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AD-A131 923 ACQUISITION IMAGE AND DATA COMPRESSION(U) CITY COLL NEW 1/2YORK D L SCHILLING ET AL. 30 APR 83 AFOSR- R-83-0505AFOSR-81-0169
UNCLASSIFIED F/G 7/2 NIL
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AFOSR-TR. 83- 0505
"oJ A ! i i l i!,i . i 'IVi E. AND DATA",lf FW-SS1 ON
-I ,h _ E L HN ] C AI.. FE P DF:Cr
i '8' 1 -- :L," E:R,87.
IGrant A2OSR-81-0159
AIR FORCE OFFICE OF SCIZINTIFIC RESEARCH
Bolling Air Force Base
Washington, D. C. 20332
Donaldi L. Schilling Georg-e Eiciirann
Professor Professor
j Co-Principal Investigator Co-Principal Investicgator
ISAUG 2 9 1983
SCO.iIiUNICATIONS SYSTE.S LABO."ATORY AUG29 _-' 3
OEPARTNIENT OF ELECTRICAL ENGINEERING A
Approved for publi 0 release "
diUtribut ion unlimited.
_ -- THE CITY COLLEGE OFI THE CITY UNIVERSITY of NEW YORKC.D
.- 0I
I 8Z3 06 21 02t
ACQ U:' ITlON , IM'-AGZ A7ND DATA ''SO
Final Technical Report
5444-23/1/81 - 2/28/83
Grant A'F0Si".-S!E1-69
A7R FOCRCE OFF'ICE OFZ SCIEr':TC RSA
Bollina Air Force --asci
Washington, D. C. 20332
Do3nald L. Schillinc Gciorce Ecnn
0 : 0s sc0r Professor
Co-Princloal InveCstIqator CC-?ri:nci 7al1Ivsiao
DEPARTMENT OF ELECTRICAL ENGINEERING
COMM', UNICATIONS SYSTEMS LABORATORY
~V 7
SECURITY CLASSIFICATION OF THIS PAGE (When Des Ftered)
REPORT DOCUMENTATION PAGE BEOR ,NSTuc'LTIO;SI F(1 dA 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER_&gEn A._ _..505 P -A / 7 ;__
4 TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED
Final To-lcnical ReportACQUISITION, IMAGE AND DATA COMPRESSION 3/1/81 - 2/28/83
6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(s) B. CONTRACT OR GRANT NUMBEReir,
Donald L. Schilling
George Eichmann AFOSR-81-0169
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREiA & WORK UNIT NUMBERS
City College of New York . \ \ y-New York, N. Y. 10031 2305/B1
II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Air Force Office of Scientific Research Apr 1 10, 198Z13. i NUMBE OF 98GEBolling, A.F.B., Washington, D. C. 20332 13. NUMBEROEPAGES
14. MONITORING AGENCY NAME & AODRESS(il different from Conttrolling Office) IS. SECURITY CLASS. (o tis report:
UNcAFSirEDISa. DECLASSIFICATION DOWNGRADING
SCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)
Apn rovee fOr 4'hl ic release Wdi:;trihution tuJliited.,
17. DISTRIBUTION STATEME.NT (of the abtract entered In Block 20, If different from Report)
1. SUPPLEMENTARY NOTES
Presentations made at MILCOM'82; Optical Society '83;
SPIE'83; Optical Computing Conference'83.
19. KEY WORDS (Continue on reverse aide It necessary end identify by block number)
Spread Spectrum, Optical Transforms, Acquisition,
Simulation, Tracking, PN Direct Sequence Frequency Hopping
20. ABSTRACT (Continue on reer. aide it nece-iary end Identify hy block nomber)This report discusses the following subjects:
A new technique to achieve a fast PN Acquisition scheme.
A comparison of schemes for coarse acquisition of frequency-hopped spread-spectrum signals.
Investigation of the tracking of frequency hopped spread
spectrum signals in adverse environments.
DD J 1413 EDITION OF I NOV 65 IS OBSOLETU
SECURITY CLASSIFICATO OF T, .
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGErIhen D.tI Ent.red)
Coherent optical production of the Hough transform.
Estimation of the closely spaced frequencies buried in whitenoise using linear programming.
)
Restoration of discrete Fourier spectra using linear programming.
Two-dimensional optical filtering of 1-D signals.
Random TDMA Access Protocol with application to multi beamsatellites.
A reservations scheme of multiple access for local networks.
A random TDMA access protocol for satellite channel has beenanalyzed. The analysis is based on obtaining the idle and busyperiod distribution. The average packet queueing delay in theuser's buffer is obtained. A modified random TDMA is suggestedto improve the performance. Implementation of the scheme in amulti-spot beam satellite is presented and the delay on board thesatellite is obtained.
A reservation scheme for multiple access suited for localnetworks of broadcast channels is considered and analyzed. Theanalysis is based on two imbedded Markov chain models, one forthe individual user and one for the entire set of users. Theaverage queue size as well as the average delay time are obtained
A fast algorithm for scene matching is studied and comparedwith other algorithms. The measure of similarity is the sum ofthe absolute values of the differences of image intensity betweenthe template and the scene. Comparison with other measures ofsimilarity is presented.
UNCLASIFI"n
SECURITY CL ASSIF A O10N Sr . AGE(Whon F. fl Fnte,e
Table of Contents
1.0 Research Objectives 1
1.1 A Fast PN Acquisition Scheme 1
1.2 Tracking of Frequency Hopped Spread Spectrum 1
Signals in Adverse Environments
1.3 Comparison of Schemes for Coarse Acquisition 1
of FH Spread Spectrum Signals
1.4 Two-Dimensional Optical Filtering of 1-D Signals 1
1.5 Cohrent Optical Production of the Hough Transform 2
1.6 Estimation of Two Closely Spaced Frequencies 2
1.7 Restoration of Discrete Fourier Spectra 2
1.8 Study of a Random Access Scheme for Multi-Beam 2
Satellites
2.0 Status of the Research Effort 3
2.1 A Fast PN Acquisition Scheme 4
2.2 A Comparison of Schemes for Coarse Acquisition of 9
Frequency-Hopped Spread-Spectrum Signals
2.3 Tracking of Frequency Hopped Spread Spectrum 16
Signals in Adverse Environments
2,4 Coherent Optical Production of the Hough Transform 22
2.5 Estimation of Two Closely Spaced Frequencies 27
Buried in White Noise Using Linear Programming
2.6 Restoration of Discrete Fourier Spectra Using 31
Linear Programming
Table of Contents (Continued) I
Page
2.7 Two-Dimensional Optical Filtering of 1-D Signals 37
2.8 Random TDMA Access Protocol with Application to 42
Multi Beam Satellites
2.9 A Reservations Scheme of Multiple Access for 50
Local Networks
2.10 An Algorithm for Scene Matching 77
. .
1.0 RESEARCH OBJECTIVES
1.1 A Fast PN Acquisition Scheme
in this project we studied acquisition techniques for
DS spread spectrum signals. The technique utilized two thres-
holds rather than a single threshold. As a result, the time
to acquire was reduced by an order of magnitude as compared to
the techniques proposed by other investigators who use single
threshold techniques. The procedure can be employed with most
acquisition techniques merely by replacing thc. single threshold
with the two threshold system.
1.2 Tracking of Frequency Hopped Spread Spectrum Signals
in Adverse Environments
The objective of this research is to design tracking
strategies in adverse environments such as fading and jamming.
This led to optimization of tracking in the sense of maximizing
the mean time to loss of lock.
1.3 Comparison of Schemes for Coarse Acquisition of FH
Spread Spectrum Signals
The objective of this research was to compare three of
the more popular acquisition techniques on the basis of prob-
ability of miss, time to acquire and complexity.
1.4 Two-Dimentional optical Filtering of 1-D Signals
The object of this study was to implement a generalized
space-spatial frequency filter and study its operation.
2)
1.5 Coherent Optical Production of the Hough Transform
The objective of this research was to study the operation
of the forward Radon transform (FRT) and compare it to the stan-
dard Fourier transform.
1.6 Estimation of two Closely Spaced Frequencies
The objective of this research was to use linear program-
ming techniques to improve the resolution of a spectrum analyzer.
1.7 Restoration of Discrete Fourier Spectra
The objective of this study is to examine the use of linear
programming for superresolution. A technique is presented which*
greatly enhances the resolution of a spectrum analyzer.
1.8 Study of Random Access Scheme for Multi-Beam Satellite
The objective of this study is to examine the study of a
reservation access scheme for local area networks. We also
examine the study of an algorithm for scene matching, using the
sum of the absolute values of the differences of image intensity
between the template and the scene as the measure of similarity.
3)
2. Status of the Research Effort
4)
A FAST PH ACQUISITION SCHEME
Sorin Davidovici Laurence B. Milstein Donald L. Schilling
TRW Department of Elec. Eng. & Computer Science Dept. of Elec. Eng.San Diego, California University of California. San Diego City College of New York
La Jolla, California New York, New York
ABSTRACTThis paper describes a new technique to achieve
rapid acquisition in a direct sequence spread YTcspectrum communication system. It is based in 9 t - JTc) g (t- iTc) dt TcYtl -I (I)part on an upper bound for the partial correlationfunction of a PH sequence, and hence it is a
fairly general procedure, applicable to a wide The output of the correlator in Fig. I isvariety of systems. YT
l RO U IN Vo Yc) " [ gl--. 8 t JToThe problem of acquisition of a spread spectrum 0o
.T signal consists of correctly estimating the phase
of.& received PH sequence of arbitrary length + n t gt - 1T dtL - 2
N - 1, where N refers to the number of delay wt)(
stages in the generating shift register or, equiv-
alently, to the degree of the generating polynom- which consists of a signal term So(YTc) and aml which describes the feedback connections, noise term no(yT) . The noise term can easilyThe acquisition process consists of repeatedly be shown to be Gaussian with zero mean andcomparing the received PH sequence g(t - JTc) , variance .2(YTc) -.! yT. where is the two-sidedwith a local generated PH sequence, g(t - iTc), power spectral density of the input noise. Theuntil the phases iTc and JTc are found to be equal. signal term, So(YTc) to given byAt that point the locally generated PH sequence
is said to be synchronized to the received PH yT1sequence and the acquisition process is ended. S c wr28 J -IThroughout this paper, the carrier phase and chip a g t - J g C dtiming are assumed to be in perfect synchronism. oThe in-phase/not In-phase decision is based on apartial correlation process. An is well known. V.S YT c if i - Jthe partial correlation function of a PN sequence (3)is not nearly as well behaved as the autocorrela- ~ ( - ition function, it being a function of the parti- 28 YT c -I if i
cular starting phases I and J. the specific feed-back connections, and the length of the correlation The synchronism/no synchronism decision is seenyrc. where y equals the number of chips (an only to involve the choice of which signal term,integer) and Tc equals the chip duration. To S (yT ), is present at the correlator's output.circumvent all of these difficulties, various De tc the presence of noise, this decision canbounds have been found ([1)-U)) to the partial only be made with a specified accuracy. If asautocorrelation function, and an approximation to shown in Fig. 2, we wait for a time T - yiT beforeone of these bounds is used in the analysis making a decision, then we can define two voltagepresented in the next section. regions such that if the correlator's output falls
in region I, we decide synchronism is attained,ANALYSIS and if we fall in region II, we decide the two PH
Reference (2] describes a recursive method for sequences are not synchronized. The probabilitygenerating a non-linear sequence which serves as of an erroneous decision, given we are not looking;n upper bound to the partial autocorrelation at the correct phase position, is equal tofunction of a PH sequence (via the shift-and-addproperty). This bound, unfortunately, is defined P - P[V('T VT]via a recursive algorithm as opposed to an analyt- e >lcic.l form. It can be shown that a good approxi- rI( > i (4)mation to this bound is given by y(1 - y/L), and - P no YlT t1 Jin what follows, we will use this approximation inorder to obtain a simple analytical result. Using where C is shown in Fig. 2 and VT, the threshold,
I the approximate bound, it follows that for i 0 J. is given below [see(7)].
