ] [ ] [ ]− − + − − (24)

Sine terms are missing from above two equations in comparison with Eqs. (10) and (20)

respectively. Without the quadrature phase shifter, measured quantities on sine

dependence no longer exist. But it will not limit our ability to resolve simultaneous

pulses.

97

1 0 G H z0.0GHz

Fig. 9 is similar to Fig.3 except that Fig. 3 is based on Eq. (10) and Fig. 9 on Eq. (23). With a help of quadrature phase shifter, there are only true peaks as in Fig.3. Without the help, there are additional image peaks as in Fig. 9.

Let us return to the case depicted in Fig. 3. From Eq. (23), we perform the same kind of

fast Fourier transformation as that leads to Fig. 3 from Eq. (10). The result is depicted in

Fig. 9. All pulse peaks in Fig. 3 also appear in Fig. 9. Besides these true peaks, there are

additional peaks. From a close examination, we find that they are mirror images of the

true peaks. If the quadrature phase shifter is available, then with the help of quadrature

detection these images will not appear.

The presence of images will contaminate the result of measurements and reduce

the resolving power on simultaneous pulses. However, the contamination can be

overcome. The alias free range in the present simulation is from 0 0. to 10 0. GHz , that is

determined by the time delay difference ( sec)tdl = −10 10 of recirculation loops. In Fig. 9,

the true peaks are

9 950 9 900 9 800 9 600 9 400 9 200. , . , . , . , . , .and GHz , (25)

and the image peaks are

0 050 0100 0 200 0 400 0 600 0 800. , . , . , . , . , ,and GHz (26)

These are also image peaks

10 050 10100 10 200 10 400 10 600 10 800. , . , . , . , . , ,and GHz (27)

that are not shown in Fig.9. The alias free range has folded image peaks in Eq. (27) to

that tin Eq. (26). If we change the time delay difference to ( . sec)tdl = × −08 10 10 , then the

alias free range will be from 0 0. to 12 5. GHz . Following similar steps as that lead to Fig.

98

9, we will get a new figure where the true peaks will remain at the same positions of Eq.

(25) while the image peaks will each move by 5000. GHz to new positions at

15050 15100 15200 15 400 15600 15 800. , . , . , . , . , ,and GHz (28)

The movement is from the reflection with respect to the upper frequency boundary of the

alias free range. A change of the time delay difference will move the upper frequency

boundary as well as the images. But the true pulse peaks will not move. From the

movement, we are able to differentiate images from the true peaks. Hence the

contamination can be removed.

Due to the absence of quadrature phase shifter, we have two alternatives. One is

to have additional set of recirculation loops for a second measurement from Eq. (23).

The loop delay difference for the first set is different from that for the second set. The

second is to have a recirculation loop, which is able to change its length. After finishing

the first measurement, the loop length is changed and the second measurement is carried

out sequentially. Both alternatives will lead to two different sets of measurements.

Positions for the true peaks should be the same in both sets of measurements, but

positions for the images are different. We remove images by examining the differences.

Eq. (24) is the basic equation for the digital receiver. It only utilizes one single

recirculation loop as depicted in Fig. 7. In the absence of quadrature phase shifter, the

experimental setup is very similar to that of Fig. 7 except that time shift sizes of two

oscilloscopes are different.

The above discussion indicates that the absence of quadrature phase shifter is not

a limiting factor.

Instability and Noise

We will consider effects of instability and noise on the channelized receiver based

on sampling oscilloscope. All pulses were assumed to have the same pulse width in the

numerical simulations. Eq. (20) can be simplified as

99

Exp in tn t

Ai Exp in t

n tA

idd

dd[ ] [ ]− − + + − − +ω α ω α01

2 2

1 02

2 2

24 4 (29)

A common constant π / A in front of each term has been omitted in the above equation.

In this case, there is only one recirculation loop. The instability in the loop length has a

little effect on the channelized receiver, if the sampling oscilloscope is triggered by

reproduced replicas. It may happen that replicas do not have sharp leading edges, which

will lead to difficulties in triggering. If it is the case, two alternatives exist. One is

through synchronization between loop circulation and sampling oscilloscope; the other is

to seed a sharp edged pulse along with simultaneous pulses of interest to yield needed

triggering replicas. The instability effects can take place, if the sampling probe and

switchable coupler have jittering variations.

Other instability might be that amplitudes of replicas cannot be equally

maintained. We model above instabilities as

( ) [ ]

( ) [ ]

14

14

11 01 01 21

2 2

1 1

12 02 01 22

2 2

2 2

± − ± − + ± +

± − ± − + ±

ε ω ω ε α ε

ε ω ω ε α ε

Exp in t i tn t

Ai

Exp in t i tn t

Ai

d dd

d dd

(30)

where ε11 and ε12 denote non-equal replica amplitudes. ε21 and ε22 are sampling

variations respectively. ε1 and ε2 are the background noises associated with each

simultaneous pulses.

We reevaluate the case of Fig. 8 again by inserting above instability and noise

parameters. First, all these parameters are allowed to vary randomly with their

magnitude equal to 0 45. . The result is depicted in Fig. 10. We observe all six peaks as in

Fig. 8 despite a noise floor. When the randomly varied magnitude becomes 05. , we

cannot observe these peaks.

Second, the amplitudes of random background noise parameters ε1 and ε2 are

allowed to increase linearly with circulation numbers from zero to one to reflect that the

background noise is accumulative. Other random parameters are fixed to a magnitude

equal to0 45. . The result is depicted in Fig. 11. We again observe all six peaks as in Figs.

100

8 and 10 despite a large noise floor. The horizontal axis has to be shifted. It is due to our

plotting program, where the highest peak is always in a specified position. The random

parameters have different effects on different peaks. This can change the highest peak

among these six peaks. The plotting program causes highest peaks in Figs. 10 and 11 to

appear at the same position in the respective figures.

9.622 GHz9.597 GHz

Fig. 10 is a reevaluation of Fig. 8 with random instability and noise parameters. All six peaks in Fig. 8 are observed here despite the noise floor.

Fig. 11 is another reevaluation of Fig. 8, but has a different set of instability and noise parameters. The background noise is allowed to increase linearly with circulation numbers. The random parameters have different effects on simultaneous pulses and cause a change in highest peak. Due to the plotting program, the highest peaks in Figs, 10 and 11 are at the same position respectively. The horizontal coordinate should shift accordingly in comparison with Fig. 10.

Third, the amplitudes of random background noise parameters ε1 and ε2 are

allowed to increase linearly with circulation numbers from zero to two, which is twice as

fast as that of the above. Other random parameters are still fixed to a magnitude of 0 45. .

The result is depicted in Fig. 12. We again observe all six peaks as in Figs. 8, 10, and 11

despite other minor noise peaks.

Fig. 12 is similar to Fig. 11, but the background noise increases faster.

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In above simulations, simultaneous pulses have the pulse length of 16. secµ .

Within the above pulse length, a digitizer with digitizing rate of100 MHz can sample

these pulses for 160 times. Such a digitizer is very common at the present. In our

consideration above, we only sample each replica generated by the RF signal train

generator once. A commercial RF signal train generator8 has a band width of 20GHz

and is able to generate 1000 replicas without noticeable degradation. Hence there should

not be any difficulties, with the aid of their replicas, to sample these simultaneous pulses

for 160,000 times. However, our examples giving in Figs. 8, 9, 10, 11, and 12 only

require the sampling of 65,538 times. Above simulations have shown that new

channelized receivers will perform well in experiments.

Conclusion

We have presented an analysis on new channelized receivers based on

recirculation loops. The resolving power of these receivers is related to the number of

replicas produced by these loops. It is feasible for the number to be in the order of

hundred thousands. A success means that, even without a bank of expensive and bulky

filters, we are still able to distinguish hundreds simultaneous pulses and to resolve them

in the neighborhood of their theoretical limits.

These new receivers operate in the RF band and do not need an intermediate

frequency. At the present state of technology, this band can be from 0 0. GHz

to 40 0. GHz . A realization of new channelized receivers discussed above will increase

our ability in electronic intelligence gathering and enhancing the art of electronic warfare.

Conventional channelized receivers rely on band limited RF components, like

filter bank, dispersive delay line, sweeping local oscillator, Bragg cell, etc. New

channelized receivers based on recirculation loops do not have such requirements. This

is the main difference between conventional and new channelized receivers. The

difference provides the latter with excellent capacities in resolution and band width.

102

The interferoceiver depicted in Fig.2 analyzes simultaneous pulses through

correlation measurements. According to the theory of radar, the most optimum receiver

for radar signals is the correlation receiver. It was impossible to store radar signals for

correlation measurements. Matched filters, which can not be perfect9, were invented to

mimic these correlation measurements. Now, the interferoceiver removes the

impossibility. Measurements as indicated by Fig. 2 are true correlation measurements.

Furthermore, the interferoceiver will automatically compress FM modulated pulses10.

Hence the interferoceiver should be the most optimum apparatus in analyzing radar

signals and simultaneous pulses.

All RF emitters have their own intrinsic modulations. These emitters may change

their modes of operation, but can not change their intrinsic modulations. With their

ability of high resolving powers, it would not be difficult for new channelized receivers

to identify intrinsic modulations. Hence they will become viable tools in countering war

time reserve modes of hostile emitters and in passive identification.

103

References:

1. E. Desurvire, M. J. F. Digonnet, J. Shaw. “In-line Fiber Optic Memory and

Method of Using Same”, US Patent No. 4,815,804, (1989); E. Desurvire, Erbium

Doped Fiber Amplifiers: Principles and Applications, (John Wiley & Sons, Inc.