20.4-1CH 1734-32.0001/0.75© 1982 IEEE
- ---- w
5)
Note that both in Fig. 2 and from Eq. (4), for This implies that the two arguments of the erfcsimplicity the decision boundary has been located ftnction mut be equal. Hencehalfway between the two signals. Therefore,
1 2'1 - ~ -vI'vT V) " 2
" 2 9 1 L ) T V 26(5). 0 J .2TT
. 2L _ 2 T or,
- -2. 2 Y2The probability of error of Eq. (4) then becomes 2 " C TO "1 -2iT (10)
2 1_ 2UIC ,2
P[no(.yTc) 2 -- _ 2 _T] .2L I Eq. (10) defines the parameter 2 such that at anyI2
E-(6) time: T T a decision can be attempted. Fig. 2shows quaItatively the decision regions at any
1 ~given Y2<o erfc • n 2 < '
age equals V * the following decisions can be mad!
and equals PTc. and with a given probability of error Pe:oC
x 2 1) Synchronization has been attained (I - J) I
erfc(x) 2 e-Y dy
V0 > V " f'/.2Tc| -r2P + '2
Finally, V T(YIT) is given by 2) Synchronization will not be attained (I J.
if
2T
V 0 < V n -/2PY2T 2
" /2P'viT c L (7) 3) No decision can be made; continue thecorrelation if
In what follows, we will show how the acquisition
process can be improved by shortening the acqui- V < V < Vsition time. n o b
Thus, Vn(2 and VsY)become
The LwAslon Process Prior to yI - Y .( Thus,"V(Y
- At times It may not be necessary to wait the Yn(Y T,ull YI chips to make a decision. The only re- n 2) - /2Ps'2 '2
quirement is that at any time a decision is made,the probability of error, Pe, be kept constant. .- Y1' Y21If any attempt at making a decision after, say, YT - ([1)Y chips is made, then the threshold voltage must -T I
set such thatp T erfc
SPIno(v2 .'21 2 [ ( j V / 2 1 -(Y2 (8) 2 2
exception of the pertinent parameters (i.e. the - yIT 2 L (I- +L1
Note ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Y thtE.2)i dntclt q 4 ih h 1yi i- (12)decision mechanism is Identical). But it theprobability of error, Pe , is to be kept constant,then From Fig. 2 the probability of no decision can be
seen to equal P(no decision)P s lat t - Y!Tc Pe lat t - 2 TcD
or (9)ad"' PrVnL < so + no < Vol
e thtY a ad V given in equations (11) and (12),
rfc 1 "2 respectively, the probability of no decision at
2 r 2,- J erfc[ 1 time Y2 T becomes
20.4-2
: 6)
V
er[ f J
2n Lr L
If 0d is defined as the probability of making
a decision at T = 12Tc
, then
Qd2 -Pnd2
The realization that a decision can be madeprior to t - Y T without compromising the system'sIcerror performance can be shown to decrease the PNacquisition time. A detailed analysis of the
actual improvement in the acquisition performanceobtained by the application of this technique isj rather lengthly and is given in £4].
NUMERICAL RESULTSFigures (3a) to (3f) show curves of probability
of decision versus the chip number (i.e. the valueof Y2) for various sets of system parameters. Theratio of energy-per-chip-to-noise spectral density
E /n has a value of -10dB, -5dB or OdB, and theprobability of making an error is taken to beeither 0.1 or 0.01. The period of the PN sequenceis 1023 (i.e., N - 10) in all the figures.
On each figure there are two curves, oneshowing the probability of making a decision at orbefore any time T, and the other showing the con-ditional probability of making the decision at anyspecific time T (i.e. Qd ). It can be seen fromthe figures that there N a high probability thata decision can be made before time y with no losqof system performance.
REFERENCES
1. S. Davidovici and D. L. Schilling, "MinimumAcquisition Time of a PN Sequence," NTC '78.pp. 35.6.1-35.6.4.
2. S. Davidovici, G. Sevaston, and D. L. Schilling."PN Sequence Acquisition." NTC '79, pp 54.5.1-54.5.4.
3. F. Hemmati and D. L. Schilling. "Worst CaseAcquisition of PN Sequences." Submitted toIEEE Trans. Comm.
4. S. Davldovici, L. B. Hilstein, and D. L.Schilling, "A New Rapid Acquisition Techniquefor DS Spread Spectrum Communications." To be
submitted to IEEE Trans. Comm.
20.4-3
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20.4-5
IEEE TRANSACTIONS ON COMMUNICA IJONS, Vo L. COM- 1, Nol 2.1 IIIUAltY 1983 183 9)
A Comparison of Schemes for Coarse Acquisition of
I L Frequency-Hopped Spread-Spectrum SignalsCLIVE A. PUTMAN, STEPHEN S. RAPPAPORT, SLNJor MIH.,tr. 1t1, ANt) DONALD L. SCHILIING. Iill.W, IEE
Ah.hsractr-Three schemes for course acquiition of spread-spectrum
signals are compared: stepped serial search. matched filler, and ,o-DOlevel. Analytical models and formulas are de seloped % hich charai -lerie performance in adserse en ironments.Coimparions are on thebasis of mi.s prohahility, mean time to search uncertainty region, andrelative complexity. control! ' T 2.Syn-h
1. INTRODUCTION l.,. atle*
1: g. 1. Stepped serial search scheme.TiE synchronization process required for spread-spectruFasystems consists of two parts- acquisition and tracking.
This paper describes and compares several methods for acquisi- ( , L,,
tion. Three schemes are considered: stepped serial search.matched filter, and two-level acquisition. These schemes havebeen well described in the technical literature 11-141. The SarD
intention here is to develop analytical models and niatheinati- , I vs .,,
cal formulas which characterize the performance of the schemesin adverse environments, and to compare the schemes in termsof miss probability, uncertainty region search time, andrelative complexity.
The investigation is primarily motivated by consideration Fig. 2. Matched filter scheme.
of ground radio frequency-hopping schemes. It is assumedthat communicators operate nt a push-to-talk mode and that a The syslem thus performs an active correlation with a receivedsynchronization process is initiated with each transnission. code sequence.
A brief description of the schemes under consideration is The matched filter, a passive correlator. searches the in-followed by a general derivation of performance measures. coming code in real time. The essential structure is shown inModels for Rician fading arid user or jammer interference Fig. 2. A sequence of M consecutive frequencies is chosen byenvironments are characterized and used to predict perform- the receiver to establish a code start epoch and the matchedance. Methods used in obtaining numerical results are out- filter performs a near optimal noncoherent detection as thelined and the resulting performance curves are discussed with sequence is received.reference to particular spread-spectrum applications. The two-level scheme combines passive and active correla-
tion and, as such, combines the capability of searching theII. GENERAL DESCRIPTION code in real time with integration over a large number of
In the stepped serial acquisition scheme shown diagramal. chips. As shown in Fig. 3, a matched filter detects a relativelyically in Fig. I. the timing epoch of the local PN code is set short 11 hop synch prefix and applies code start signals to aand the locally generated signal correlated with the incoming bank of c active correlators. Each signal causes the next idlesignal. If, at the end of an examination interval, the threshold correlator to cycle through K hops at the end of which anyis not exceeded, the search control inhibits a clock pulse to the correlator output exceeding the second threshold causes aPN code generator so that the local code phase slips to the synch indication; otherwise, the correlator is again madenext cell, n cells per chip, and the process is then repeated. available to the common bank.
Ill. PERFORMANCE MEASURESPaper approved by the Editor for Communication Systems I)i%-
ciplines of the IEEE Communications Society for publication after Detection Probabilitiespresentation at the International Conference on Communication% For ground radio systems operating in a push-o-talk modeDenver, CO, June 1981. Manuscript received May 30. 1981; revi sd
f July 20, 1982. This work was supported in part by S Consulting Serv- the prime concern is to establish the desired signal code phaseices (now SCS Telecom. Inc.), Sands Point, NY. during some brief initial acquisition period. We thus charac-
C. A. Putman and D. L. Schilling are with the Department of Elec- terize the performance of an acquisition scheme in terms oftrical Engineering, City College of New York, New York, NY 1003 1.
S. S. Rappaport is with the Department of Electrical Engineering. the probability that, at the end of the examination interval,StateUniversityof NewYork atStony Brook, Stony Brook, NY 11794. the desired signal code sequence is not detected. This is a func-
0090-6778/83/0200-01 83$01.00 © 1983 IEEE
184 IFEE TRANSA( lINS ON COMMUNICATIONS, VOL. COM.3i, NO. 2, FEBRUARY 1983 10)
where
r.e. Synch QMa )= xxa M l-a*2/1; (ax) dx.
r~m Its complement is the miss probability expressed as
Fig. 3. Two-level scheme. ,'I -Pi = Q4"(N/V, v' ). (6)
tion of the threshold level setting, and for the purposes of The false alarm probability is the probability that the
comparing schemes we set this parameter by specifying the threshold is exceeded when the correct code sequence is not
probability that the threshold is exceeded %hen the desired present In the case of the benign environment (noise only-signal code sequence is not present. The performance meas- no fading ot interferers) considered thus far, the false alarm
ure is then the miss probability for a given false alarm proba- probability isbility.
The input waveform in the ith hop interval can be charac- PFA = QA (0, /). (7)
terized as a narrow-band waveform Search Time
SR(t)=A 1 costw,{t)+Oj +gjt) (I It is to be emphasized that the acquisition scheme is re-quired to reliably detect the first occurrence of the correct
where wi is determined by a known pseudorandom sequence, code epoch during the search process. We do not considerand g,(t) is a Gaussian process with variance u2. This signal systems which can tolerate one or more misses of the codecan be expressed as epoch and continue to seek acquisition. Let the uncertainty
region length AXc be the maximum expected delay between theSR(t) = r1 t) cos (wi{t) + 01 + OAOt) (2) locally generated and desired signal code phases measured in
hop intervals. Thus, a second performance measure whichwhere r,{t) is a Rician distributed process, must be considered is the time taken to search the uncertainty
In the case of an active correlator the local waveform is region, T.j hopped at the same rate as the input signal and the detector IV. ENVIRONMENT MODELING
output is integrated over Al hop periods. In the case of a pas-sive correlator the outputs from Jf similar mixers and de- Fading Channel Modeltectors for successive frequencies are stored and added. 1he We consider a Rician fading channel so that the receivedsystem thus forms the sum of squares of independent Rician signal with which synchronization is desired is given during thevariates and the output Z can be shown 15) to be noncentral ith hop interval bychi-squared distributed with 2M degrees of freedom. For con-venience we define a normalized variable Z0 , such that Z0 SR(t) = v' cos (wit - 0)Z/0 2. Its first-order probability density is then given by
+ cIiP, cos (wit - 0,) + ,lw(t) (9)[pZo(Zo) = (ZO/S) (M - I)J2e-(_o+S)12jA_ tS- o (3) (90 2 1 (yZ 0 )
where the noncentral parameter consisting of specular, scatter, and noise components, respec-gt tively. The scatter and noise components are Gaussian and
S Aindependent and the scatter component is signal dependent.s= A2/o21= Equation (9) defines the fading model, and for a, = 0, is
identical to that given in 171.and Im . is the modified Bessel function of the first kind, For comparative analysis we equate the sum of powers inorder Af- I. the specular and scatter components to a constant P, repre-
If the output exceeds a threshold level V7 a synch detec- senting the average received signal power over many hops. Intion is declared. The detection probability is then particular we take
"+b= I (10)PD -- P{Z > V f Pzo(zo)dzo (4)i Lwhere
whereL = VT/a 2 . This can be expressed in terms of the gen- and b=eralized Marcum Q-function [6) as and b
PD QM(VV/- ) (5) Ilere a is summed over many hops and b is summed over many
PUTMAN e ai.: COARSE ACQUISITION OF" FREQUENCY-ItOP'PED SPtHiEA D.SPECTRUM SIGNALS 185
multipaths. The quantities i and b are indicative of the relative I he false alarm probability given / slots are jammed is thenstrength of the specular and scatter components, respectively.It is assumed that these parameters are fixed over the acquisi- P (false alarm I/) Qi(Vi, N4) (14)tion period, although they may have long-term slow variations.
Following a derivation similar to that in 17), the above withsignal can be expressed in the form of (2). The Rician dis-tributed envelope at the input to the detector now has param- Si= 2JPs/NoB1F =/ 2JEJN 0eters
where E, is the received energy per hop, J is the jammerA %~i aP =power (or limited signal power) to desired signal power ratio,
and we assume that BIF = IlTh, T being the hop interval.and The overall false alarm probability is then determined by multi-
pl) ing by the probability off slots being jammed and summing
o2 = NoBF + bP. (11) over/.
where No is the one-sided noise power spectral density and hi
BIF is the detector bandpass filter bandwidth. PFA b(/. Al.p)Q 3 (.,f , -, F) " (15)/=0
Multiple Access ModelSimilarly, the miss probability is
We now consider the interference generated by otherusers who are hopping over the set of F frequency slots. The Al
probability that no other user causes interference during a ' , = E b(i, MP)QftC ( /S-, vTP) (16)
given hop when N other users are active is (I - I/F)N . The 1-o
probability that at least one other user causes interference is with
p = I - (I - I IF)N . (12) [i J(J +a) + (At-i)a I2P [iJ + A I2EINoSd=-
Since we are integrating over Al hops, the probability that j NoBlIF + bPs I + bEc/No
slots have at least one user present isanid
[M IL' =LI(I + bQeNo).
b(IM,p)-()p /(I -p)m/. (13)V. ACQUISITION SCI EME PERFORMANCE
Jamming Model Stepped Serial Search
In application, the receiver could provide for limiting of There are many ways to control the search process (8]. in
received signal strength to slightly above that of the desired the scheme considered here it is assumed that three successivesignal. This mitigates the effects of CWjamming. An optimized threshold exceedences, or hits, are required to initiate thejammer would be forced to jam a random selectioi of fre- tiacking phase. Any one miss causes the cell to be rejected.quencies. When the number of frequencies available F is hence, the overall probability of false alarm is
large, the above model is reasonable for random hop or combjamming if N is taken as the number of interferers or inter- p FA = PFI 3 (17)fering tones, respectively.