New York, 1994); M. Eiselt, W. Pieper, G. Grobkopt, R. Lugwig, and H. G.

Weber, IEEE Photonic Technology Letters, 4 (1993) 422.

2. M. C. Li, "Impact of Optical Fibers on Bistatic Radars," in Proceedings of 1990

Government Microcircuit Applications Conference, Las Vegas, Nevada,

(November, 1990), 475; T. L. Lane, N. T. Alexander, and C. A. Blevins,

“Overview of the Bistatic Coherent Measurement Systems”, in the SPIE

Proceedings Vol. 3395, (April, 1998), 37.

3. M. C. Li, Journal of Electronic Defense, (January, 1993) 60; D. Gobel, Journal of

Electronic Defense, (February, 1999) 45.

4. M. C. Li, " Optical Fiber Based Interferoceiver" in IEEE Technical Digest on RF

Optoelectronics, Keystone, Colorado, (August, 1995) 72; “Interferometric

Radars,” in Proceedings of Thirty-Third Annual Association of Old Crows

International Electronic Warfare Technical Symposium and Convention,

Washington, D.C., (September, 1996).

5. Y. Yin, A. Chen, W. S. Zhang, G. Chen, X. Z. Wang, “Multichannel Single-shot

Transient Signal Measurements with an Optical Fiber Recirculation Delay Line

Loop”, in Proceedings of 2003 Particle Accelerator Conference, Stanford,

California, (May, 2003) ; Nuclear Instruments and Methods in Physics Research,

A517 (2004) 343–348.

6. J. B. Tsui, Microwave Receivers and Related Components, (National Technical

Information Service, 1983).

7. Mathematica 3.0 is used for computation here, (Wolfram Research, Inc.

Campaign, IL 1998).

104

8. A product from YY Labs. The price for each generator is twenty thousand dollars.

To configure a new channelized receiver as describe in the present paper, we only

need subsystems off the commercial shelve. Hence, its cost will be far less than

that of conventional channelized receivers. .

9. J. V. DiFranco, and W. L. Rubin, Radar Detection, (Artech House Inc., Delham,

Massachusetts, 1980).

10. M. C. Li, IEEE Transactions on Antennas and Propagation, (to be published).

105

The Design and Analysis of a Space-Based Experiment for Inflatable Structures

David C. Moody, MSEE and Richard A. Raines, Ph.D.

Department of Electrical and Computer Engineering

Richard G. Cobb, Ph.D. and Anthony N. Palazotto, Ph.D. Department of Aeronautics and Astronautics

Air Force Institute of Technology

2950 Hobson Way, Bldg 642 Wright Patterson Air Force Base, Ohio 45433

Contact author: Richard A. Raines Voice: 937.255.6565 extension 4278 Fax: 937.656.4055 E-mail: richard.raines@afit.edu

106

Abstract

As the demand for space based communications increases and the need for faster data

throughput, satellites are becoming larger. Larger satellite antennas help to provide the

needed gain to increase communications in space. Compounding the performance and

size trade-offs are the payload weight and size limits imposed by the launch vehicles.

Inflatable structures offer a cost saving opportunity since the structure is significantly

lighter and has a reduced storage volume. This allows for smaller launch vehicles and/or

increased performance capabilities. Inflatable structures offer possibilities for increased

satellite lifetimes, increased communications capacity, and reduce launch costs. This

paper presents the results of research to support the design, development, and deployment

of inflatable structures through the Rigidized Inflatable Get-Away-Experiment (RIGEX).

The RIGEX is an autonomous computer and sensor system used to control the flow of the

experiment while at the same time collecting and recording temperature, pressure,

vibration, and image data. The computer system consists of two processors, one for

experiment control and sensor data collection and the second for image data collection.

The results of this research lay the foundation for future inflatable structures research.

This pioneering effort has been selected for flight testing on-board the U.S. Space

Shuttle.

107

I. Introduction

Over the recent decades the demand for satellite communications has increased

dramatically. Among the demands are telephony, television, and the new computer

networks in the sky. As the demand rose, more and more geostationary satellites have

been placed into orbit with longitudinal spacing of only 2º [PrB03]. The frequency

spectrum is also becoming very limited. New satellites are being designed to optimize use

of orthogonal antenna polarizations and the use of spot beams to provide for spatial

division multiple access (SDMA). SDMA allows for the frequency spectrum to be reused

[Skl00]. To provide tight (small beamwidth) beams, the antennas must have relatively

large apertures. The problems with large antennas are their weight and the amount of

space needed for launch.

The solution to the weight/space problem of mechanically built satellite antennas

is the use of inflatable structures. These inflatable structures can be used for building

large antennas for communications, radar, and electronic warfare purposes. They can also

be used for building large trusses for manned space stations and support structures for

large solar sails. The primary benefit of the inflatable structures is the cost savings.

Folded tubes can be packed tighter and lighter than solid beams that must be folded and

hinged together for launch. Also, compressed gas can be used to produce the force

needed to form the structure instead of relying on motors. Some examples of research in

inflatable antennas can be found in [FrBi92, Fr97, Hu01].

The Air Force Institute of Technology (AFIT) has developed an experiment called

the Rigidized Inflatable Get-Away-Special Experiment (RIGEX). RIGEX is the first step

towards a future of space inflatable structures. This experiment is planned for flight on-

108

board a future Space Shuttle mission. The data collected from RIGEX will help further

the development of rigidizable inflatable structures. RIGEX will also be the first space

experiment involving the deployment of a rigidizable inflatable structure.

The objective of RIGEX is to design an autonomous control and inflation system

for experimentation on the Space Shuttle. The goal of this experimentation is to verify

and validate ground testing of inflation and rigidization methods for inflatable space

structures against zero-gravity space environment. The inflatable tube structures under

testing are shown in Figure 1. L’Garde Incorporated is the manufacturer of the tubes

being used in the experiment. The RIGEX experiment contains an aluminum structure

housing the equipment required to inflate and rigidize three of the tubes. The conceptual

experiment structure can be seen in Figure 2.

Figure 1: Folded rigidizable inflatable tubes

109

Figure 2: Conceptual design of RIGEX structure with its equipment

This paper presents the design of the RIGEX system. The paper is organized into

five sections. Section 2 describes how the experiment will execute. The third section

describes the experimental sensor and operation requirements. The fourth section

presents the results of the design testing. The fifth section discusses the conclusions of

the research.

II. Flow of Operations

The RIGEX experiment consists of computer boot-up, initialization, and tube

excitation and inflation. The tube excitation and inflation processes are repeated for each

of the three tubes. The initialization of the experiment includes everything leading up to

the start of a tube process. Initialization has the following steps.

§ Shuttle launches: The experiment is carried aboard the shuttle in a Get-Away-

Special (GAS) container.

110

§ Altitude of 50,000 ft: A barometric switch on board the shuttle closes a relay

allowing power to flow to environmental heaters. The heaters are controlled to

maintain the temperature of the computer system and cameras at 20°C.

§ On orbit: The shuttle crew closes relay for power flow and computer system boot-

up.

§ Initialization: The computer initializes memory and sets the on-board timers.

§ Failsafe check: The computer system checks to determine the boot-up iteration that

is occurring. If it is the second or subsequent time, the computer jumps to the

appropriate location in the experiment program to continue where it left off.

After the initialization and indication of first boot-up, the computer proceeds to

inflate the first tube in the experiment. The inflation process is required to heat the tube

in the oven above 125°C, inflate it, allow it to cool, and vent the gas from the tube. To

execute this process the following procedure must be performed by the computer system:

§ Generate failsafe marker to indicate that the computer has entered into a tube’s

inflation process.

§ Activate heaters and lights: The computer allows current to the ThermoFoilTM

[MI99] heaters in the oven. The interior of the experiment is lighted with super-

bright LEDs.

§ Check temperature: The computer system monitors the temperature of the tube

using two thermocouples. The coldest of the two temperatures is compared to the

threshold temperature of 125°C.

§ Generate failsafe marker indicating that the tube was completely heated.

111

§ Inflation: Once the transition temperature has been reached, the computer opens the

oven doors and allows nitrogen to flow into the tube. During the inflation of the tube,

the gas pressure inside the tube and the vibrations of the top flange of the tube are

measured. Black-and-white digital images are taken during the inflation of the tube.

A preset timer controls the amount of time for inflation.

§ Venting: After the tube has inflated and the inflation timer has expired, the computer

pauses to allow the tube to cool. Once the tube cools, the computer vents the gas

from the tube giving the tube equal pressure inside as outside.

§ Generate failsafe marker indicating the completion of the inflation process.

After tube inflation, the computer system collects data in order to measure the

structural characteristics of the inflated tubes. The structural characteristics of interest

are the modal frequencies. The following routine is used to excite the tube in order to

collect the required data.

§ Image the tube: Five tube images are taken before the excitation process begins.

§ Generate failsafe marker before vibrating the tube.

§ Excite the tube: The computer transmits a waveform into the tube. The waveform is

injected into the tube using a pair of cross-connected piezo actuator patches located at

the base of the tube. The patches contract or expand oppositely depending on voltage

level in order to vibrate the tube. The computer measures the vibrations felt at the top

of the inflated tube. The measurements are taken along three coordinate axes.

§ Generate failsafe marker indicating completion of the tube excitation.

§ Image the tube: Five tube images are taken after the excitation process ends.

112

§ Generate failsafe marker indicating the end of an individual tube analysis. The

computer moves onto the next tube, or shuts down if the third tube analysis is

complete.

From the experiment’s flow of operation, all the design requirements are

determined. The only requirement not readily apparent is power. The experiment is

powered using bank of 30V battery cells. Each 30V battery cell consists of 20 D-Cell

batteries. These batteries are chosen due to other successful experiments flown on the

Space Shuttle. Section 3 addresses the remainder of the experimental requirements as

well as the chosen solutions.

III. Requirements and Solutions

This section presents the experiment sensor and computer requirements. Under

each topic, the requirement is presented as well as the solution.

Temperature

The experiment has seven places that require temperature monitoring. Each tube

requires two temperature sensors, and the encasement structure requires one. The two

temperature sensors are placed inside the two major folds of the tube. The last

temperature sensor is placed on the encasement structure. The exact location is not

crucial because there is nothing aboard the experiment that is capable of heating the

structure to a significant temperature.

113

Thermocouples are the sensors of choice. They have a relatively high

temperature range and do not require much space for adhering to the tube. Due to the

choice of thermocouples, a custom analog-to-digital (ADC) converter board is required.

Pressure

Two pressure measurements are required. The first requirement is to be able to

measure the pressure during the inflation of the tube. The pressure during inflation will

be less than 15 psig. The second pressure requirement is to be able to measure the

pressure in the nitrogen gas storage tank (less than 500 psig). This requirement exists in

case the experiment returns from the flight and the tubes are not inflated. The recorded

pressure provides an indicator for any potential gas storage failures.

Piezo pressure transducers provide an experimental solution. The transducers are

made for different pressure ranges. Each transducer requires a +5 VDC supply and the

output voltage is given using a differential pair. The differential pair requires an ADC

board capable of handling differential inputs.

Vibration

Vibrations experienced by the tubes during inflation and excitation must be

measured. The vibrations are detected by measuring the acceleration in a particular

direction. The accelerations must be measured in all three dimensions. This requirement

dictates the use of a triaxial accelerometer. The accelerometer must be small and

lightweight, since it is placed on the top of the tube. It must also be able to produce the

output voltages without the use of signal conditioners. The primary purpose of the

114

accelerometer is to measure the vibrations experienced during the tube excitation. The

level of vibration felt by the accelerometer will be less than 1 g. This low level of

vibration requires a fairly high sensitivity.

The secondary requirement for the accelerometer and computer system is to

measure the vibrations felt at the top of the tube during inflation. The accelerations felt

during inflation are recorded in an attempt to model how the tube inflates. The model

requires the displacement signal and not the acceleration. To attain the displacement

signal, the acceleration data is digitally lowpass filtered to reduce the amount of noise

present. Then along each axis, the acceleration data is numerically integrated twice.

Equations 1 and 2 show how the integration process is performed for each axis where Ts

is the sampling period and N is the number of samples.

where n = 0,1, … N-1 (1)

where n = 0,1, … N-1 (2)

The resulting displacement signals are combined to show the coordinate point where the

tube is located at each sample point. This method of modeling movement is accurate

provided the coordinate axes of the accelerometer are not rotated. The amount of rotation

present in the top flange of the tube determines the model accuracy. Due to the size of

the tube, it is unfeasible to place a gyro or another accelerometer at the top of the tube to

measure rotation.

∑=

⋅=n

isnn Tav

0

∑=

⋅=n

isnn Tvs

0

115

Tube Excitation

Part of the RIGEX experiment is to measure the modal characteristics of the tubes

after they have been inflated. To determine the modes of the tubes, the transfer function

of the tubes must be determined. To find the transfer function, the tubes must be excited

by some vibrating source using some waveform that represents all frequencies between 0

Hz and 1000 Hz. The waveform chosen to excite the tubes is a linear up-chirp waveform

whose frequency ranges from 5 – 1000 Hz and has duration of 1 second. Equation 3

defines the excitation waveform.

(3)

The waveform is generated using a digital-to-analog converter (DAC). The discrete

amplitude waveform is then filtered using an analog 8th order lowpass Butterworth filter.

The filtered waveform is relay switched to a step-up transformer based on what tube is

being excited. The transformer applies voltage gain raising the waveform voltage from 5

volts to hundreds of volts. This step-up in voltage is needed to apply enough voltage to

excite the piezo actuators attached at the bottom of the tube. The large voltage is needed

to move the tube enough for the accelerometer to detect the vibration. The actuators are

located at the base of the tube and are placed on opposite sides with their leads cross-

connected. The cross-connection is used to make one actuator contract and the other

expand for a given voltage across their leads.

Once the tube is vibrating, the resulting data is then used to determine the transfer

function of the tubes using Matlab®’s transfer function estimation program called tfe.

))9955(2cos(5)( ttty ⋅⋅+⋅⋅= π

116

The tfe program uses power spectral densities in its theory to estimate the transfer

function (H(f)). This ratio consists of the cross power spectral density of the input and

output and the power spectral density of the input signal. To perform this estimation, the

input and output (x(t) and y(t) respectively) signals are assumed to be random. The

power spectral density of the input signal is found by taking the Fourier transform of the

autocorrelation of the input signal [ShB88].

(4)

(5)

The cross power spectral density is found by taking the Fourier transform of the cross-

correlation of the input signal and the output signal [ShB88].

(6)

(7)

It is shown in [ShB88], that the cross power spectral density is equal to the transfer

function of the system multiplied by the power spectral density of the input signal.

(8)

tdtxtxRX ∫∞

∞−

−⋅= )()()( ττ

ττ π deRfS fjXX ⋅⋅= ∫

∞

∞−

2)()(

tdtytxRXY ∫∞

∞−

−⋅= )()()( ττ

ττ π deRfS fjXYXY ⋅⋅= ∫

∞

∞−

2)()(

)()()( fSfHfS XXY ⋅=

117

The transfer function can be estimated by forming the following ratio:

(9)

The TFE program performs some additional signal conditioning before finding the power

spectral densities.

Experimentally, it has been shown that the first and second modes of a fully

inflated tube lie at 62Hz and 660Hz [SiT02]. Figure 3 shows the transfer function of a

fully inflated tube that is excited using an up-chirp signal and shows the desired results.

The accelerometer used in this case had a sensitivity of 10mV/g.

Figure 3: Transfer function of a fully inflated tube

)()(

)(ˆfSfS

fHX

XY=

118

Imaging

The imaging system has two purposes. The first purpose is to capture pictures of

the tube as it is inflating. The second purpose is to determine the height of inflation and

any tilt the tube might have after rigidizing. The solution for the first purpose is to use a

digital camera that is controlled by the computer system. The testing involved for the

image capture is to determine how quickly the computer can capture the images and put

them into memory.

The second purpose poses a little more difficulty. To determine height from the

camera based on the images alone, a white circular target with single peg at the center is

placed on the top of the tube. To determine height, a relationship between the tube’s

white circular target diameter (in pixels) to its true diameter must be determined. This

relationship is determined by setting the tube with its target between 1.66-9.91 inches in

0.25 inch intervals from the camera lens. At each of these distances, 10 pictures are

taken. Each image is then preprocessed to remove all the background light reflections

leaving the target and a dark background. After each image has been preprocessed, the

Matlab® program thresholds the image and fills in any holes, leaving only a circle or an

ellipse. Once the circle/ellipse is determined, Matlab® uses regionprops script to

determine the major axis length in pixels. The script uses the second moments of the

image pixel distribution to determine the major and minor axes of the ellipse [Ma03].

The results of each of the 10 images are averaged and a ratio formed based on a known

diameter of the target and the averaged pixel diameter from the images. Each known

distance from the camera lens has an averaged diameter/pixel data point. The 36 data

points have a least squares line fit in hopes that a linear relationship exists. Once the

119

relationship of the diameters to height has been established, future images can have

height measured by evaluating the ratio of the true diameter to pixel diameter using the

least squares result.

For angle determination, the target peg is the primary tool used. After the image

is thresholded, the algorithm subtracts the thresholded image with the peg from the filled

in image (ellipse only). This results in leaving only the peg in the image. To get only the

outline of the peg, the image is edge-detected. This edge-detection is accomplished using

a Laplacian-based two-dimensional filter [Lim90]. Once the outline of the peg has been

found, it is necessary to center the peg on an origin. This is accomplished by subtracting

out the mean value of the edge pixels of the peg. To determine tilt angle, it is necessary

to measure the length in pixels of the peg in the picture. This measure in pixels is then

converted to physical distance, using the results distance determination. Based on the tilt

angle of the tube, the measured length of the peg gets larger as the angle increases up to

90º. At 90º, the measured length of the peg is the length of the peg at that height. This

gives a sine relationship between the tilt angle and the ratio of the peg lengths. From

Figure 4, the tilt angle is determined by taking the inverse sine of measured peg length

over the true peg length.

(10)

= −

truemeasured1sinα

120

α

TruePeg

Length

Measured PegLength

Camera Lens

Distance from

Cam

era

Figure 4: Peg Length determination of tilt angle

The shape of the peg in the image after edge-detection has a distorted, non-

rectangular shape. This irregular shape makes measuring the length of the peg much

more difficult. To determine the length of the peg in the image, it is necessary to find the

major and minor axes of the peg. To do this, the covariance matrix (K) of the zero-

meaned edge pixels is found using the following formula [ShB88]:

(11)

where A is a 2xN matrix containing the coordinates of each of the N edge pixels. The

major axis of the peg is the eigenvector of K corresponding to smallest eigenvalue

[Str88]. The minor axis is the eigenvector of K corresponding to the largest eigenvalue.