where PxFl, the probability of a false hit, is given by (15), and
The Combined Model the overall miss probability is
To quantify performance for a fading environment withinterferers present it is analytically convenient to assume that tif = I-Pif3 (18)
the power of interfering signals lies in the specular compo- where PH is the probability of a hit.nents, and that hopped interfering signals are (hop) synchro. The magnitude of the detected energy is dependent on thenous with the desired signal. In addition, all interferers are relative delay T between the locally generated and desiredassumed to be of equal strength as "heard" at the receiver. signlA code sequences. This can be seen from the correlationInterferers are assumed to be statistically independent RF diagranm in Fig. 4. Due to the stepping nature of this scheme,sources with random phase. Our approximate performance the worst case correlation will occur for a delay of ±Tt/2n.analysis proceeds by noting that the sum of interfering and Assuming equally likely delays, the average loss from fulldesired sinusoids produces a resultant sinusoidal signal at the correlation can be found. In the calculation of the parameterreceiver. Because of the assumptions above, the average power Sj, the desired signal energy must be multiplied by the factorin the resultant is the sum of the powers in the components, 2n - Iand we proceed by using this effective sinusoid as the resultant d _ _ ' (19)specular component at the receiver. 2n
V________I11II ... " , l
12)186 IEEE TRANSACTWS ON COMMUNICATIONS. VOL. COM-31, NO. 2, FEBRUARY 1953
Peak correlo L The detection and false alarm probabilities for the sub-systems are provided by (1 5) and (16) with K substituted for
Av.erag leve M in the case of the active correlators
worst Com The search time is given by
Ts = (NC + At + K)Th. (24)
~VI. RESULTS
Starting with the serial search scheme, the basic perform-- - -delay T ance curve is presented in Fig. 5 as the miss probability versus
1h -- T energy per chip to noise density ratio for various integrationin 2f%
Fig. 4. Signal corelation diagram, periods At (hops) in a benign environment. It is seen, for ex-ample, that a miss probability of about I-3 is obtainable for,
It should be noted that this scaling is used only for the stepped say, ECIN o = 5 dB with Al of 20. To take full advantage ofIshoald each noe Th tis scangiuse on rac the spped this scheme, performance can be improved by increasing Atserial search scheme. This is because, in practical implementa- without increasing hardware requirements but increasingtions of both the matched filter and two level schemes, the
largest output of the passive correlator (in the interval where an environment where half the received energy is in the scat-that output exceeds the threshold) would be used to deter- ter component and there are five interferers present shows thatmine the coarse code epoch, so there will be essentially no longer integration times are in fact necessary to produce misscorrelation loss.
cea ls emprobabilities of the same order.Since each cell is examined for a period MTh the search The performance curves for a matched filter scheme are
time for no false hits would be provided in Fig. 6. If the matched filter complexity M is
equated to the stepped serial search integration time in hops,Ts = (n' + 2)MTh. (20) we find that the serial search scheme is slightly superior in
False hits tend to increase this time, but since PFt is small performance. This is due to the fact that the serial searchthe effect will not be significant scheme considered requires three successive hits for a synchindication, whereas the matched filter relies on a single thres-
Matched Filter Acquisition hold exceedence.
The analysis is straightforward. False alarm and miss prob- With the two-level scheme it is meaningful to seek thoseThe nalsisis traghtorwad. als alrm nd is rb thresholds L1 and L2 which minimize the miss probability,abilities are given by (15) and (16). Since the search time whie the flar probabilityv
justthetim taen t serchtheuncrtaity egin oce, while the false alarm probability is held fixed. For each givenjust the time taken to search the uncertainty region once, prmtrstcnitn fAK ,EIo eaiesrntparameter set consisting of At, K, c, Ec/No , relative strength
TS = (NI + M)Th. (21) of scatter component b, and number of users N, the speci.field false alarm probability establishes L2 for a fixed LI..
Thus, each computation of miss probability is minimized overLI subject to the constraint of a given false alarm probability.
This scheme is analyzed in [4). A false alarm occurs when Curves for the two-level scheme are presented in Fig. 7. Herethe matched filter gives a false start signal which finds an idle schemes with one and three active correlators are compared.correlator, which in tt,' generates the false synch indication. The active correlation period is 10 hops. It can be seen thatThe probability of this occurring is an improvement in performance of 1-2 dB in the region of
interest can be gained by using a bank of three active correla-PFA =PFAIPFA2( 1 -B(c,g)) ,.(22) tors as opposed to one. Comparison with Zhe matched filter
results show that, for similar performance, the two-levelwhere PFAt and PFA2 are the false alarms conditions for the scheme with K = 10 requires a passive filter complexity onlypassive and active correlators, respectively, and B(c, g) is the half of the matched filter on its own.blocking probability for the queue ofc active correlators given VII. CONCLUSIONSby the Erlang loss formula with g = KPFA I.
A correct code acquisition epoch will be missed either when The serial search scheme has been shown to provide goodit is missed by the matched filter, or when it is correctly de- detection capabilities even in adverse environments and istected by the matched filter but is either missed by an active relatively simple to implement. The cost is a long uncertaintycorrelator or finds all active correlators engaged. The proba- region search time compared to that for the matched filter orbility of this occurring is two-level scheme. If this can be made small, such as in systems
with reasonably fast hop rates and short codes, or if somePM =-PM1 + (I -PMi)[B(cg) + (1 -B(c,g))P 1 2 ) (23) rough code epoch can be maintained with clocks, then the ac-
quisition times obtained can be acceptable, and the system Iswhere PM and PMvt 2 are the miss probabilities for the passive preferable in terms of detection reliability.and active correlators, respectively, Matched filter detection requires a complex hardware
PUTMAN e aL. COARSE ACQUISITION OF FRLQUENCY-1101ILL) SP'REZAD ti-CTRUMI SIG;NALS 187 13)
ci.
lk 4
74-
.10.
188 IIJA! TRANSA( rlONS ON COMMUNICAI IONS, VOL. C)M31. NO. 2, FEBIRUARY 1983
- TWO-LEIFI. ACQUISITI N
PF 109 M-5 K .10 ~ i -00*-..j -A iOda
10
I~ I.
DEIGN~ ENV.
N0!-5,i c-0.
-4L..J. C-33 4 5 6 7 a 9 to
E /N149)i
Fig. 7. Miss probabilities for two-level acqusition.
structure for Similar performance. hlowever, the matclicd what higher complexity, which, however, is not necessarilyfilter searches the uncertainty region in real ltle, offering formidable.a clear advantage where it performs adequately. It istherefore preferable in systems operating in less adverseenvironments where simpler filters will provide the desired REFERENCESperformance. III R C DIsi(n. Spic,id Spwrurm S irtrni Ne% Yolrk Wile). 1976
The two-level scheme combines the advantages of the 121 C R (dhtn. "Sjrcad ..pettriiiii applialn aind ..tate-othc*r
rapid search time of the matched filter with the detection cquipnicnIv" (Agaied I cclurc %cirN Nil 5it)reliability or an active corelator. It would thscrefore lind 131 P I trrti iit/ . "Appli~atiuns iii spread spettuunt radiio it) wirclc.%
tonul{.aorl.- presenied at the Nil Teleciininiun Con(.. Dc.application in systems which do not rely on a time referencec4(and which utilize long codes but require rapid and reliable 141 S % Rtappapurt and 1) t. Schlli~ng. -A twoirtc~ci coarse codeacquisition in adverse environments. In situations with im A(iim it'll crlic for 'ptcad ..jccrurii raiou.* l.AE Tranr. Corn-
iu,, "Ii (4OM.29. pp 1714-174K, Scpi. 19X11paired signal detectability due to interference and/or scat- 151 A D Whalen. Detecion of Signuls in Noise. New York:ter, the parallel bank of active correlators in this scheme AcadeiniL , 1'971provides additional versatility. Roughly, even while some cot- 161 C W itcI.tr. Sta,,iita Theoryof Signat Degecoon. London.
Fn~ifand Pecrganriin. 1968relators which may have been falsely engaged are operatinig, 171 J NI Woiecaict and I M iio..Principles of Communicationsuccessful signal acquisition is still possible with an idle cot- fnigoincae Ne* Yorlk Wile). 146i5
relator. This parallelism is not available in thie other schecmes ht iiipi..Aute ni.i.u sul.,i. .nhrnVconsidered here. These advantages are at the expense of sorine- CON11-25. Aug P177.
15)PUTMAN etal.: COARSE ACQUISITION OF IREQUENCY-IiOlPED SPREADJ-SPECTRUM SIGNALS 189
* C3i~g A. Putmian was born in Durbicn, South Secionr Chairman of the L. 1. Communications, Society Chapter, andt Ariaon Ositober 10. 1952 He received the Asocritie Fditor for CONIUNICA~ri)NS MAGAZINE, as well as Session
B cIng deerce from the University of Natal. Chairian, Orcganizeer. and Technical Progratit Committee Member furSouth Atfrica, and the NI Sc Unrg deeree front someri M ilir Sott ety' miajor con let erwes lie is currenrtly Assrciate Editor
h e C r le d Institute oft echn oloy , C ran t e d, lr t It-ia C o m murrr n ict iois S siertis fr ti c I LL " ( N sSA ( IO N S ONUngli. ii 1.474 and 1977, tespctively ((tvt1ii. NI Al I NS
Fon1197S it 1976 lie % a, with HiarlowsTcetion Coittiny South Altta. atid in 1978
51 Ih jie Marcont Instrurients Ltd , Lirgland~ ~ Sinc 1979 he has been emiployed by lirliws
(ritrtrrrratosSrouith Attica. ird is Lurrently _____ Donald L. Schillinig (S'56-Mv'58-SMl'69-F*75)working toward the Ph D) degree at the City ( ollege of New Yrrik. New was brn in Brooklyn. NY. on June 11, 1935. HeCYork. NY received tire H F 1. degree fromt the City College
1'. ^.t i t New Vork, New Yirrk. NY, the M S E E.degee frotr Columbriia Uiversity, New York,
* and the Phi D degree itt electrical engineeringAilfrom the Polytechnic Intie of hirookly n,
I Bruooklyn. NY. in 1956, 1958, and 1962, re-spcttvely.
Stephen S. Rappoport IS*iS-NI'65-SNi76i re- Fromrr 1956 to 1962 he was a Lecturer in thecerved the B L: 1. degree tront the Cooper UAnion. Depirinteui of Elecirieal Engineering, Ctty Col-Nrew Not,r NY., iii lIhr(l. tire Ni S I 'IL degree lcgc ofi New York. a Lecturer itt tire Department of P'hystcs, Brooklynfrt he Uirsrry ii Southern Calirrii, Los Crillcge. Brooklyn, and a nienirber of the Techntcal Staff of the ElectronicAngels, in 1962. and tire Ph [D degree tn dlec Rcseareh Laboratriis, Coilumthra Unrtversity. In 1962 he was appointedtriil engineertng frontr New York Utrersity, Asststait( Prorfessor in the Departmrent of Electrical Engineering. Poily-New Yoirk, in 1965 techiric Institute rif Brooklyn, and in 1966 he became an Associate
Front61 to 1962 he was a member of the lrrolesir He is currently with the Departitent of Electrical Engineering.c ehni Sal Saf at ifugh % Airraft Company, City College ol New York. and was appointed Distinguished Professor of
wher he was engiged inl Sstertis anjlysib tir Electrical Engineering through the award of the Herbert G Kayser Chatroinurricaiirs and Widt Hec was tin tire laculty iii September 19801 tie ts also P'resident rif SCS Telecorm, Inc (formerly S
at N Y. U. fronm 39h4 to 190~5. aird in 1905 joinetd Bell Labrries. where Cons ulting Servicesl, Sands Point, NY lit this positton he superviseshe performed research and prelrrninary desclrrpirerr of stgnal processing resech: and develiptient tn tire ilitary and cortirtercial aspects oiflor radar and digital data commicrtrationts Stince 190K he bras been tin tire telecoitntiuntcahioiis. as well as dircting an elaborate short course pro.Faculty of Electrtcal Engineering. Statt! University rib New Yorrk at Storny gram in telecommrnunicatioins. ire comtputer. and tndnagenrent He hasBroork. where he has 5erved variously as Graduate Prrrgrarrr Dirctor and coastiored forur textboroks, Electronic Circuits: Discrete and Integratedas Undergraduate Prigramt Directorr His presen intnerest% are in corn- I 196s. 2nd ed. 1979), Principles oif Communicatioin Syaremns (197 1i,rtunicaitons systems and techrrrrues. irutiple acs, sy stern inr...d10r1ig. IVigitcil and Analng Swerirts, Circuiti and Devimes (1973). .nd Digitalq~ueuing, comtniciatitns trait ic. and p~rrcd-,pCitrUIi teIriiques A Inrrh trnid Electrmnitrs ( 1977), and is currently writing a book otnregular contributoir to the technical literature. he his been a econsultarni It, a t1irrciomputer applications, Ile has published morre than 100 papers in thenumber of industrial tirn,, telecc-mntnicatrionn field.
Dr, Rappaport is a nmemrber of Tau Beta Pi, Eta Kappa Nu. and Sgrtra lDr Schilling is a tmemiber of the Board of Directors of the IEEE He wasXi. He has been active in ILE affatrs Hec was elected it) the Ads isriry President of the IEEE Cotmnrrications Society from 1979 to 1981 andCouncil of the Cmrnmrunicairrutr Sciciety for a twit-year termt which began w as Director of Publicatioins and E 'ditor of the IFIEE TRANSACTIOiNS ONJanuary 1981. and he has served it rte Ccntiinunitiirns Sorciety ('tin- COMMINUNICATIONS fromi 190hl:. 197&. HeI is a mnember of Stgnra Xi andference Board. In the past he served as Treasurer of the Loing Island Eta Karppa Na.