The edge-pixels of the peg are now aligned onto their major and minor axes.

AAN

K T ⋅⋅=1

121

It is expected that the base of the peg will have a convex curve and the tip of the

peg a concave edge. This distortion is due to the round shape of the peg and the fact that

the camera is viewing the peg from above. To get an accurate measure of the pixel length

of the peg, the Euclidean distance between the center points of the tip and base must be

found. The center of the base is found using the point with the largest vector projection

onto the major axis of the peg. To find the center point of the tip, all the edge points are

projected onto the major axis. A 1000 bin histogram of the projections is used to find the

point on the major axis with the most edge-points in a single bin. This is assuming the

concaved edge of the tip is smooth with many pixels having relatively the same

projection value. To get the pixel closest to the major axis, the pixels in the bin are

projected onto the minor axis. The pixel with the smallest projection is declared the

center of the tip. The measured pixel length is found using the height relationship from

distance determination, and the tilt angle is found using Equation 10.

Computer Systems

From the above requirements, it is apparent that the computer system must be able

to do the following:

§ Take and store black-and-white digital images

§ Sample analog signals with an ADC

§ Generate analog signals with a DAC

§ Sample thermocouple signals with a thermocouple ADC

§ Control functions like turning on the power for the ovens and activating the gas valve

§ Failsafe system to recover in case of power outages

122

To accomplish these functions, a PC/104 computer system [PCCon96] was

chosen. To ease implementation, it was decided to split the computer processing

requirements between two separate computers. This tandem operation allows the

individual computers to operate simultaneously while controlling the experiment and

recording the needed data. The first computer is the primary (data collection) system to

control the experiment and collect the analog signal data. The second system is

responsible for collecting all the digital images and storing them to secondary memory.

The data collection computer is composed of the following:

§ Processor board (100 MHz Pentium, 64 MB RAM, 1 GB FlashDisk)

§ 16 relay board

§ 32-channel 16-bit ADC board that also has 12-bit DAC

§ 12-bit 8-channel thermocouple ADC board

§ Timer/digital input/output (I/O) board

§ Power supply board with outputs ±5 VDC and ±12 VDC

The imaging computer consists of the same processor, timer/digital I/O, and power

supply boards, but also has three digital image capture boards.

In order for the operation of the computers to work successfully, they must have a

handshaking process. The handshaking allows them to keep up with each other during

the course of the experiment. As the computers approach a point in the experiment

routine where images must be taken, the computers follow the below pattern to

communicate the need for image collection.

123

§ The data collection computer addresses the tube to be imaged.

§ The data collection computer polls the imaging computer.

§ The imaging computer responds with a ready ACK/NACK which in turn causes the

data collection computer to signal picture capture when ready.

§ The imaging computer begins its imaging routine.

§ The imaging computer checks for completion and exits appropriately.

The next responsibility of the computer systems is to implement a failsafe routine.

As mentioned previously, the failsafe markers are used to indicate how far along the

experiment has gone. The failsafe routine for the data collection computer queries an

ASCII data file to examine/set markers. The failsafe system for the imaging computer is

implemented differently. The imaging computer performs its job only when it is signaled

to do so and on the tube that is addressed. This paradigm of operation does not allow for

a sequential recording of experiment flow. Instead, the failsafe system is implemented as

a data file containing the number of imaging notifications per tube. By examining these

notification numbers, it is possible to determine the number of images taken for each

tube.

IV. Results

This section summarizes the results from tests and experiments conducted to

validate the operation of the RIGEX computer system. The topics to be discussed are

temperature, pressure, vibration measurements, inflation modeling, imaging, computer

integration, and the results from a single tube experiment.

124

Temperature

To test the accuracy of the thermocouple ADC board, a process calibrator was

used to simulate a K-type thermocouple signal. The computer was programmed to take

25 samples at 1 sample/sec for temperature ranges of 0 – 200°C in increments of 10°C.

This produces a stair-step look to a temperature plot. Equation 11 was implemented into

the programming to adjust for circuit board biasing that was observed. The 21°C was

experimentally chosen to adjust the bias to the input temperature.

(12)

A noise problem was discovered that caused a large variance in the data. To

mitigate the variance, a 10-point moving average filter was implemented in the

temperature data collection routine. This filter smoothes the data before the temperatures

are compared to the threshold temperature. Figure 5 shows the effects of the smoothing

filter.

Pressure

To test the pressure transducers and the ability of the computer to sample the data,

positive and negative pressure was applied to the sensor. The data was recorded at 5000

samples/sec. Figure 6 shows the results from the test.

)21( Cboardmeasureddata °−−=

125

Figure 5: Temperature testing results

Figure 6: Pressure sensor test

126

Vibration

As described earlier, a 3.15VAC/230VAC step-up transformer was used to

provide the voltage gain needed to vibrate the tube. The triaxial accelerometer chosen for

the experiment was used. Due to the change in mass between the piezo accelerometer

and the one chosen for the experiment, the modal frequencies lowered. The resulting

transfer function using the up-chirp and the computer sampled data clearly identifies the

1st, 2nd, and 3rd modes of the inflated tube. Figure 7 shows one of the transfer function

results, where the Y axis is the axis along which the actuators are aligned. The modal

frequencies were later validated using a laser vibrometer and exciting the tube with a high

power amplifier.

Figure 7: Transfer function results

Inflation Modeling

The inflation modeling was conducted to compare the accuracy of moving the

accelerometer without rotating the coordinate system and moving the accelerometer

while applying rotation. The first test involved just moving the accelerometer up and

down and changing directions. To apply rotation, the accelerometer was placed on top of

127

a rubber tube (similar to the ones for the experiment) and the tube was inflated. Results

of this testing revealed the tube motion during inflation and provided a high degree of

confidence in the accuracy of the measurement devices.

Imaging

Testing the imaging system involved the image capture/timing test and the

distance/tilt angle measurements. For the image capture rate, the imaging computer was

programmed to take five images and measure the amount of time it took to capture the

images and store them into RAM. The capture time was 7 sec for five images. Each

image is stored at a 486 x 1134 pixel resolution. This odd picture size distorts the image

data shapes. To fix this problem, a simple form of interlacing is applied to each image.

This interlacing involves replicating each row and placing it below the original. This

creates a 972 x 1134 pixel image. After this interlacing, the shapes in the images are

back to their original proportions. The camera can be programmed to do interlacing on

its own, but this involves the camera having to take two images.

To determine the height of the inflated tube, ten images are taken at thirty-six

different evenly spaced distances from the camera lens. The 360 images are then

imported into Matlab® for analysis. For each of the thirty-six distances, the ten images’

major axes are averaged. From the average, a ratio is formed using the known diameter

of the target to the average measured pixel length. Each of the thirty-six data points are

then plotted against their true distance from the lens. Once the data points are plotted, a

least squares line fit is performed. Equation 13 gives the least squares line fit equation to

the thirty-six data points.

128

(13)

The meter value is the diameter of the white circle in meters. The pixel value is the pixel

length measurement of the major axis of the ellipse in the image under test. The 360

images are analyzed individually and their distances averaged with the deviation statistics

also determined. The maximum deviation was approximately 1.5 mm and an average

deviation of about 0.1 mm. Figure 8 displays the results from the validation set of images

compared to the true distance from the camera lens. The average deviation of the nine

images is 0.922 mm.

Figure 8: Tube distance measurement validation test results

0241.04.2156 −⋅=pixelmeter

d

129

For angle measurements, the procedure outlined above was implemented with 150

test images. Ten images were taken at fifteen different angles for a total of 150 test

images. The tested tilt angles ranged between 0.14º to 15.4º at a distance of 9.5 cm. The

average error from the true angle was determined to be 0.916º. To validate the process,

seven images were taken at angles not measured before in the first test. The seven

images were processed according to the defined algorithm and resulted in an average

error of 0.09º. Figure 9 shows the validation results.

Figure 9: Tube angle measurement validation results

Computer Integration

Once the operation of both computer systems was validated, system integration

and testing followed. These tests began with system communications handshaking

followed by single tube inflation, and failsafe mode testing.

130

Single Tube Experiment

This experiment performed the entire RIGEX process on one tube. The tube was

heated to 130°C to insure it met its transition temperature. It was inflated and excited

upside down so as to not work against gravity. It is understood that inflation upside gives

the tube an unfair advantage, but in space no gravity will be present. Unfortunately, there

is no happy medium for ground testing. The results below show the temperature plot and

pressure.

Figure 10: Single tube heating

131

Figure 11: Inflation pressure

The inflation modeling was accurate since the tube was inflating after falling from

inside the oven. Figure 12 shows the 3D inflation model with the diamond marking the

start point.

V. Conclusion

Inflatable structures offer a cost saving opportunity since the structure is

significantly lighter and has a reduced storage volume. This allows for smaller launch

vehicles and/or increased performance capabilities. The use of these inflatable structures

will allow for larger antennas providing increased capacity for communications, radar,

and electronic warfare. The structures will also allow for larger solar sails for the

collection of more solar energy for transmitter power.

132

Figure 12: Inflation model

This paper has presented the results of on-going research at the Air Force Institute

of Technology into the design, development, and deployment of inflatable structure for

space-based applications. This work specifically addresses the systems engineering

involved in developing the computing and sensor systems for the Rigidized Inflatable

Get-Away-Special Experiment (RIGEX). RIGEX is a novel and first-of-its-kind

approach to lay the foundation for future inflatable structures research. This pioneering

effort has been selected by the DoD for flight testing on-board the U.S. Space Shuttle.

133

VI. References

[FrBi92] Freeland, R. E. and G. Bilyeu. “In-Step Inflatable Antenna Experiment,”

International Astronautical Federation IAF 92-0301. 1992. [Fr97] Freeland, R.E. ”Large Inflatable Deployable Antenna Flight Experiment

Results,” 48th International Astronautical Federation Congress. 1997. [Hu01] Huang, John. “The Development of Inflatable Array Antennas,” IEEE

Antennas and Propagation Magazine, Vol. 43, No. 4. August 2001. [Lim90] Lim, Jae S. Two-Dimensional Signal and Image Processing. 1990. [Ma03] Matlab®. regionprops Help File. No date listed [MI99] MINCO Products Inc. 250ºC All-Polymide Thermofoil® Heaters. Jan.

1999. [PCCon96] PC/104 Consortium. PC/104 Specifications, Version 2.3. June 1996 [PrB03] Pratt, Timothy, Bostian, Charles W. Satellite Communications 2nd Ed.

2003. [ShB88] Shanmugan, K. Sam and Breipohl, A. M. Random Signals, Detection,

Estimation and Data Analysis. 1988. [SiT02] Single, Thomas G. Experimental Vibration Analysis of Inflatable Beams

for AFIT Space Shuttle Experiment. Master’s Thesis, Air Force Institute of Technology, Dayton, OH, March 2002.

[Skl00] Sklar, Bernard. Digital Communications, Fundamentals and Applications

2nd Edition. 2000. [Str88] Strang, Gilbert. Linear Algebra and its Applications 3rd Ed. 1988.

134

A COMPARISON BETWEEN THE SPECTRAL FIDELITY OF THREE DIGITAL RADIO FREQUENCY MEMORY

ARCHITECTURES

T. W. Küsel, M. R. Inggs and J. E. Pienaar

P.O. Box 395, CSIR, Pretoria 0001 (SOUTH AFRICA) Tel No. +2712-841-3493 Fax No. +2712-841-4015

e-mail: tkusel@csir.co.za

ABSTRACT

Comparisons are made with regard to spectral impurities in the output spectrum introduced by the sampling and coarse quantisation of Digital Radio Frequency Memory (DRFM) systems. The spectral fidelity of the Amplitude Encoded, Quadrature Encoded, and Phase Encoded DRFM architectures are considered. Existing techniques of reducing these spectral impurities are investigated and their effectiveness evaluated. Other sources of spectral impurities in DRFM’s are briefly discussed.

Key words: DRFM, Radar evaluation

INTRODUCTION

Recent advances in digital technology resulted in the advent of high-density, very large-scale and very high-speed integrated circuit devices. For the first time it has become viable to store and process sufficient lengths of broadband signals digitally. The digital capturing, storage and processing of a signal has some obvious advantages over traditional analogue methods. Given such a digital receiver technology, the system designer has at his disposal complete, real-time, receiver and processor reconfigurability.

The Digital Radio Frequency Memory (DRFM) [1], is a specialised hardware that is able to store very broad bandwidth signals digitally and recall them at any desired time. It is commonly employed in Electronic Counter-Measures (ECM) systems and in radar test and evaluation equipment, where it is used to store and re-transmit radar signals coherently [2].

The fidelity of the recalled signal depends on numerous system factors, the most significant of which are discussed in this paper. After storage, the digital data may be manipulated as desired. One of the great advantages is that complex, phase encoded waveforms may be handled with ease, making it useful for the analysis and synthesis of such signals. By applying the appropriate digital transforms, filter functions can also be created that have no simple analogue equivalent for handling special receiver and environment interaction.

Several DRFM architectures have been proposed [3]-[6] using amplitude and/or phase encoding techniques. These encoding techniques can be reduced to three basic configurations: the Amplitude Encoding (AE), Quadrature Encoding (QE) and Phase Encoding (PE) architectures. The fidelity of the repeated signal (i.e. the 'transparency' of the repeater) can be defined in terms of several properties of the repeated waveform [7]. One of the most important measures of transparency is the level of spectral impurities, or spurious signal level of the output signal.

According to the Nyquist sampling theorem, a minimum sampling rate of 2B Hz is required for the sampling of a signal with a B Hz bandwidth. For broadband input signals, this requires high speed analogue-to-digital (A/D) conversion, digital-to analogue (D/A) conversion and digital data processing. The instantaneous bandwidth of the DRFM is therefore usually limited by the A/D conversion speed. Flash A/D converters are commonly used to obtain a fast conversion speed.

135

The hardware complexity of flash converters increases rapidly with an increase in the number of bits used. As a result, coarse quantisation is usually used in wideband DRFM architectures.

While reducing A/D conversion complexity, signal fidelity is lost due to the coarse quantisation. The coarse quantisation, together with the sampling, induces unwanted spurious signals, which cause the recalled signal to be of poorer quality than the incoming signal.

Several techniques of reducing spurious signal levels in DRFMs have been proposed [8]-[12]. The effectiveness of these methods can be evaluated by comparing the output spectra of DRFMs using these techniques to the output spectra of conventional DRFM architectures.

Simulations were used to compare the output spectra, spurious signal levels and other aspects of signal fidelity. Although theoretical methods exist for the calculation of output Power Density Spectrum (PDS) and spurious signal levels ( [13] for the AE architecture), these theoretical methods were not used because they restrict the analysis in terms of investigating various types of input signal waveforms, DRFM architectures, sprious reduction techniques and component imperfections.

For the purpose of simulation verification, the simulation results were compared to theoretical results on a sample basis. It was found that the simulations produce virtually identical results to theoretical calculations [7]. The simulation results were also validated by comparing them to data measured on operational DRFM systems on a sample basis, also yielding highly correlated results.

In this paper the three basic DRFM architectures for the digital capture, storage and reconstruction of broadband signals are investigated. A comparison of these architectures is made, based on the spectral impurities (caused by sampling and coarse quantisation) of the reconstructed signal. Techniques for improving the quality of the reconstructed signal are evaluated. Other sources of spectral impurities in DRFMs are briefly discussed.

The results from the analysis are intended to help the users of DRFM systems to understand the causes and magnitude of spectral impurities in DRFM’s.

DRFM ARCHITECTURES

Simplified block diagrams of the three proposed DRFM architectures are illustrated in Fig. 1. The corresponding signal space diagrams for the special case of 4-bit quantisation are shown in Fig. 2, illustrating how each architecture quantises and digitally encodes the amplitude/phase information of the signal in terms of the in-phase (I) and quadrature (Q) channels. It can be seen that the number of quantisation levels of the DRFMs were chosen so that all architectures exhibit the same number of quantisation points on the signal space diagram. The architectures are described in more detail below.

Amplitude Encoding

Fig. 1 (a). shows a simplified block diagram of the AE architecture [6]. It consists of a down-conversion stage, an intermediate frequency (IF) stage and an up-conversion stage. The incoming signal is demodulated to IF by a mixer and local oscillator. The IF signal is then filtered by a low pass filter to prevent aliasing in the sampling process. It is then sampled and quantised by a conventional amplitude encoding A/D. After digitisation, the data is sometimes de-multiplexed or pipelined (not shown in the block diagram) in order to reduce data rates, thus making it more compatible with the operating speed of the memory module. The data is then stored in random access memory (RAM), which is usually managed by a memory controller to regulate storage space, addresses, and possibly changes to the stored data. In the recalling process, the data is multiplexed into the D/A and filtered to eliminate sampling images. The IF signal is then modulated by the mixer and local oscillator. After the first up-conversion stage, the lower sideband is filtered out by a band pass filter.

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LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

BANDPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

LOWPASSFILTER

A/D

t

t

t

t

t

ω

ω

ω

ω

ω

ο

ο

ο

ο

ο

A/D

A/D

COS( )

COS( )

COS( )

SIN( )

SIN( )

MEMORY

MEMORY

MEMORY

PH

AS

E

PH

AS

E

MEMORY

D/A

D/A +

+

+

+

-

-

Σ

Σ

D/A

A/D

D/A

(a)

(b)

(c)

Fig. 1: DRFM Architectures. (a) Amplitude Encoded -AE , (b) Quadrature Encoded - QE , (c) Phase Encoded - PE

Q

(a) (b) (c)

II I

Q Q

Fig. 2: Signal space diagrams. (a) AE (b) QE (c) PE

Quadrature Encoding

In the QE configuration (Fig. 1 (b)) [5], the band pass filter is not needed, since it is not necessary to filter out the sideband. This is done by adding or subtracting the quadrature channels in order to cancel the upper or lower sidebands. The incoming signal is split by a power divider and quadrature mixer into an I and Q channel. Since the sideband is suppressed by the mixing process, the input signal is down-converted to baseband rather than IF. The operation of the baseband stages of the two QE channels are identical to that of the IF stage of the AE configuration, as described earlier. After quadrature up-conversion, the sum/difference of the I/Q channels yields the desired sideband. The cancellation of sidebands by quadrature up- and down-conversion provides the QE configuration with a factor two bandwidth advantage over the AE configuration (i.e. the information bandwidth of the QE architecture is equal to the sampling frequency, fs ).

137

Phase Encoding

The mixer up- and down-conversion stages of the PE (Fig. 1 (c)) and QE configurations are identical, and again the input signal is down-converted to baseband. The ‘phase A/D’ converter then compares the relative amplitudes of the I and Q channels to encode the phase of the incoming signal digitally. Amplitude information is discarded. In the recalling process, the ‘phase D/A’ converts the digitally encoded phase into two quadrature amplitude signals. These two amplitude signals are then added to eliminate one of the sidebands. As for the QE, the PE architecture has an information bandwidth equal to the sampling frequency fs.