16
TRACKING; OF FREQUENCY HWTI FD sniiAD C7ECTRUMSIGNALS IN ADVERSE ENVIRONWE~NT
C.A. pUtIAMN S.S. RAPPA lST P.L. 'CHILLIN?Dept. of Elec. Engg. Dept. of EL-c. Engg. Pept. of Flee. Edz,.City College of N.Y. State Univ. of N.Y. City College of N.Y.New York, N.Y. 10031 Stony Brook, N.Y. 2179L New York, N.Y. 10031
ABSTRACT will provide acquisition of the receivo1 pseul, -random modulating sequence to within ablaut half a
Tracking requirements for a frequency hopped code bit, or chip. Cince finer acqui; ition isspread spectrum system with a view to application needed, and since reliance cn clock stability toin mobile radio are investigated. Models charac- maintain accurate synchronization is generally notterizing Rician fading and multiple access or Jam- sufficient, even for short transmission rush tozing environments are used to formulate signal talk operation, a tracking loop is usually employed.detection probabilities. These determine the mean The early-late gate trackinr loop as used fortime to loss of lock. Optimized tracking in these range-gate tracking in ralar systecs !'), lends it-envirooments is then considered, self well to the freq.ency hcppe! signal arpli'a-
,i n. This loop aol Itn i ociltei w'efors arehown in Firs. la,h. 71;n r'tine wav'-f, m. "t) from
.1. INTRODUCTION the VCO is alternately neE tive wnd ;'sitive wits
The frequency hopping (FH) waveform as a pseudo- the minus to plus transitions coinciding with the
random modulation technique has the attraction of chip to chip frequency hopping instants of the
ease of implementation and inherent frequency div- locally generated waveform S (t). The error signalerslty [1]. A central fcature of a FH system is is generated by integrating he multiplied gatingthe pseudorandom code generators at both the trans- signal and envelope detector output v(t) and ismitter and receiver, capable of producing identical proportional to the delay T between the local andcodes with proper synchronization. The pseudoran- !ncoming code phases. The resulting discriminatordom code sequence is used to switch the carrier characteristic Is sh:z! in Fig. Ic. 7he error
frequency via a frequency synthesizer and videband voltage is used to advance or retard the local
mixer. When the synthesizer in the receiver is waveform into aligorrent.switched with the synchronized sequence the fre- Inherent in the loop operation will be some phasequency hops on the received signal will be removed, Jitter which will increase with decreasing signalleaving the original modulated signal. .o noise ratio. The variance of this phase error
The synchronization of FH spread spectrum signals is given in (31 for a single received pulse as
can be broadly divided Into two phases: acquisition 2 T Tand tracking. A typical system would first obtain T Ccoarse acquisition of the received pseudo-noise 8E/Noencoded signal (21, followed by fine acquisitionby te tackng oop Attheendof he cqusi- where T is the gate width, T is the pulse widthbythe tracking loop. At the end of the aquli and /Nis the pulse ener ' o single-sided noise
tion period loop parameters would be adjusted from Oacquisition to tracking settings and the system Is ;.wer spectral density ratio. For a frequency hop-said to be In lock. There are two possibilities pod system T is equal to the chip duration Tc, andat this point. Depending on the scheme used, assuming thaf the channel correlation bandwidth is
* either one or a series of false alarms may have Preater than the hop rate, Tp Is also equal to Tc.
causea the system to incorrectly enter the lock Hencestate, or the correct code epoch may have been 2 T )detected causing the system to correctly enter the oT ' c (I)lock state. The requirements for the tracking BmEc/Nophase are to maintain tracking of the correct codephase in the latter case, yet to quickly reject wre m Is the nuher ef hpn In the intecrationthe false lock condition to allow resume-d search p-riod and Fr Is the enerwy per chip. Thlt resultfor the correct code phase in the latter case. als agrees wit:. that in (4] with the loop band-Although this paper focuses primarily on tracking width written as PL = i/mTc .of F signals, much of the analysis arplies direct- 3. IN-LOCK MONITORINGly or with little modification to other spread Once thp syetem enters the lock state, it becomesspectrum schemes such as direct sequence or hybrid ncessnry to monitor the tracking of the receivedsystems. In-lock detection reliability is charre- code phase. This Is achieved using the detectorterized by mean time to loss of lock when mean of Fig. Za. The input waveform in the ith hoptims to reject false lock is given. Interval can be characterized as a narrowband wave-
2. THE TRACKING LOOP form.
The coarse acquisition schemes described in (21 5R(t) - ,'TFcos(VI(t-iT ) + g1 (t-iTc) (2)
27.4-1CH 1734-3/1124O,$oo IS, 1982 IEEE - -
~17)
where w is determined by a known pseudorandom Be- Combining the tracking loop with a stepped serialquence, and gi(t) Is a gausslan process with vari- search acquisition loop results in the circuit ofonce o
. The envelope of this waveform is then , Fig. 2c, with each loop providing fine and coarse
Rician distributed process, and since Integratiot VCO control rczcctlvcly. Ceparate envoeore detec-is over M hops, the output Z is the sum of squares tors are used for tracking and signal detectionof M independent Rician variates. For convenience since It is likely that their characteristics willwe define a normalized variable Zo, such that differ. It can be noted that the In-lock detectorZo = Z/0
2. Its first order probability density is is inherent in the same circuit with probable para-
then given by [5] meter changes. However, if matched filter acquisi-
(M-1)/2 -(zo+S)/2 tion were utilized, such circuitry would be addi-= l(z /S) e 1 z) tional.
0 05. COntROL STRATEGYwhere I is the Modified Bessel f'unction of the Information received from the threshold detector,first ktn , order M-1, and the noncentrality para- that is whether or not the threshold is being ex-meter is ceeded, is interpreted by a control system which
decides whether or not to continue tracking or re-S = 2WP /02. (3) turn to the search phase. This decision will be
s based on some form of control strategy [61. In itsIf after MT seconds the output exceeds a thresh- simplest form, a single occurrence of the thresholdold level VT a synch detection is declared. not being exceeded, a miss, causes the return todetection probability is then the search phase. Alternatively a series of such
occurrences may be required before this actionP D
= P(Z(VT} =) (zo)dzo takes place as shown in Figs. 3a,b. Here n is the
L o~ number of additional in-lock states. The differ-
ence between the two strategies shown is that awhere L = V TI
2. threshold exceedence, or hit, either causes control
This can be expressed in terms of the generalized to return to the first in-lock state or the previ-Marcum Q-functIon as ous one.
The control strateg can be represented by aPD = QM(rS,'7j) (5) finite Markov chain [7]. It is assumed that the
2a2 probabilities of a hit or a miss are constants.where QM(a,b) = f x(x/aM1 e(a+x)21 (ax)ix, The Markov chains for the strategies of Figs. 3a,b
b - are shown in Figs. ia,b respectively, where p isIts complement is the miss probability expressed the probability of a hit and q is the miss proba-as bility. The first state has a reflecting wal, and
the last state is an absorbing state representingPM = 1 - P = (6) rejection of the current code phase. The Markov
chain can be described in terms of its transitionThe false alarm probability for this system is matrix P whose element Pij is the probability of
the probability that the threshold is exceeded transition from state i to state J. For the Markov,,hen the correct code sequence is not present. In chain of Fig. 4a, the (n+2)x(n+2) transition matrixthe case of the benign environment considered thus Isfar, the false alarm probability is
PFA QM(0, 4) . 7) p0 q
p0
The tracking loop phase error will reduce the p =correlation between received and locally generated awaveforms, thereby degrading in-lock detector per- p0qformance, and must be included in the detection 0 ......... 0 0 1probability formulations. This is done by assum-ing that the phase error is approximately Gaussian and for that of Fig. 4b,distributed (3] so that 90 percent of the tlme the p q 0.........phase error is less than
3oT. It can then be seen p 0 qfrom the diagram of Fig. 2b that the correlation p 0 0is reduced by a factor less than 3oT/Tc and hnce .the signal power at the detector output is scaled b "0 q 0by a factor greater than 0 0 q
3ST/To) 0 .........0 01d -(13T3/T)
2
respectively. In each case the matrix can be parti-Hence the miss probability is given by (6) with, tioned as
from (3),2MdP P =
B (8)IF". so that the (n+l)x(nI) matrix Q describes the
where B17 is the system noise bandwidth. Markov chain without the absorbing state, R is acolumn vector of n zeros and a q, and 0 is a vector
27.4-2
of 11+1 Zeros.ec.t 1'a'-Te r n trwansiticz t,.ne fro= &aLy state i to ha- n r. -- e,.tnet c
sorbing state n.+! Is given by the jth element of of 1: Elen tallecoret C..i. ejCcb. ha. hethe vector [7 ee'. or tta - ~ ~ ~ *.j.
- -~tcte [:,I f ,;r n ex-c . cale orT t zI N QI fo tn t z~r-
where I Is an identity m.atrix and t is a vector I - ' s t:., rnear tIm 1.a fwv.3ae elmns are th-e tim-es sln in state ge raI.n r entnThe mean tin-e to 1j flock TI, jthos given by-the first eeetci'f T, the *.e3L' tru -L;t1On time ____
from state 0 to state r.'KI. Using~ eltncritury --trix agebra, We ":i-:3 :r a hoia: fa 21ji ngca.n .a ta
re ce l:e - signaEL w. :..och z-,n-1:rn1 ntiLon is de-s~re is li d~ln:-.e th interval by
L i=3 etNSt,2aP esK,-.(~ §cs
where Cij istthe cofac:tor of ni ~.if ti , the ~lthreshol detector irnteerution imes, are a con-ofs zar* .t adolecmo-1stant KM70 then. nerts rvspectivel.'. The s~rJer find n-oe
n ne riS 3 r f,..ar a:-. j:.1n~~~dn n the EScatter
L icj= det Nfinies &C drgmde. o for a =0 Is i dentical to
This formula. ayi ies to any corntrol ctrutegy an,! t.at ;,.u r, 131is easily calcuolated wh~ec matrix aagebra routines Wo ~rA.o~ay~ e ecqoate tl., non: ofare available. power, .nte otsLra. catter c nr--.e:.t3 to ,a
A slightly di fferer.t appqroach yielsce form P.r'e, ntn the uvreerccveexpressions for a Fivec,. cntrol &trategy ofn '' F~nal. rY3.rer over- nary hopj. 7, particularstages. The rtate tranisition. -iagra.-5 can itetransformed int.) signal flow grajhL 7li Te a+ b 1strategies of F6 S.- 3a,b car, thien also, be relre- er a=11 anbsented by the signali flcv grah3 of F;;o ab.eea aIaiIb=ZcJwhere the variathie t r.; .rc vntso the unit pa~th leng-th, in this case the fixed I ntegriiliar. tin:. he Here a is sli.-nei o-.'vr =!,r.y i. is sundc-eiciean rath leiietn from n 0e to nILde 11+1 in thenj Over ::,:.y 7%~ahs.Te ftr~n !,terQtht mean tin-e toD loss of lok. Cns e t!,e m anth indiCative of, ti hn relaive 'L, ~ ho' 0 ' ;oulength 1 is ioot the soos of "Il path ln thstier and s52011cr tornet repoie:,ntheir probability 'it can te obtul.,neJ frz= the that t5 Protr are fo 3305 Over "e,~-graph transfer function l~t y differentiating ticn perod atho'ngh they' n--y h ave lonc tr S"with respect to t and settin~g t=1 [311: variations.
TL'1r L t) 0) Fcl2.:win g a deiaio l rtotaTL t= at ',; 2i teao s.-lca:. I - expr.:- i:-..........ci
(C. TeL. S r:rlt'.o~e in -lie eqiiationThie graph transfer functioin for Fie. 5b in ob-- for thec miss ;robah1i.ity (t6) the: beoc
tained by inspection using M'ason's form~ula 191 as
G (t) = + nv n+l. I S
pt (Iher a +o IF ,
1=3 i ti' 5C5 I'e r.-; coonider th e interference gene ratei I)y
A-,plying equation (10) and using p l-q it is other users wh-o are !'nppirg ove-r the set of' F freq-foun'd that uency slots. The Trcrtbility that no oth~er user
causes interference dui nw a given hoT wheOn N other
L c -qr ~ at least one other user cqu!cs Intcri'erence is
late that as the miss probability approaches unity Sinc we are inteegrating. over M hops, the probaibil-
ye m T indn ity th-it j slots hrve att le as5t otie usier pr se n-t, Is
Am'L P(l (1
q-1~',p 3l-P) M"j (15)
This result will be used an a basis for comparison Jfvyflnif MoIdeof various control strategies. A general cooced In applicatiorn the receiver could provide rorform expression for --L for the graph of Fig. 5a limitlng of received signal strengeth nbcove tl.tat ofhas not been found, but the san.e techniique yields the donired sig-nal. -'-.5 rmitigaqtes, the effects ofa expression for each value of n. In each case CXW Jamming. An o;%timiz-ed jammer would be forced
27.4-3
19)
to jam a random selection of frequencies. When the sufficient in a benign environment. However, when
number of frequencies available F Is large, the the f-nvironrent crnsists of a fading chnnel where
above model is reasonable for random hop or comb half the received en,-rPV is in th,. scatter corpo-
jammiug, if N is taken as the number of interfer- nent FuiJ thre are 50 inturfertcr: present, cne orers or interfering tones respectively, two additional in-lock states are needed.The Combined Model Minirm, tracking system requirements for a speci-
To quantify performance for a fading environment fied m,:an hold-in time In a given environment are
with interferers present it is analytically conven- determined by the specified mean time to rejectlent to assume that the power of interfering sig- false lock. To see this consider the curves of
nals lies in the specular components and hopped Fig. 7, log of normalized mean hold-in time vs log
interfering signals are assumed to be synchronous of normalized false lock time, which show thewith the desired signal. In addition, all inter- trade-off in performance for redu'ced mean false
ferers are assumed to be of equal strength, and lock time. With reference to the results for anwhere interferers and desired signal are present envirorent where half the received energy is in
in the same slot their powers are assumed to be the scar*Cr comporents and there are 50 interferersadditive. present, we see that for the parameters chosen, if
The false alarm probability given j slots are T-F is required to be less than lOms then two addi-FLjammed is then tional in-lock states should be used.
aThe curves of Figs. 8 and 9 show the effect ofalamIJ) Qm( ,) (6 the relative strength of scatter component and num-
with ber of interferers respectively on the mean hold-in
Si J2JP [N B j2JE IN time. The effect of the fading environment is seenj ao IF c to be more severe than that of the utiple-user or
where J is the jamer power (or limited signal jamming environment, and control strategy require-ments can be determined from these results.
power) to desired signal power ratio and we assume Similar results were obtained for a system usingthat B IF T the control strategy of Fig. 3b, and were found tois then determined by multiplying by the probabil- have little significant difference from the aboveity of j slots being jammed and summing over J results for the values of n considered. Either
M strategy would thus suffice.