SIGNAL FIDELITY: SINGLE-TONE INPUT SIGNAL

The special case of a single-tone input is considered important, as it best illustrates the magnitude and location of the spurious signals (created by sampling and coarse quantisation) in the output spectrum.

Sampling a quantised signal

Quantisation is a highly non-linear process, which creates harmonics at odd multiples of the input signal. Quantisation thus effectively increases the bandwidth of the input signal by adding harmonics at higher frequencies. When this quantised signal is now sampled (at a rate which would have been sufficient for the original input signal), spectral aliasing occurs, resulting in a fold-back of harmonics into the baseband. For the single-tone input, this means that a large number of spectral lines fill the information band of the sampled-and-quantised signal. An example of the output PDS of the AE architecture using 4-bit quantisation, and with a single-tone input (in this case f = 0.139 × fs) is illustrated in Fig. 3.

Maximum spurious signal level vs. input frequency

From Fig. 3, it can be seen that unwanted spurs appear over the entire information band. The magnitude and position (frequency) of these spurs depend on the relative input- and sampling frequencies. As explained in the previous section, harmonics from the replicated spectra fold back into the baseband after sampling and quantisation. With a change in input frequency, the position of these harmonics change in relation to each other, causing harmonics to overlap and add/subtract (depending on their relative phases) at different input frequencies. The maximum spurious signal level will therefore not be constant as a function of input frequency.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−70

−60

−50

−40

−30

−20

−10

0

frequency (f/fs)

Pow

er d

ensi

ty (

dBc)

Fig. 3: Output spectrum of a 4-bit AE DRFM for single-tone input

Since the maximum spurious signal level, Spmax, is a very important parameter, it is desirable to know how this level changes as a function of the input single-tone frequency, fin.

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Simulation results showing this relationship are illustrated in Fig. 4 for all three DRFM architectures and for different quantisation coarseness.

The results show how the highest spurious signals in the band add/subtract with overlapping spurious signals, resulting in a different value of highest spurious signal for each input frequency. Table 1 summarises the results by giving the mean and maximum values of Spmax, as defined below:

( ) ( )(Mn S E Sp fin= max ) ....................(1)

where E(x) denotes the expected value of x.

( ) ( )(Mx S Sp fin= max max ) ....................(2)

−0.5 0 0.5−70

−60

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4−bit

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max

(dB

)

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−0.5 0 0.5

−60

−40

−20

0

2−bit

4−bit

6−bit

8−bit

f_in ((f−fo)/fs)

Sp_

max

(dB

)

(c)

Fig. 4: Highest spurious signal vs. Single-tone input frequency: (a) AE, (b) QE, (c) PE

Output spurious signal levels (dB) AE QE PE

quanti-sation

Mn(S)

Mx(S)

Mn(S)

Mx(S)

Mn(S)

Mx(S)

1-bit -9.6 -5.8

2-bit -19.3 -11.4 -9.7 -5.8 -9.7 -5.8

3-bit -26.9 -19.0 -17.3 -13.4

4-bit -34.2 -24.8 -18.3 -11.4 -23.6 -18.7

5-bit -41.9 -28.8 -29.8 -24.5

6-bit -48.3 -38.0 -26.4 -18.3 -35.8 -29.9

7-bit -55.4 -42.5 -41.7 -36.5

8-bit -61.7 -47.2 -34.4 -22.2 -47.7 -42.0

Table 1. Output spurious signal levels for single-tone input

Table 1 shows that the AE architecture produces the lowest spurious signal levels. The PE architecture produces slightly higher spurious signals, but shows a smaller fluctuation in spurious

139

signal levels as a function of input frequency. The QE architecture produces high levels of spurious signals, which are comparable to those of an AE architecture using half the number of quantisation bits. In other words, the spurious level of a 4-bit AE architecture is equivalent to that of a 8-bit QE architecture.

SIGNAL FIDELITY: TWO-TONE OR WIDEBAND INPUT SIGNAL

In order to see the effect of the different DRFM architectures on amplitude and frequency modulated signals, the output spectra for a two-tone input signal and an amplitude weighted, linear FM-chirp input signal were calculated. The simulation results for 4-bit quantisation are shown in Fig. 5 and Fig. 6 respectively.

It can be noted that the AE architecture shows superior results in terms of spectral purity of the reconstructed signal, followed by the QE architecture. As can be expected, the PE architecture produces higher levels of spurious signals because it fails to reproduce amplitude information. From Fig. 5 (c), it can be seen that the PE architecture can induce high levels of spurious signals in the output if the input signal is amplitude modulated.

−0.5 0 0.5−40

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Mag

nitu

de (

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(d)

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nitu

de (

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(c)

Fig. 5: Input/Output spectra for two-tone input: (a) Input, (b) Output:AE, (c) Output:PE, (d) Output:QE

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−30

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Mag

nitu

de (

dB)

(d)

Fig. 6: Input/Output spectra for amplitude weighted FM-chirp input: (a) Input, (b) Output:AE, (c) Output:PE, (d) Output:QE

SPURIOUS SIGNAL REDUCTION TECHNIQUES

A number of methods for reducing spurious signal levels in the output spectrum of DRFMs have been proposed. Two basic techniques are discussed here:

• Local Oscillator Modulation (LOM) [9]-[12] : This technique uses random modulation of the Local Oscillator (LO) to decorrelate output spurious signals and sideband levels and can be used in conjunction with any one of the three DRFM architectures.

• Random Linearisation (RL) [8] : This technique adds an identical random waveform to the I and Q channels of the QE or PE architectures to reduce output spurious signal levels.

Wiegand [9] claims that modulation of the local oscillator (in this case of the AE architecture) by a random waveform could substantially lower the output spurious signal level. In addition, it is claimed that this technique decorrelates the sideband level of the AE architecture, thereby transforming the AE architecture into a single-sideband architecture and thus increasing the bandwidth of the system by a factor of two.

A number of simulations were done [7] to assess the extent of spurious signal reduction capabilities of this LOM and other techniques. A simplified block diagram of the LOM technique as applied to the AE architecture is shown in Fig. 7.

By choosing a suitable modulator, the output PDS can be calculated for different random waveforms until the best results are obtained [7]. For a given input signal, the spurious signal reduction capability of the LOM and RL techniques depend on a number of system factors:

• Type of modulation (phase/frequency modulation)

• Modulation waveform (usually band-limited noise)

• Modulation amplitude and bandwidth

• DRFM architecture

• Recorded pulse width

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LOWPASSFILTER

PHASE /FREQUENCYMODULATOR

RANDOMWAVEFORMGENERATOR

LOWPASSFILTER

BANDPASSFILTER

A/D

LO

MEMORY D/A

Fig. 7: Block diagram of LOM technique

The systems were simulated with both a single-tone and a wideband FM chirp as input. Simulations using a number of different modulation waveforms for both the LOM and RL techniques, were concluded with similar results.

Figure 8 illustrates typical results which were obtained from such simulations. The results show the output spectra of both the normal 1-bit AE architecture and the 1-bit AE architecture with LOM, for both inputs. Phase modulation was used with a band-limited, Gaussian distributed white noise as modulation waveform.

-0.5 0 0.5-30

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Mag

nitu

de (

dB)

Linear FM chirp: AE with modulated LO

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Linear FM chirp: Normal AE

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Single frequency input: Normal AE

-0.5 0 0.5-30

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nitu

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dB)

Single frequency: AE with modulated LO

Fig. 8: Output spectra: conventional vs. LOM techniques. Single-tone and FM chirp input.

As can be seen from Fig. 8, it is evident that these techniques have significant advantages, in terms of output spurious signal levels (and bandwidth for the AE architecture), for the capturing of periodic data such as a single-tone signal. In this case the spurious signal level is reduced from -8dB to -15dB, and the image from 0dB to -15dB. In such cases, a suitable modulation waveform has to be found, which is dependent on a number of system factors.

From Fig. 8 it is also clear that this technique has no noticeable benefits for wideband signals such as a FM chirp. Complexity is added to the system without significantly improving the output signal fidelity.

142

OTHER SOURCES OF SPECTRAL IMPURITIES IN DRFMs

I/Q imbalance

Like other single sideband systems, the QE and PE systems remove the unwanted sideband by a “phasing out” process. The output of the I and Q channels are added or subtracted to produce a single sideband output signal. The requirements for the relative I/Q phase difference of 90° can become stringent for wideband systems. For example, the relative timing precision between the I digital memory and Q digital memory must be equivalent to a value of the order of two degrees (2°) or less to maintain a sideband suppression of better than 35dB [6]. This means that if the I/Q memories are operating at a clock rate of fs Hz, a relative timing difference of 2/(360 × fs) seconds is required to maintain a sideband suppression of better than 35dB. This becomes a very stringent requirement for high speed systems [7].

Analogue component non-linearities

Non-linearities in the amplitude transfer function of any analogue devices in the signal path (e.g. amplifier compression) create harmonics similar to the distortions caused by quantisation.

Dynamic Range

For input signals with amplitude sufficiently large to saturate the A/D converters, clipping occurs. This clipping causes the output spurious signal levels to increase substantially [7]. For input signals with amplitude too small to utilise the full range of quantisation levels, output spurious signal levels also increase, since less quantisation levels are used in the capturing and reconstruction of the signal [7]. In general the PE architecture is much less sensitive to this problem, since all amplitude information is discarded in the encoding process.