PFA = t b(J,M,p)QM( /S I, ,rf). (17) Finally, there were also found to be little signi-j=0 ficant difference in the performance obtained from
a system with n=O and M=30 compared to a systemSimilarly the miss probability is with n=2 and M=l (application of equation (11)
M yields a normalized mean time to loss of lock of
= E btjMp)QMc(/S, VV) (18) 30 in each case). This implies that adequate per-j=O formance can always be obtained simply by increas-
with ing the in-lock detection integration time, provi-ded the constraint TL /Tc - Pi is met. However,
sd = [J(J+a)+(M-J)a Ps = [jJ+Ma12Ec/N O the analytical results predict average performanceand do not reflect the disastrous effect of events
NoB + bP l+bE No such as deep fado which are often prevalent in a
and mobile environment. The in-lock states and shorterIntegration times would be preferable since a
L' = L/(I + bEe/No), single miss does not indicate loss of lock.
6. RESULTS 7. CONCLUSIONSA system for the tracking of frequency hopped
The performance parameter of interest is the signals has been evaluated in terms of mean hold
mean hold-in time p given the mean time for false in time for given mean time to reject false lock.lock TFL. For grapical convenience we generate It was found that tracking system requirements forthe logarithm of these times normalized to the hop a benign environment are greatly extended by theperid T* Ths gven FL M Im, nd nP Fis ician fading environment, and only to a limited
speciftea so that for any set of parameters- 0y;N, degree by a multiple access or jamming environment.relative strength of scatter component b, and nu-relativestrength of interfee sc ter deco t te d l l System design should therefore utilize the predict-her of Interferers N, the detector threshold level ed results for an expected environment.
and hence PM and finally TH can be calculated. In the case of a beniln environment, performanceConsider a tracking system employing the strat- of the tracking system in terms of mean hold-in
egy of Fig. 5a. The basic performance curve is time can always be improved by increasing the in-
presented in Fig. 6 as the log of normalized mean lock detector integration time, providing that the
hold-in time vs energy per chip to noise density mean time to reject false lock is acceptable. How-
ratio for varying number of control strategy ever, a search lock strategy with one or more addi-tional in-lock states gives similar performance
states n. For the loop parameters M and m chosen, for the same mean time to reject false lock and Isthe need for a control strategy with one or more preferable in the case of the fading environment.additional states is evident. For example, If themean hold in time is required to be greater thanan hour at an E IN of 6dB with a hop rate of10KHz and a mean tme to reject false lock of lOmS,then a yalue of zero for n (i.e., a single isscauses immediate return to the search phase) Is
27.4-4
20)
1IM.CIS 0rUTi W F. Utlaiut, 'Spread fpectru.', Ifli Teieccm- ,,F
2".o Iationa Journml, Vol. 45, Jan. .o'11C.A. Putman, S.S. Ra;Fapcr an Z. L. cilr.,-
A Ccmparl~on cat Schenes tar the 7jarse AcV.-s. ition of Frequency HKp~vd pread £f.rrtr n A-nals' , !,, Conrerser~c 'norl, j-x~ I l 2a. in-lock d~ttctot.
1]) D.K. Bart..n, 'hu~ar Cyo-ternaY 15ri , irtte-H all: Englewood Clfs, 4.J .
1N M.X Simon, '4onliressr Analysis of an AbsoluteValue Type of an Early-lAte C;ate Bit Cynchron- 1tier' . IEEE Transa. .n C--.s. e.* V.1- CM4-i tILrclt
No.~e .%Cctoi~er, lQ7C.(A.D. Va~en, ' etection of Signals in Noil;e' ,
Academic Press: Ne~w York, 1971l,(6) P.M4. bOpei1rs. 'A Unifi,-1 Analysis of Pseuo-
noise Synctircnilzation ty Ei.vel41. C~rrelaticn',IEEE Trans. cn i-n .ch. , VoDl . CM- N..August, 1977.
17)l M. Iositnsz.., 'Finite Markav Chains at-1 Tt~eirApplications' , Jchr. Wiley: fj-ns lD ~______ .
181 J.K. Hle3 and C.C. Chenr., 'Acq~iait4 rs Ti~'- TpassoPerfomnanL of FN L~read .r1,ctri _stens',! EZE Trans. cn Crns . Teonh. , Vol. CM-:5 Na.
* August, 19"7. 2b. Correlation diAgram.
( 91 V.',. Eveli,rh, ' Introdoctiori to Control CystemDesign', XM Iraw-flill: New York, 1',1:2.
1101 J... Wozr'rcrai't ani 1.M. jacots, ' 'Erinciples aofCaourictsn Engireering', W.ley: lNi '-, . X r
195 MV -lui.
x Oil)
1 - f
Ia. early-late gate tracking loop. 2c. Seriai search acquisitbon and tracking system.
rig. 3aic,.Sac
lb. Loop veforua. rig. 3b wos -osrh
3a, b. Control strategies.
q q
daisy 1. T P 1g. 4.
1C. isciniato chaactcisic.4A, L. Macboa chain diagrams for Figs. 3s, b, romp.
27.4-5
I ;2]
II
-7-L - - -- -s
h, i '5, [a, b. iidosi finw :tq.S f,.{ nij. 3*, DreF
o . . ...,---- -
* . .
4--
ri
I..
. .,
-- " -C " ....
• f. e c n,falz d ear - ihd- In tri s r nefti O f
[Lo o f no malied ean hold-ln t i me 1e o c*neonusr.
pet chip to meoie densit" rtO.
1 - ,,.1 I- --.-
.. / -- - -. - -.. . . .
I-
noilalized mean time to teject 141 f~ .o~e ue
727.4-6
- ... .. -.
,- - .. .. . . --- 0'-
Coherent optical production of the Hough transform 2
George Eichmann and B. Z. Dong
tulrL I .it Irai~ i 0 t I 'sI l 1i it lit d.el Iite -ru tIa I,.,1 i t i(.h t- , i iri Ih .; til ,, kI p rm thre,
I, r- )I,:v i t-, d e nt i tcr hut it-1 p tud mit h an t tlt-1 r,, r W i-..n ftaii r th ui , -r. 1 iftin -i t m u ti
I. Inrdu to tor ofc-iltf- ,Ilga h n i 11 and- recm Itit1,t a w Isi oI t he ,o p, rat fi , fi vi i r
I w tt is uss d n the t it e t If ila e ttr af ilt' i I' hav aI I,, l i, I t, ke %i l fuss t 1., -1 r nill v I it-sis Tlh Alug thil irnt I ) i cti Ift a ptiibit t the,, tkttiitrehorent Itdl: itcl p ifeir fil opt-ile Iecits.uci ltmi ich
Il -I lst iIto a sin curv It the t , tansf -[,orm te I.III II it pI tlit I a vaulahle ij i or recent til - l ,d lyllII, iatt111I if viiusing thea fom l (I I I Io virnt AI straish Iilin t ( IY r , iscl It~ use at iit ret otcal F R%,'1;. !W ,,.jI ' rIfe)'r 'MfIr
Jt h -ipln sit ilple10iti plitit. in t Ihe Ftan form I perfirm itimii munl-ill il Imptieiti itt I-'ct ,"l~utiitv to t ctir rau t.
pl. Introdu ictur egelntt.wiittahe ttiil peanlcntusig t (iof -he a o tiali aon
sis. ind re its g a itrlengthrmtiosgne it s apint ilitti translorme prcsos tn this inrcoherent optical eosrkdIfltt 1a _ pn int tsncuvil the 1) transformt pltte lleiliiui d e n itie ise iltabil f e. Mre aredc ssed. Tirte" pur
using ~ itt~ thit formula pii)t xcs+Y inthe sraigtfiirni hiist,' ofse thie Iin-er optical R processrsi t tuet s 1hand
platte. Ever toaliturer ofg enleettl collittma lte aidsth a aonaesiso tipicga setape f 1-1)toiticait well t
segnitents formn a histogram at at pointt iut the transformt itt the use oft 1 -1) coit-rettt opt ical signal priicesiitg ofplante. A single long or ntaitv short collittear lite seig- illtltidintieitsioital signtals.
Ii'is(,i idtli an itsi vate. IiagKes thfat 11. Radon Transform itanittritspcsittContin ege evillnts nlYthe oldsappoxtte llttc 'Ili F T can litdfndt aumeofscsad
lutle segients. Tlhe Houtght t ransformt ntaps these line' itt iti arbitrary inte fdnetin. If'/i.i hIs thleseginttnts tlt,) a sitialler featutre space ito form thle Houttghi 2-1) intputt di I~tribuio lith le 1l' of this Iaitction is tie-transform shape dlescriptors.fitd15usnveortttti.
Recently it has been pointed out by D eants' that theilt-~i ~u. iHiought transform for a bintary intage is equivalent to tlte i'=-
forwatrd Radon trnfr" WWIR'') of' thIis imitage, it wheretransfornt that has been kntown ftor It half-cent ur v. Infact thle generalized versionis of the H ought t rantsfotrmnx I 'a , a,. I -
Ithat detect other shapes such its reeltangles, ellipses, or c"0,+ sit.k = "'a, + "u.k ilparabotlas can lbe shown tt ble equivaleint to it genteralized iitht
version of thle FR''. lin thtis case, t lie generalized F"Rl' 'I'lte t ransfiormt paramteters ( 100 callt lhe ciiitsidlered atsitaps tlte 2-1) shape functiont into I tmultidiimenisiotnal defited onl it rectangttlar grid ilt the QQil plane. A point
featture spaice. T1he Hdndtt transform theory htas not (pc.1 1 il iltt - transform plante ctirresptids to it linie inlWily applicatiotns inl pattern recognitiont bui t also inl thex .r N, platte where pit is t lte short est d istance bet wecitntuclear mtedicitte, imaging by ntuclear magnetic resti- a pint on tlie line and the itrigviti anid I/, is tile anglett it cc. deterinmatIioni of' thIe veti-ct roti-nint eitumi d is- fo rmitetd by t Itis shItort est clhiord to tilt ite, iteasai ret! fri ntrihutitm itt solids, scatttering thteory, and plastta diatg- tile piositive x axis (F~ig. R) ('inversel v, littes piassintgItiistics.8 " While dligital mtethtods for bo~th consi rut- throtugh the point (x(,,vI,) illt tlt, x-Y plane correspotnd
to t ite sin u soldal Ic~urye
I'lii' uthors tire "tit Civ t'iilegr ofi'l(li iv tUtiversit 'v of Nit t , ('010 i 11 Jij()i(rk. IDepa ri tment of Elect riciti Eng ineerintg. New YoriTk. New Yotr k inte(p,) dai.'I'esinwrl'strettcbisfte
i te-ei,ed 7 Septetmtber IS2. lite detect ion capabtlilityv of' tile FUT'l. While (ite presetMA0: tFIisIit it r, o'$iit ill ii discuissiitn dteals with 1tIi1te (ttectiott itt nloiseless, ittages.c I9S3 Opticli Soctie y if Ame ruia itiit aaIs is i curve detect iotn ii ntoisy imitages is a vail-
830 APPLIED OPTICS / Vol. 22, No. 6 / 15 March 1983
23)
able."' Some of the elementary properties of the 'inallv the FlIT of the 2-I) convolution of two inputsFRT are now described, f i(x) and f[(x) in the space domain
Translation of the input transparency by a distance ftx) ti /W E /,( Y)/,(x dy tillXg., i.e.,
l ( + 1o,) +- v) , i S
yields a FRT - lo'W1AIP - ti1W, 112)
= li, + W .I a I-1) convolut ion (if the FIl'l' of the inputs in the Radontransform (RT'l') domain. This property of the l0',Taking the directional derivative of thle ilnit, namely, that it maps 2-1) into 1 -1) convolution, can be
a, £ftx) (7) exten~ded into higher dimensions. Also this propertyallows reduction of molt idimensional correlation, con-where a is a constant vector, yields as its i{' volution, and spectral analysis to he performed using
)= (a. -~dAY,. 81 1-1) acoustoi)tical signal processing architectures.while the linear multiplication of the input III. Optical Methods of Generating the Hough
ft() = is .X)fhX) TransformThe starting point for the coherent optical generationvields a directional derivative in the angle FRT ( of the FRT is the direct implementation of the inte-
plane: gral
ho,., f=x, )(p - X cos -,Y siih)dxd, (l)
y where /(.y) is the input and J(p,0) is the output witha delta function as the space-variant impulse resq)onsekernel function. A l)articular way to implement this as
N, well as other similar kernel functions is to combinesuitable linear and/or circular motions that will map thespace-variant kernel into a space-invariant kernel. Ifthe coordinate axes tx,y) are rotated by an angle 0, thenew coordinate axes (u,t,) are
14 = Icoso +. sinO I' = -x sint? +.% cosfi. (13)
, _In the rotated coordinates the input transparency is0 f -(,t,. Thus by rotating the input, the output FRT
Fig. 1. ('artesian input and radon tran-trr plane coordinate can be expressed in terms of a space-invariant kernelgcrmetries. The shortest distance to the straight line is p. while 0 operation
is the angle formed by the shortest distance cord and thet axis. l(p,) = J [-(u.c)b(p - u Ididv. (14)
P, P P This integration can be performed either in the space2 or in the spatial frequency domain. In the space do.