Sampling Images

Another source of spurious signals in practical implementations is the non-ideal filtering of the information band in order to eliminate the sampling images and sidebands. Low-pass and band-pass filters are used to filter the reconstructed signal (see Fig. 1) to eliminate the images created by sampling, as well the unwanted sideband in the AE architecture. As the image/sideband approach the edge of the information band, they can take on significantly high levels [7].

Mixer imperfections

Mixer LO leakage and mixer inter-modulation products can induce significant levels of spurious signals in the output spectrum of DRFM modules [7].

Interlaced Sampling

As mentioned earlier, the instantaneous bandwidth of wideband systems is usually limited by the state-of-the-art technology in terms of the sampling speed of the A/D conversion and digital processing speed. One way to overcome this problem is by using interlaced sampling to reduce the sampling- and data speeds. The incoming signal is divided into a number of channels (N). Each channel has a separate A/D and is clocked in turn by a time-delayed version of a low duty-cycle clock. The time delays induced in the different channels are multiples of Ts/N where Ts is the clock period. Each A/D therefore reads only one sample every Ts seconds. The system as a whole, however, reads a sample every Ts/N seconds, thus increasing the Nyquist bandwidth from 1/2Ts to N/2Ts . The data processing is done in parallel, therefore decreasing the data speeds to 1/N times the normal rate. A theoretical analysis was done [7] on spectral impurities induced by timing errors in the read-in and read-out process of a system using interlaced sampling. This analysis showed that very accurate timing is required to prevent spurious signals or 'interlaced sampling images' in the output spectrum of such a system. It was also found [7] that the timing accuracy requirements increase with an increase in the number of channels (N) used.

143

CONCLUSIONS

Three DRFM architectures have been presented, and it has been shown that the process of sampling and coarse quantisation induces unwanted components in the spectrum of the output signal of such DRFM systems. The spurious signal levels in the output PDS have been quantified for these architectures and for a number of quantisation levels.

It was shown that the AE repeater shows generally better output signal fidelity, but that the QE and PE architectures have a factor two bandwidth advantage. The PE architecture shows better spurious signal suppression than the QE architecture for a single-tone input, but induces severe levels of spurious signal and other spectral deformations for amplitude modulated input signals.

Existing techniques of reducing spurious signal levels in the output spectrum have significant advantages in terms of spurious signal levels, for the capturing of periodic data such as a single-tone signal. In such cases, a suitable modulation waveform has to be found, which is dependent on a number of system factors. It was also found that these techniques have no meaningful benefits for wideband input signals.

A number of other sources of spurious signals exist in DRFM systems, and such sources of spurious have been briefly discussed.

REFERENCES

1. J.L. Dautremont, "Digital storage system for high frequency signals," US Patent No. 3,947,827, Mar. 1976.

2. (Author not given), "Digital RF Memories," Journal of Electronic Defence Supplement, Jan. 1994.

3. T.T. Vu et al., "A GaAs phase digitising and summing system for microwave signal storage," IEEE J. Solid-State Circuits, vol. 24, no. 1, pp. 104-117, Feb. 1989.

4. G. Webber et al., "DRFM requirements demand innovative technology," Microwave Journal, pp. 91-93,98,102,104, Feb. 1986.

5. W.J. Schneider, "Digital countermeasures memories: new techniques possible," The International Countermeasures Handbook, pp. 367-373, 1986.

6. E. Koos, "Digital RF memories enter second decade," Journal of Electronic Defence, pp. 49-51, Aug. 1985.

7. T.W. Küsel, "On the waveform fidelity of broadband digital storage architectures", Dissertation submitted to the Faculty of Engineering, Univ. Of Cape Town, Mar. 1995.

8. W.N. Barnes et al., "A technique for reduction of spurious signals in a DRFM system," NAECON, pp 483-491, 1985.

9. R.J. Wiegand, “Modulated digital radio frequency memory,” US Patent no. 4,713,662, Dec. 1987.

10. R.J. Wiegand, “Multibit decorrelated spur DRFM,” US Patent no. 4,885,587, Dec. 1989.

11. R.J. Wiegand, “Spur reduction system for DRFM,” US Patent no. 4,933,677, June 1990.

12. R.J. Wiegand, “Modulated single channel digital radio frequency memory,” US Patent no. 4,891,646, Jan. 1990.

13. S.D. Berger et al., “An expression for the frequency spectrum of a digital radio frequency memory signal,” NAECON, pp90-93, 1990.

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Expendables and Ship Protection – Asset Coordination is Critical By Dr Arthur G Self, B, Sc, P.Eng

1.0 Introduction

Recent years have seen a number of remotely piloted vehicle (RPV) and expendable EW system programs come to fruition. RPVs for surveillance and other missions have been of the preprogrammed type as well as remotely piloted and tethered. Recent shipborne systems such as with NULKA, towed decoys and seduction decoys such as the DLH/SIREN system for the Royal Navy have confirmed their places in today’s naval platform systems as cost effective and necessary adjuncts to sophisticated onboard systems. Additionally, navies are moving increasingly towards fully integrated sensor/weapon suites (such as AIEWS, HORIZON, and other programs), where integrated assets (sensors; weapons) are a principal feature.

Littoral environments create new hazards for naval forces as the constituents can be from several Nations’ resources as well as the fact of their close physical proximity (as compared to the more traditional Blue water deep ocean scenarios where forces are separated by over the radio horizon distances). Thus, in the littoral, platform self-protection is still fundamental but also being aware of one’s neighbors is becoming increasingly important (e.g. Atlantic Conveyor incident in the Falklands). The US Navy has embarked on a very ambitious program called the Cooperative Engagement Capability (CEC) as the first known program addressing multi-platform sensors and weapons protection.

Thus, part of today’s complement of sensors/weapons for shipborne uses is the expendable countermeasure (active and passive). In this paper, we consider typical expendable EJ roles (non-comms and IR) in a range of scenarios together with onboard assets. Performance characteristics are reviewed and current expendables and integrated EW programs examined. 2.0 Expendable Jammers (EJ)

For the purposes of this paper, expendable will be assumed to include concepts of low intrinsic value and thus are able to be thrown away after usage. Expendable concepts could be both tethered and free-mounted. In reality, expendable usually means recoverable in peacetime operations and throwaway during wartime (ref 1). In this paper, we restrict ourselves to communications (1Mhz-1Ghz), radar (1Ghz-100Ghz) and IR bands using delivery means such as hand-emplaced, RPV/UAV and launched (rocket, mortar, shell). 2.1 Roles The principal roles for EJs in the shipborne context can be categorized as follows:-

- platform self-protection - area protection

However, the use of EJs has historically raised a number of key issues such as: how would they be used?; how would they be delivered?; how would they be controlled?; and, what about mutual interference?. Examining likely ship and missile engagement scenarios, EJs offer promise in a number of engagement phases, including:-

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- distraction/confusion: tactics are directed against the launch platform/search platform

- dilution: tactics are directed during the missile target acquisition phase - seduction: tactics are directed during the missile tracking phase

Tables 1 and 2 summarize the advantages and disadvantages of EJs in ship self-protection and area protection roles, respectively [Ref .3]. 2.2 Applications Table 3 summarizes the principal applications for EJs in naval scenarios. 2.3 Performance requirements In naval scenarios, principally radar and IR-related EJs are the most common and relevant; communications EJs are not in use due to the ship’s dependence on communications, the need to remain silent etc (comms EJs have most applicability in Army scenarios). Table 4 summarizes the results of some radar range calculations for a typical ASM threat to a ship and the required EJ ERPs in order to provide protection for the ship. In this table, an ASM ERP of 85dBw is chosen and the ship’s radar cross section (RCS) is varied from large (e.g. 10,000 sq. m) down to a very stealthy platform (of say less than 1,000 sq. m RCS). Given a defined jammer-to-radar signal (J/S) level, then the required EJ ERP can be derived for a given range (ASM to ship). It is clear that the EJ ERP has to be in excess of a few kilowatts – also, stealth works in favor of the EJ. Typical EJ performance in a seduction role can be listed as follows:- ERP Several 10’s KWs

Frequency bands I/J bands Deployment means Rocket/parachute; RPV; Hovering platform Deployment time within 10 secs Recoverable? Possibly

EJs can be deployed via a range of different delivery means. Table 5 summarizes some of the advantages and disadvantages for such ECMs. In Table 6, we show some of the deployment and ship interface issues for such an offboard active EJ system. 3.0 Some current EJ products and programs Table 7.1 and 7.2 show some current shipborne radar and IR-related EJs either in service or coming into service shortly. SIREN and NULKA are the two newest active offboard EJs; summarized below are some further details on these two new systems:-

(a) GEC-MDS DLH decoy - for UK Royal Navy - 28 Kg, 130x1700mm storable munition compatible with Sea Gnat launcher - I/J band radar ECM payload

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- targeting information via a control processor prior to launch - reaction time to deploy is 10 sec - up to 180 sec active life - export version is called SIREN - SIREN being considered for French Navy also.