main, for example, one could electronically integrate therotated function along the t axis and then use a 2-D
- coherent optical processor to implement the secondintegration. This integration can readily be performedLusing an astigmatic processor.2,5 Or one can use a 2-Dmultiplexed holographic processor to implement di-rectly the space-variant impulse response function ofEq. (1). Alternatively, since the kernel of Eq. (14) isspace-invariant, the filtering operation can be per-
f f formed in the spatial frequency plane. Letting the, Fourier transform of the rotated input transparency be
Fig. 2. ('Cherent Fourier optical lens configuration used to record F-(zw), the FRT isthe FlT,,f a 2-1I input. The input is lwated in phcne P,. A verticalslit is placed in the Fourier transform plane P.. while the output FRT ?(pll J F tz.,.)tt -r.cv;p.t)dzdw. t15)is recorded in the inverse Fiurier transform plane 1P:1. For a fixedangle of the inlput in the plane Ig. a Cartesian coo)rdinate slice, with where the filter function H- isFIRT oc rdinates as a Cartesian grid. is generated at the output plane t-(z. j)pnlj = (I2r)tw) exp(-j,p). It;)I',. Rotating the input in plane P, by an incremental angle while
simultanceusly linearly translating the output storage material in the This filter can be implemented using a slit and lineart output plane P., a new slice of the FRT picture is generated. phase-shifting wedge. An experimental implementa.
15 March 1983 / Vol. 22, No. 6 / APPLIED OPTICS 831
24)
'1 y
0 \
aa
Pf - -9 --
P
b bFig. 3. Experimental results generated using the optical setup de- Fig. 5. Experimental results generated using the optical arrangementsitriled in Fig 2. In a) the input hue i ihi the Fl{]' slices of )a) shown in Fig. 2. In (a) the straight lines input, while in (h) the FRTare shown, ']'hle vert ica laxis in ij is ), % hile Ifiv h.ri-i intal is the ) ottput slices of the input of (a) are shown.
axis. H|ere the angle is ieutsrel trorn the (',rtesian) ixis.
3 60 P2. The 1-D linear phase shift is implemented using
a second Fourier transform lens together with a I-D slit35. in the output plane Pj. In the film plane P3 , for a given
angle 0, a slice of the FRT plane is recorded. By ro-tating the input in plane P1, a new angle is selected.
27o' The new angle is recorded by linearly translating thefilm to a new Cartesian grid position in the FRT plane.In this fashion, the whole FRT output plane is filled
225* sequentially. In Fig. 3(a), a particular input containingboth horizontal and angled lines is shown. The re-suiting FRT slices are depicted in Fig. 3(b). The ver-tical isp, while the horizontal is the 0 axis. The FRTslices are spaced 45' apart with zero at the left and 360"at the right. In Fig. 4, results of the computer generated
135* slices for the input of Fig. 3 are shown. Note: to beconsistent with the experimental arrangement, theangle 0 is measured from the vertical axis. While the
90 - results are sensitive to the choice of the origin [see Eq.(6)], these results do match the experimentally obtainedresults of Fig. 3(b). In Fig. 5, a second input with the
45" --------- corresponding output FRT slices are shown. Here, asin the previous figure, the straight lines are thickenedfor the FRT analog computer. While the ideal HT
8 o" descriptors deal with idealized line segments, practical_ boundaries are always filled for better detection. Thick
-04 -02 00 02 04 boundaries are also necessary for the optical analogFig. 4. Computer generated FRT slices for the input of Fig. 3(a). computer to improve the detection SNR. In Fig. 6,
computer generated FRT slices of the input of Fig. 5(a)tion of this scheme using a coherent Fourier optical are shown. With the identical caveat mentioned pre-processor is now presented. viously, these results match the experimental results of
In Fig. 2 a coherent Fourier optical lens configuration Fig. 5(a).is used to record the FRT slices. The input transpar- Various other expressions for generating represen-ency is located in the plane Il1 . A narrow vertical slit tations of the FRT can be derived by using differentprescribed in Eq. (16) is placed in the transform plane representations of the )irac delta function. Thesc
832 APPLIED OPTICS / Vol. 22, No. 6 / 15 March 1983
2
representations can lead to new optical implementation /p/.i = . exp(-ji)j,, (p1 (2)methods for the analog evaluation of the FRT operation....The kernel of the FRT, a space variant impulse response wherefunction, is a function of four variables. One may iso-late any particular variable and derive new represen- - (241tations for the FRT. Consider a change from Cartesian " j /,,r)(! - /r21 exl-.n cos'(p/rlldr(x,y) to cylindrical coordinates (rs), where
The FRT is expressed as periodic Fourier series in theX = r cos, y= r sin,;. (17 1 angular variable and a forward circular transform in the
In cylindrical coordinates the FRT kernel function is radial direction. The inverse cirvular transform hasbeen discussed in the context of computed tomography
htr~o;p,0) = blp - r costfI - o'}. (18) and in terms of coherent optical implementation byUsing elementary properties of the delta function, Eq. Hansen and Goodman."' The forward and the inverse(18) can be expressed in terms of the radial variable circular transform kernels are quite similar, and,as therefore, both of these transforms can be performed
with the same optical configuration.h lr.%O;p.l~l = bjr - p/c's)) - /of,,,s) - ,,'I I )i We n()w use the second radial expression of the kernel
or in terms of the angular variable as function of Eq. (19) to derive a different representationof the FRIT. Let the input transparency be representedh~,~.|= blo - Il + cus-(p/rhl/r sqrtI - )/r-21. (20) aas
These representations of the delta functions are nowused to derive new expressions for the FRT. /Iro, = .irJ JG0 (;.( expot rd* . (25)
Let the input function f(r,(p) be expressed as an an- wheregular Fourier series
", . f,(r( exI{-/fl&G (21 1'.0) = 3 /(r,,, exp( -u r)dr (21}f(r,oi 21
where is the Laplace transform of the input transparency inthe radial direction and Br stands for the Bromwich
,= I' f(rs.( expjinP)do. (22) path necessary for evaluation of the inverse Laplace" transform. Substituting Eqs. (25) and (19) into Eq. (1)
Substituting Eqs. (20) and (21) into Eq. (1) we have we have a new representation:
lip.0= '%r J ;-(0 ;p'Odu , (27)
36o" -where
X exp(-puw )du (291
2 7 0'F Here again the 2-1) integration has been decomposedinto two 1-D integrations. The optical evaluation of theLaplace transform operation has been discussed.2" The
2 25 second integration indicated by Eq. (28) can be per-formed by an astigmatic processor.26
Another method of optically performing the FRT istoo* based on a coordinate distortion technique. Coordinate
distortion in optics has been discussed,2 7 .28 and in par-ticular use of coherent optical IRT has been describedby Hofer. 17 The method is based on a 2-D Cartesianform of the kernel function (1). Consider the followingexpansion of the delta function kernel:
h5 (p - x cosOf - y' sinfl) - 1/2 exlji - X cos? -YIi 5"- X sin#)ld,.°- 129)
Using this representation of the delta function we can. . "represent the FRT as
-04 -02 00 0.2 04 E = ' cos,., sinO) exp-up)du. (30)
Fig. 6. Computer generated FRT slices for the input of Fig. 5(a). where
15 March 1983 / Vol. 22. No. 6 / APPLIED OPTICS 833
I
L
26)S.,., = ( ,.i,,,1,tpi ,t +, ,,(I,,, 1, those proposed for evaluation of the ierse Radon
transform known in computed tomography.Here the 2-1) Fourier translormed input is used ,as I Ti k was supputd in part under a grai frombuilding block to perform the FIN!'. 'I'he 2-1) Fourier tis w Forkce ffice oil Scientific Research AFr)Stransform is performed by a convyention.alI cohere it l tr 81 6nheH A r AI-()p ieFourier optic processor. Bi performing a 1 -1) Fourier enter F t9ft8-e rC-e095transform in the radial and imaging in the II direction o i enerthe 2-D Fourier transformed input, the FRT can beformed. The coordinate distortion using an astigmatic Referencesoptical process,,r muuist be used to perform tile radial to i . . V. t I'. u,u NI , chd ad N , t,,r Ieiti/ rI ,pexCartesian ciirdiinate conversion followed by a standard Pattern,." U.S. 'atent :.1 li9i. 4 W !!i2,astigmatic optical protcessor that iinages in one and 2 W. K. I'ratt. Iijal I nua:c I'r, ,'..i . ;r il I tr.., ,Iv rie. Ne("takes the Fourier transform in the other direction. 'ork, 9T.
Finally another delta function representation can be 3. D.1).Shapirand A iath.. IEEE Tran'. Pan-rn Anti Nt h.useful ill the case where both the input and its FRT are himtl. IAMI-I,:li i 14.7!1.limited to a finite region ill both the input and FRT .. I (). Duda and 1'. F. Hart. C,,irrimn ACM I-.,I 1 . 97plane. In this case. both the input and its FR'!' can be ,. I I Hcar. IFE l'ra,,s. Iatierrn Ajd Mal |, Inrit. I'A.MI-2,considered a periodic function. ('tnsider the following Is.', (I s I.
resolution of tile delta ' function: 6 I. Had,,n. l1.r..ari h- Akal i\' I.e ii 61, It 2119171
K. It J 1 , ;, 11. -Ir. i -E1': r.ir. Piollr n And N ilr 1i 1IAM I-I, s7 1,S2.,oo - AAl A, t .. , n %.1 '. 1 o . trA''//.. (3 21 He (od,~ {v hr m ;'' ,rimij..I Ih ,,r hl . . 1471
where . . r '. I II,, gril I, Y F,,,. IEKIK "r.i - Arni ,,-
A,, = 14,Ifn = o 1n'2I, ti x . (AA I Pr,,pag. A '-9 3 i ' t
li . C. 'Chase. F. II. Stonmi. M. L:rm--inierk,.gn II Igt r.g ,,.Substituting Eq. (32) into Eq. II) we have a periodic 'roc. S'.',, Phh(pt. lrh.trrim. I':n, 2:11. 2t;.'; i l-function description in the p direction of the FRT: II. it. M. Merseregt' and A. V. Oilit-iicm. Pri IEEE 62. i.w)
tjipH) = A,, ,I , ) 1 I 2. 7 t hrm a l 7' j ,/ Me, tii : ,,, Irrimnm: Ir,,.,.:, 1,,r2-) trd,-) J3 g i ,Ho lrI i 11n' r,, l J'r,,.u (1i,,,. II (tinl Jr i.t-t
where ,I Anerica. Washir,,Iin. D.(C.., 197,1:3, Z. I| (lr,. ;I:.. ittI usi n i, I'hv.in .gI gun (trlnrLtV 1r1 Mid
(,.) = e Ig, . JA CA uH + sirr(ilufxdv A-.iign is : i) Iimpgge Ietitongstritii w.. IEEl-:l-: 'I ri-. Nul I Si.- "l " NS-2 1 lume 171).
is the 2-1) cosine transform of the inpul. The 2-I) co- 1.1 .itH ,r.rt, A u Al m. 12 1" ." .1, I't itI ari, C'om l}I 1 I , I It v- 12, 17 -, 197.!1sine transform can he generated using the conventional I,. 1I.11. 11 1 rre I mIi \ S%'t Nl ii. Ir,,c I'EE -, s" I P72-1) coherent optical processor.." '
I6. F,. \v gr |han and I \V. lG, idman. I'r,,,. ., I'Ih. t, tOi. Il.