(b) BAe Australia’s NULKA hovering rocket - 2m long rocket - preprogrammed hovering - Sippican-developed ECM and electronics payload - Phase I contract in 1994 for fire control system development and installation on-board one Adelaide class frigate in the Australian Navy - Phases II/III contracts awarded in 1996 with first system delivery for 1998 - US Navy awarded AWADI $24.9M contract for engineering development and sea trials

4.0 Some current integrated EW programs Listed below are some key programs for shipborne applications that are based on integrated assets, both onboard and offboard. 4.1 Advanced Integrated EW System (AIEWS), SLY-2V - “shipboard system for the future” for the 21st century US Navy combatant - requirements for : situational awareness; support to the weapons systems; countering of ASM; counter targeting - Phase 1- new computer architecture and console upgrade to SLQ-32 (Raytheon contract)- NULKA decoy system introduction - Phase 2- inject a new sensing capability with better RF front ends

- Phase 3- enhance softkill firepower with IR and advanced onboard ECM - ECM Transmitter ATD (Westinghouse) with H/I/J bands

4.2 SYQ-17 Rapid Anti-Ship Missile Defense System (RAIDS) - initiated in light of ASM strike on USS Stark in Persian Gulf - an automated tactical decision aid for missile defense applications for the US Navy - a near-term, low cost situation aid (RAIDS 1) to - coordinate ship sensor information - provide threat identification and evaluation - assess the ship’s defensive readiness - recommend optimized defensive tactics - RAIDS will prioritize up to 6 threats on multiple bearings and suggest a maneuver and CM response. A “passive” aid (i.e. makes recommendations only) - RAIDS 2 pre-planned product improvement phase (integrated Sea Sparrow, 5” guns and EW) 4.3 US Navy’s Ship Self Defense System (SSDS)

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- integrates existing weapons and sensors within an open, distributed architecture; fuses track data from multiple sensors

- SSDS effectively forms Phase 2 of RAIDS integrating sensors and automating hard and soft kill systems. - SSDS supports six distinct functions:-

- control and integration of sensors/weapons for sufficient protection from ASM threat

- special sensor processing capabilities under stressing conditions - adequate reaction time for auto/semi-auto operations including multi sensor integration

- coordination of HK and SK weapons - resistance to CM and environmental effects - high operational availability 4.4 US Cooperative Engagement Capability (CEC) Program This is part of the USN Integrated Ship Defense System (ISDS) program (an umbrella program aimed at the development of a system to integrate existing stand-alone systems (which do not provide the complete detect-control-engage capabilities) into one that can control and integrate all sensors and weapons (ref. 2). - ISDS will comprise:

- Ship Self Defense System (SSDS) (local area fibre optic network for all ship’s sensors and weapons systems)

- Evolved Sea Sparrow Missile - CEC - MFR - AIEWS - AN/SPQ-9B new horizon search radar - UPX-30 next generation IFF - IRST - PESM - NULKA 5.0 Conclusions Offboard EJs have a number of clear and important roles to play in today’s’ naval scenarios. In conjunction with onboard countermeasures, EJs will enhance ship survivability in today’s’ and future high density, multiple missile attack scenarios. The principal role for active offboard EJs is in the ASM seduction role as part of a layered defence concept with onboard ECM, hard-kill weapons, and chaff and IR flares. Additionally, EJ costs are typically << 1% of the platform costs; indeed, EJs are only a few % of the costs of onboard ECM systems. Chaff and IR EJ costs are significantly lower than for active RF EJs. Finally, the inexorable push for lower platform detectabilities through such concepts as stealth work in favor of the EJ. It is very clear that expendables are here to stay as an important shipborne protection measure.

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References 1. "Elements of EW Expendables" by D. Herskovitz, Journal of Electronic Defense, December 1993 2. “Cooperative engagement capability leading the information warfare revolution” by M. O’Driscoll, IMDEX ‘95, 28-31 March 1995, London, UK 3. ‘Ship Survivability and EW’ Technical Seminar by Dr A G Self Advantages Disadvantages Large, Onboard ECM

High power (for extended range effectiveness). Sophisticated ECM techniques generation.

Limited effectiveness in distraction role due to (range )^-2 losses. Beacon for HOJ and enemy EW. EMC (own/friendly emitters; ESM). Optimum efficiency requires ESM. Trained personnel and expensive maintenance. Ability to cope with large number of simultaneous threats.

Small, Expendable ECM

Low ERP. Inexpensive. Increases scope for Layered Defence concepts. Range and angle jamming accomplished by actual displacement of the EJ relative to the ship.

System integration and operational implications.

Table 1 Self Protection role: advantages and disadvantages of EJ vs. onboard ECM [3] Advantages Disadvantages Large ECM Systems

Necessary ERP for range needs. Multi-target handling capability.

Situation Assessment. Optimum response. Impractical in dense scenarios?. Terrain effects (e.g. littoral scenarios).

Small, Expendable ECM systems

Can be used to seed large areas.

Moving threats and moving own forces make re-seeding a major task.

Table 2 Area Protection role: advantages and disadvantages of EJ vs. onboard ECM [3]

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Table 3 Summary of EJ roles for shipborne protection [3]

Table 4 Some calculations on required EJ ERP (dBw) [3]

• Long range– position EJ 15-20km in front of ship (long endurance RPV?)– confusion and anti-targeting– responsive, non-coherent modes– confusion in depth (in concert with onboard ECM)

• Near Range– confusion and anti-targeting– counter acquisition of ships by missile seekers– 130mm or long endurance platform

• Seduction– near ship deployment against active seekers– in concert with onboard ECM, chaff, etc– 1:1 most likely

• Anti- ARM– near ship deployment most likely– competes with own radar SL– emcon policy– problem of identifying ARM– cheap, compact, wide band, high power decoy

Naval EJs in a layered defense concept

ThreatRangetoownship

Naval Decoy effectiveness

Range (km)

5 8 10 15 20

RadarERP (dBw)

J/S (dB)

RCS (m )2

8585858585858585

3333361010

50,00010,0005,0001,0008001,0001,00010,000

5043403332364050

4639362928323646

4437342726303444

4033302322263040

3831282120242838

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Table 5 Advantages/Disadvantages [3]

• chaff decoys Used extensivelyLimitations in the face of improved missile seekers

• floating reflectors CheapDifficult to deploy quickly enoughWave obscuration effects

• floating active decoys Difficult to deploy quickly enoughWave obscuration effectsCurrent products are 360 deg repeaters

• towed decoys Can be towed or remotely controlledDo not provide 360 deg coverageReplenishmentBoats are large if ocean usage (Blue water)

Naval Decoys - some advantages/disadvantages

• UAV-borne decoys Launch/control/recovery (poss.) issuesLikely not 360 deg instantaneous coverageRelatively slow movementActivation time

• hovering rockets Precise positioning capabilityPhysical “walk off”Expensive?

• parachute borne Standard chaff launcherLimited lifetime

Naval Decoys - some advantages/disadvantages (cont’d)

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Table 6 Some EJ deployment and ship interface issues [3]

Name Description Type of CM Platform Manufacturer Country LAD (Leurre Actif Decale)

ASM seduction decoy

Active RF Shipborne Dassault/GEC France/UK

Dagaie CM Decoy

IR or Chaff projectiles

Passive RF Shipborne CSEE – Defense

France

Sagaie Decoy IR + chaff Passive EW Shipborne CSEE – Defense

France

Rafael Chaff Rocket

Chaff Passive EW Shipborne Rafael Israel

Sclar Chaff/IR + flare rocket system

Passive/Active Shipborne Breda Italy

Saab EWS-900E

Automated chaff and IR decoy

Passive/Active Shipborne Saab Missiles AB

Sweden

CARMEN ASM seduction decoy

Active RF Shipborne Thorn – EMI UK

SIREN ASM decoy Active RF Shipborne GEC – Marconi

UK

Alex Decoy System

Launches chaff/IR

Passive/Active Shipborne Loral USA

MK 36 SRBOC

Chaff/IR Passive/Active Shipborne Lundy USA

Active Electronic Decoy (AED)

Radar decoy Active RF Shipborne Litton ATD USA

AN/SSQ – 95 (V)

Radar decoy Active RF Ship or Aircraft

Litton ATD USA

LURES Radar decoy Active RF Shipborne Litton ATD USA

Naval Decoy effectiveness

Offboard Decoy Needs

- threat information: Relies on ship ESM to measure RF, PRI, PWData relayed to decoy launcherDecoy has onboard memory

- threat bearing: Ship ESM measures threat bearing wrt NData relayed to decoy launcherDecoy has onboard memory

- ability to measure bearing: Decoy needs an N reference and a DF capabilityDF measurement likely needed on a timeframe upto single pulse

- ability to keep its ECM Bearing reference-to-ECM antennapower in the direction of ECM antenna pointing technologythe threat: (eg fast servo; fast moving reflector

in front of horn; etc)

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Name Description Type of CM Platform Manufacturer Country NULKA Radar decoy

system Active RF payload

Shipborne/ Rocket (hovering)

BAe Australia Australia

Table 7.1 Some existing shipborne EJ systems [3]

Name Description Type of CM Platform Manufacturer Country Heatrap IR decoy Distraction or

Seduction IR Shipborne RAFAEL Israel

BT-2/1; -4/1 Seduction decoys (chaff)

Passive Shipborne RAFAEL Israel

MRCR Distraction decoy (chaff)

Passive Shipborne RAFAEL Israel

Hot Dog/Silver Dog

Automated RF (Silver Dog) and IR (Hot Dog) decoy system

Passive (chaff) Active (IR)

Shipborne Buck Germany

REPLICA Floating reflector decoy

Passive (reflector)

Shipborne Irvin Aerospace

USA

Towed Offboard Active Decoy (TOAD)

Seduction decoy

Active RF (I/J bands)

Shipborne GEC-Marconi UK

Table 7.2 Some existing shipborne EJ systems [3](continued)

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