., Irn . E~ng. 211, 222 W ,1ii)
IV. Summary and Conclusions 17.1 . Ihir. ()li.('iiini. 29,"22 P1IIx. P 1dul .I.(.I~ it- .a dI I.a ,,I \, xh -M d H ,it.[
Analog coherent optical generatiion of the HT shape 2:1. Ili I I7s).descriptors has been discussed. HT shape descriptors 1I'.. II. V'lof/r and ii Whiun hr. l,,m r.,i It.,ii-ri, \ hlare one of the most efficient boundary descriptors in hh, g.'r..Ii M'l, , i,,'A/rnIh \m AI., i,,'w nrd I,Iu.pattern recognition. A number of possible analog imi- G. V. Hily. Ki. lSprigr. New Y,,rk. I97t I.p 117 12A.plementations have been descrihed. The.optical im- 2 , A ".L (;iiro, ,5 . r rlkanr. V. Sv mhigl. II II Ilirrelt. N.
plementations are based on the various representations '. Ch d . K (,rd, (ill .7, g . 201 21n8 i!11
of the Dirac delta (identity) function, which is the 21. It. I- irre. ()Ilp. 1, 1.248 I I ,V2. C.M (; flaid. iC I. (Gra,. and N It k Iti. ;(,,nfir/ih:,idspace-variant impulse response of the Itl' kernel. Since F.it, ( i , (M(; raw Hull. Ne, ',,rk. I; V.,I ".
the HT and the FR' are identical for binary images, the 2: 1. S. II S i.,ir,,. i ,, Ir' , ,171 i i tIm.ig, l'r, i- - . _ ,,Il2 1 75i
above implementations are identical to the optical 24. .1. Skl,re.k\, I:l-iran'. (rr,,,iir ('11-27., _'t tI )7,generation of the FIT. Because the Fl' of multidi- 2,. .J. W ,-,dinan, "Imunar SIat \:irt.mgi hugiial I'r-'-.in." inmensional convolution in the space domain is a 1-I) O(plt, al Ilti 'r,,, ,inmg, wuriarrm,tl,S. I.-.u, Fi (.Sprnrrger.convolution in the Radon transform domain, multidi- Ne, 'urk. I i8)).mensional coherent Fl0' processors can act as optical 2_. I. .I Marks II. .1 F. Walkin. .) IMgr. aigicr F mlT. Krte. AIy l
)reprocessors for coherent 1-1) acoustooptic signal ()I). Ifi, 7.19 1977i.processing architectures. Experimental results using 27. 1). (a,.gtgr ami I). J)'.,;lti. AIpI I),,i 1. 17 I .2-1) coherent Fourier optical transform lens configu- 28. 1) (Vi-;ii- and I), I-.is. " i.rr,gti''r, Irvnirgtn. S;'m-t
Vairiant ()pi ag I'alt ri It'r 'lk r'.hi " ii I'r,,zr,,. I pit i ,. f ,l',iration together with rotational and linear motion have Is. K'i. W i, I Ni rtldlamnd. Ann-terdam. I 17.s1been presented. A good match between computer 211. (; T\ r,is. .1idiat'ri,, n / Yl'r,, mi, an ,, g i tr,n'fin r ,t ti, WIn,,generated and experimental results is s own. A n11W. (At ahviii. N ',rk, 91.t11. (hap Sher of other coherent optical I I"T architectures are :io. ;. 1':1 Ri. StI. It S ri. tI IC lmg ,1i ,i'. I'r. ,. 0it,,I ) lproposed. Some of these architectures are similar to istririm. Eng.,. 2tI, 21i19!7!o
834 APPLIED OPTICS / Vol. 22. No. 6 / 15 March 1983
n| n I I I I n , i
27)
WAI8-1
Estimation of Two Closely Spaced Frequencies Buriedin White Noise Using Linear Programming
Jaroslav Keybl and George FichmannDepartment of Electrical Engineering
The City College of the City University of New YorkNew York, N. Y. 10031
A RSTRACT
Linear programming is used to estimate the spectrum oftwo sinusoids signals closely-spaced in frequency buried in deepwhite gaussian noise by employing a, priori knowledge of thespectrum. The method will be illustrated by a number ofexamples.
INTRODUCTION
In the calculation of the spectrum of a discrete-timesignal consisting of two sinusoids with closely-spacedfrequencies embedded in white gaussian noise, problems arisewhen only a small portion of the signal is available. This isknown as windowing of the data and becomes evident as "leakage"in the spectral domain, i.e. energy in the main lobe of aspectral response "leaks" into the sidelobes obscuring otherpresent spectral responses. There are many methods used toestimate the spectrum. The periodogram Il] performance is poorfor short data lengths. The Blackman-Tukey method r!] is a!sohampered by spectral distortion. The Burg method r3], a highfrequency-resolution technique, even in the absence of noise,yields spectral line splitting. The recent algorithm of Cadzow(4] outperforms the Burg method in a low noise environment.
In this paper, we estimate the spectrum of twosinusoidal signals closely-spaced in frequency by employinga-priori knowledge of the Fourier spectrum of the signal in theform of linear inequalities. The advantage of imposingconstraints in spectral restoration process has been pointed out[5]. In the linear programming formulation, there are manysolutions. Here, we select a solution which minimizes the IInorm of the Discrete Fourier transform(DFT) of the measuredsignal and its estimate.
DETERMINATION OF THE SPECTRUM
We assume that the signal estimate s(k) can herepresented by a weighted sum of past signals
?p
s(k) = - a(i)s(k - i) + e(k) k = 1,2,.. ,m (1)j=1
where a(l) are unknown weighting coefficients and e(k) is anerror term. This expression is a linear predictor[6]. By taking
28)
WA18-2
the DFT of Eq.(1), we have
S(n) = - a(1)exp(-j2iin/N)S(n) + E(n) 1 <n< N (2)i=1
where S(n) and E(n) are the DFT of s(k) ;,rv" ' , r*',. ivoly.Because S(n) and E(n) are complex, n,',] tu;~.,j 1.ul!r's frni7tj'a, werewrite Eq.(2) as
pSR(n) = - Fi a(I)[SR(n)cos v + SI(n)sin v] + ER(n) (3a)
i=1Sr(n) = - _= a(i)[S T(n)cos v -. SR(n)sin v] + EI(n) (3b)
where v - 2rin/N. We have 2N equations with p unknowns. When thesignal is real, we can reduce it to N equations with p unknowns.Assuming that two sinusoidal signals are present, it can beshown, the a(i) coefficients are the pole coefficients of thez-transform of two sinusoidal signals. Therefore, p = 4 and thcrange of the a(i)'s are
-4 < a(1)< 4, -2< a(2) 6, a(3) = a(l), a(4) = 3 (4)
Eq.(4) adds 6 more equations, to the N equation generated fromEq.(3), for a total of N + 6 equations used in the linearprogramming formulation. From Eq.(3)and (4), we can now solvefor the a(l) coefficients by minimizing the 11 norm of the error- E(n).
NUMERICAL RESULTS
To test this method we have used the time series
s(k) = Alcos(2vf 1 k) + A2cos(2ff2k +4 ) + w(k) (5)
with 15 k S and w(n) as white Gaussian noise with zero mean andvariance a . The two sinusoidal frequencies are normalized sothat f = 0.5 corresponds to the Nyquist rate. The individualsingsoidal signal-to-noise ratio's (SNR) is given by 2Olog (Ak/
/Vo ) for k = 1,2. For all of our examples, we chose the signalamplitudes A = y/ A = &- and the signal frequencies f,0.2168 and f = 0.224. . In two examples we introduced aforty-five d3gree phase difference between the two sinusoids.
In Fig. I. the variance of of the noise o2 = 0.94 andM = 192. This corresponds to 0.54 db SNR on the weaker signaland an time-bandwidth product (TBP) of 1.50. The spectrumcalculated using the periodogram shows random fluctuations andit resolves both frequencies. Our method yields peaks atfrequencies f = 0.2168 and f2 = 0.2265 which shows the abilityto resolve thl frequencies with a small error in such a low SNRenvironment without fluctuation. It should be pointed out thatthe reason both peaks are of equal amplitude is because we arecalculating the poles which make the function blow up. In Fig.2, we introduce a forty-five degree phase shift between the
29)
WAI-3
sinusoids. Here the SNR is - 0.2f 9b. We see that there is noline splitting as is often the case with other algorithms. Thepeaks occur at f = 0.2207 and f. = 0.2265, again resolving thefrequencies in a low SNR environment. Fig. 3 shows the estimatedresults when the SNR is - 14.2 db. The freauencies f = 0.218Pand f = 0.2305 yield good estimates for a very low NR. InFig.4, we added a forty-five degree phase shift with a SNR - 15db. V'e see that the periodogram is unable to resolve the twofrequencies but our method peaked at f1 = 0.218 and f =0.2324. Again we note that there is no line splitting. In thenext set of experiments,the TBP is reduced. In Fig.5, the SNR is0 db and the TBP is 1.00. The periodogram shows peaks while ourmethod gives the frequency estimates f, = 0.21f8 and f = 0.2305showing that we get good quality estimites even when tge numberof samples is reduced. Finally, in Fig.6, we plot the spectrumfor a SNR of - 14.5 db. The estimated frequencies are f0.2168 and f = 0.2393. The larger frequency error canattributed the very low SNR and the small TBP environment.
SUMMARY
We have shown a new method for determining thespectrum of two sinusoldal signals closely-spaced in frequency,where the weaker signal has a very low SNR. We show that themethod is not affected by spectral line splitting when the twosinusoids have a phase difference of 45 degres. Computergenerated results of this method have been presented.
ACKNOWLEDGEMENT
This work is supported in part by a grant from the AirForce Office Scientific R earch AFOSR 91-0169 and a contractfrom the Rome Air Development Center F1962S-80-C-0095
REFERENCES
[1] S.M. Kay and S.L. Marple,Jr., "Spectrum Analysis - AModern Perspective", Proc. IEEE, vol.69,pp. 13C0-IA19, Nov.1931.(2] R.B. Blackman and J.W. Tukey, The Measurement of PowerSpectra From the Point of View of Communications Engineering(NewYork:Dover 1959)(3] A. Papoulis, "Maximum Entropy and Spectral Estimation: AReview," IEEE Trans. Acoust. Speech Signal Proc.,vol.ASSP-29,pp. 117C-1186,1981(4] J.A. Cadzow, "High Performance. Spectral Estimation - ANew ARMA Method", IEEE Trans. Acoust.Speech Signal Proc.,vol.ASSP-28, 524-529, Oct. 1980.f5] R.J. Mammone and G. Eichmann, "Restoration of the discreteFourier spectra using linear programming", J. Opt. Soc. America,vol.72, No.8, August 1982[r) J. Makhoul, "Linear Prediction: A Tutorial Review,"Proc.IEEE, vol. 63, pp. 561-580, April 1975.
30)
WA18-4
o0
o
0
0.0 ~0.5 0.0 .0 SPECTRUM-DB VS. FREQUENCY SPECTRUM-DB VS. FREQUENCYFig. 1. Spectrum using new Fig. 2. Spectrum using new
method and periodogram method and periodogramuk ='O.q4 m=lq2 g]-1.03 m-192 W/ 4
C.N N
0 0
0 D 0.5 0.0 " 0.5SPECTRUM-OB VS. FREQUENCY SPECTRUM-DB VS. FREQUENCY
Fig. 3. Spectrum using new Fig. 4. spectrum using newmethod and periodogram method and periodogram
qO r" - .12 m=192 -5. 60 m =1 ]q2 *-TT/4o 0o 0
0
Cr.
0.0 SPECTRUM-DB VS, FREQUENCY SPECTRUM-DB VS. FREQUENCY 0.5
Fig. 5. Spectrum using new Fig. 6. Spectrum using newmethod end periodogrpm method nnd periodogramTI-.00 m=128 -5.27 m--128 O= I/4
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Restoration of discrete Fourier spectra using linearprogramming
R. Mammone" mid G. Eichmmin
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INTRODUCTION I ormu I at i( in as I hat ( i I it it i ned for D I, I inave rust i rat ion %k i Ill
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its a Continuation (it thespatial Foilnurspvc1ruill hevoild I Ill, lbough the reslorcd 01, object k llonllc .Il I\c. flit, rc-lol"d
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merical m0hods ot'DI. itriage restoration are high1v mislid, I v p(merful nonnt-gatiN.itt, contrimit kicd tit the I M, itijige rAl,
in I he presence tit iticastircinvilt noise. 'I'll(- resto .a I it .11 py o I orit I i o I I ('it it i lilt I it, d I rect I \ k I sed.
'I'll(, tit (it ivi I I it lit tit I I I is ruv; i r( It is to I i r( % it It -h lj it rt ,, 0 111 It '11Ces'; CMI be stabilimd liv imposing additional constraini,. frvfIlit'llcy ('stinialv%; (4 datil avallallic onk till All 1110,1111J,Many restoration methods art, availithiv. stich it., thoe it'
hin,-' Schell,' 11irmi(LI Jimson. - and Hurg.' Revet .Illservalion ltin,,Ili It * v lvmg Ow complit;1lionilk ( lll I. 1 1Frie( Ill\.. . I To Ifit. di,Howar(F demonstrattid it complit;itionalk intiNilit-iiiN. v list - Fourivir-I rimsform ahorif lims.
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obtamitig an increase (if' image rc ohjtion uing linva , r pro spaliA frukimint\ rt-ohition i himlvd h\ flit, unt, riiiiiii,
gramming luchniques was ret oilk, giveil hi, Matrinlone Ind princildc. "' The lack of lick I tit) rt-oho i, it) 1, ri-I'lli'd to I It,-Infillitt, limits of imcgralloll Ill lilt, dulillilloll ot lilt, FooriurI ranlkirm. Silicc I fiv Ituait, i, it\ 10.1111t, over it tilill c ollrr
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st ifat h i n i r (iji t x -~to I e, Il t'Isi itt- c st-', w ich asv 'ulgtie r, \S flit' zer this- i som one t anlI i,( 11.'2 -t'I jliise c isv li 1 ik
ltperai n, th a t~ti' ) its aI sitrank Httiist rtltll se i l'f is Ckm n s and,- one '',%iin 2t it' 1, i 2 , , ***,N ,i1, ( 5) I(
Ilchi er ept rie ttxt m istatii used lt ' sfite juil st'tf lt iifll seq'uit e il' itttrti l i -if tpo s itiv flit f ruiti ist m o iii itr lii
poshed, ie .,en 'tart' r a l d fitcit'nt. theeii'an tth't if flit'nc tt't-fillglt't iik+ \ real.
11 Afalmrntt andi~ G. F), itittit \',.I '72, N,. W!iigijt 1IM2,'. Opt l Smt Ami 989 33)
where, if %e diecompose tile tiegrodioliti matirix, then cle('nits will lhe eqIil to zeroi. Ili a vettr-spaee interpretll-
A' It lioni of 1.1', thec incIiliii tieljint iii-trtilg It yperplaties,
ra d i e iona.tl lit lie or it lt 'I'll-sia ll i lix. itt
h~er' he .ni'rt'iit t eI e rtt'~' iri'.llili t ll i te po ,1 ii tile tirof itilt .c lir p c lii T t' t t le es ar
ctII (-- - - Ilc \ '511 I t" iIIIvI-v1)1 1 ,r iIi\u :iIi ~ it a u
I,) li-mii mm lt c s I kinc -~ p er lne u I ( alwhere s a 1. I r 2pr+e t I. f 1, It ilts it it 2I. 1,12iit- c ,ii l i mjcx Th or nae of ad v ;(-"Jltlil11 Ciit i t I) litre t lie 2 t-Ie lil .it2ni I rIlsd il , 4 If ti i'r t ) ri fm hvN ')''t Nmdv -niio( k ;t l vI(1I
Ie, Midait lare tm irctlto'alming \e 1.1') Ititti l I iir flit-illlll n i lil tl ,1 h m \ith aP -t~ti -l o +i tt trhitrVirv e o an itili i h\ ~ It mi (:', rvp- iuv n tli d eo lt n prgv , to__d~c-a d t l tI
t I I A I ilt' ltt l 1Iith rc li tll\ tit rt i(- t t ero r u ti intl tai tc
ett~slritiltetir. nd in X~~t tl'.ftjll iiltrx.nilt iiti lt- I'tj W) toi li. pit't ot tirleicoinstraint tueix 'Ispec
wher tie mpersrip R enols r~er , oncr.the (arm tvc1h~i al flt- \ m.J)vs rv n mm-atie, ane tssa
34)
lisheti compluter rouines are avail;1ille. '" I'lti* 1 iluliv
5sIm itl tliitionst skich as E'll. Ii ). Fhr t.\iiiillt Ilie
mum Ipiirvs urrir. Howxard(I has 1cii'iiiiiii r~titilstratintil luts? siiares aipplrotach lor Owi. ilil irlbiriti Ii ,r
nomIticmii v. lRusirt 11 t (11di.cx,'.cIltiil this ,tpijrii;a.'Ii forI
It'rt,, ire maximum ~ilikelhoodiu estimittvs witi th wii iis .tixtlY." It was funld, hAtwex'er, Ilit 1(r thli1- iwiii situa- -Si -
t ion of' interest. t he pjId' t' t he notise dhuii rit ajiprti'ial liv alt ?''
1ritirmni tsisttttlvo titervd thelovst tstimate, irri'sjki.'tv to Il i i2i 'i iii'' 'i"ii-:itK
file noiise pit. TIhis lact Canl lit attribled toi the, rilatilv isiit' iiiriiiii{
hewermitumrical oijerit institeeded istwill as Ili t li st ri iigett
bounidtis onl thll ind~ividui~l spiei'rat componeniittts hor Ilic 11 ('s-
eraiitis. Alsii. tiliniziiig tile I- noirm iscqi\-tinsiit il mini-
miin~g the tiital notise spectral energy. W'heri'as I i i s- ihtesirattle t'eatiure inl genl,('l fiir the purpliill it o irt'~istt Ire-quenilcy resitlut ion it tends toi he i-iiiiterprtriil~cti . The
111 ltl Itizaloltl iif The gloial spcttral-errir tiiii.elii r,,%- dhis-
tributioin tenils tx-ens' it) snioiii h tile estinmte. leailing toi a/decrease inl frequency res~itit in
NUMERICAL RESULTS
shectiiserigur S Ittiti if.tc niithet seinad ut tl (I' isiotI /
all subtsequnt figures. thle ltoiviittuie ittuI'mirier sitvitriti -~ 1-ret1i 7-l stittit t
is plottted. The doftftti linte rejirtseis a 12 pintt Ill of' 23t u~lfim njl
samjtes tf a iislt- quen5(iiiit ivnre toI is fill iimidiult;Ittifriieouec 27r/32. Heri, utly s~een dlita sinitles lis'v, liven actual ,pekiI rut,,. Figiire 2 illtistia isc Ow Iipurliritii .mi? Ilii-
replaced biy zerois. 'The soilid line illustrates the exact spvc- I1 irniOw l it nv 7 if the it sattit' dl tlti Illitimii l
trn itt' thle full :12-sample seqiteinic. The Lack oft resilhitiut eilsittusii are availahie. 'Ihi uhitteilo hu curs e ri'lircstmsCtitceals the fact that only the fuittlatental 'reuwiii\c is thet txerly smimiti estittitd it ft (firet I sTi-ur rifil '.
prnesent. 'Ihe si-im f the FF1' as,,well as Ilie niunler it sittithis tiimattl I1 i-st inat-. tli hut-lttel hut, is ti tii t- Ilit' inigitiuaiciitfireil is knownt a iriurl. [his intuiriiutiiln t~iie i-%il I splt Irin. 'I'lit tIilo line (u-ur%(- hti- i. -jlde-k- in icl
thle houindso(n the amplitude spectrumt. is used toestImate thte siilit-linie 'itrxa' 'Iius a ,-idoehit ri "tirt? i is pi-ilh-\41iten unkv 7 ii' :12 sanu I-in- iari avlIt'. I-.gur ri ilir Ist
sit: txit 1i itlc t 1;ie txt imillu irt vq lvlle I c llu.luu iI-Ii o1:
ti-ivt n OwetSclidtiixv1\O ~~(i o
suutuiulu) iofasti Ow m i ,rsn i-si iO l'iiIll.. ili
lit ttrut-r tit aahii oif2 itt r(.ihiiiii-it is idiiaielh ext-l it, tilie
dasheil timid tatd the exact l)Vl sjieitrtiu usll filit, tevi, I cttr\xv lsolid (u- clsv. Ili ttaii iatotho'itit is it
I'P ' I Allot Am 99 1
I i it. I r I. 11, - i I % c I lo, i i, i v I I I 1 4 4 1, 1 loot W [I I'm t-d ( lose
lot H., pcrI,,rmahf- tit Ili(- 1,
ok it 1, 1);f 'I Lt. din
I'll, oil I I it, I' F I ,It - tilt- dljff, (I -p( I IT till It'd, ln-d
k I!,, Jmli d''I td ,I I A lit, t.-iiiiiot-d I -iri, r -1wi, trun,
pl, d -!A :it Ow
\,I, 1 .1, , Ifulli ,, t IL,1 till. mth-lmgui h-
SUMNIARY AND CONCLUSIONS
-I-,;I !k it Ml--" '1 1, JrA'-, 1,11"i A I I,' MCI I,, I, I d IT t't d)
. 1, 1, 11 1 , - I - I I i - I - , i I I , -I I I I . i t I I it I , t - , , I I r I. o , ( i . , I I I I I t I I I e I ) I . A ,
) I I I I; Ili 1, -ihi, t tri- ,,.nIrjtrwd it, halite
r, I,, J.. . I I i I i r I rt-i I , it- it, v ( , -Ili I)( tie n t ,
hltc I %miallo I. trik jpjor-ik It li.i liven tak-(-n,lt it ,rt-d lot. lruri, I, 'I. j", 1, d I., Ol,
t 'I iit rr,,i i it-rin, .%t ro, d t-, i- . d I tit- i'l, I... and
Ill r 11 11 i- it vt it, ii I it r I I i i I tit at Id iti,,ii (,I o lier lint-ar
-. [, I A 'll.l.
-:w 'I'l, I,,- I Ili q tl ll it .1 ll lr o low .it I lit- (-th cr si tie
j;i to- oid , I I :it- jilt jo;,ion .: t )I till., tN p e
!).I, Ili I : I I Illid t -Iti I iii/f. tilt- -,th o.fo, ;-c lilt-
\M111 I l to il)!, do I i, Ili, rUAI(' Ili rc oolulitln as
%% , ; ; ,, i I I . I i i ! , o.. it i I I I it I it it, ii-v . I it ter1w , 4
rt t lot, norni I%,j I I. I. onsi t eiltl% the b"t
r"l
ACKNO11TEDGMEXI'S
'I lit, pip( r oil A p.,ruoi ,I it di- crtaiwii stilinutted
/ J1 1 h N .1. 1,, I!x Cit\ I*mt r it \ od Nvo, Y(,r , in
oil (-,, I ho:n I ir o I ;it 11w 'I"oll i pirij.il Ildlillmi, ill ,I' tilt- To wirunt tit, i(,r tilt- dugreeIli ciwim--i iii . 'I Ili, %k.,t %,i, kipprit-d Ili p.irt undur grant
,I J 1- 1o, d ijrv iiiiii, ill III(- in, -iipl, 1, J-o r'. it OM I I, A F( )S H, ,, I I) 10 ) 11 ''lil I Ili \it F oa t I. Ic(. )I it'litilik.
IM"llilril Ill, lit I rr,,r I Hot vxtr.iI,, iit,! , ir t o, l-,:i dm d lit,, kvvm It mid k onmi, I o,(1-C oo.-) tr(on the konle :%IT
mv jklu.0 %oavli.rin (-hd 1111v Oj%( I , !-,k ("cillcr.
* I 'rt -cm itd r( -. I It I iti ri munt (It Flc( I m al F.tiginecring,
Hillk 1 1 1 ii;%(-r W\.
Pwi.oi, -plicriiiid %av hill,iiin, I i.ow% I.' B, o3
2 1 t u r,I r I I I r -
to " ifill -m oil tilt oollercill
I -i t., i i ii iw I to, to oil% im , too r
\ 'I , .1 1,\ , I . I - I I.- ,
i im ill -t p, ri," .1 )1 1 So A li, im . -.,o) I illto I ttll , % 1.1\ 11,ittlit I I it 'p i i 1 1.11 11.,11k -I- , lilt lilt 41 .11
III I I I lit I i \ ;,w i. I I I I I I I: --I ,, , I o 1, -r. I I i I
g t r, I I to t i 1 2 1 1111 0 1 ,, 1 11, 1. , I - 1, 1., 1, '1 ( G 1. , I" Il , , .
I To ali 'I'm 1.11 0, am plv pool ill V I I I I I - I i I 1 0 7 1 1 1 , -" , I I . , ( , i , I I t d I , I d I , F, I I I, i I,, ri tim g a
4 1, d I I I th ,bcd I ;irt,. m d Ow rvi-,i t d %% I % I r i I i I I to -% I I I till 11 111111111 111 1:. 11 1 1 A 1 7 1 s
lir,
S 0 hN- I\ ~ i~ I i. i. I ,tI,*-i T ~ i i IrIr Ir If- I- F I r.wru . wi '.AS 1' .* v .I
r r r . I t ) S o , A i l 7 , rr1 I , I k Ihr ' I rn .o n xr t h .r r ' .1 Il l . d . r .
rr -i ti~rrrr rrrr ior rrrr mr i lgr.- Airii Opti 21, 4-Il rnl 1ir:116f A All- it 1''..
Ir Ir II rII I\ - I , rI nri r,411,, I IrrI rIIrI ) i - I~ li' I' I wI I ,~r~ II ri- hr I, Ii , rI In~ r I t
v-nnI Ir .~. I fi \ ir Ir Ir , i. I- Kir IvIi ' I n I ; ' I ,'. I ,Vc. I 1 .1 NI 1,'9rlr , Iriflirrr: rinrin iI i mnnir V.rrn rrr nr in, \fi"11:~5, nnn
13. 1 1) A. M, damr .r s Ill i r,i (t tI\ iIrw(. 1 ir1i1r 1rinrn i rr~r rn inr I i !1. rr ii r ir - i i
vI.l l" 1 0 . - ~ 0 ' 1 :I111) 0 S % " , ' . 1 q ,! -111. '
11. I . A Mat aridi, a d W K.I 'rt t. Iligf~d c N att. I m It e-adi ,-,m wn. lu ;- .1 ,4
Two-dimensional optical filtering of 1 -1) signals
G. Etchrnann and B. Z. Dong
I. Inrdutinir ;li,, rol ir unh i,,iid p tu cil t \lt I- ii p- tw . tRe
(lit ('%l ,vtel lidS l)'tt 'Ij'tl 11iiiil 1s I t iti s iih' t\rdI t- I k t, 1711 1 1 . hf , ulilt , I he Iait too ; iol n.Itr rvii te p tnilhrj .r1l g. iet di"j'.J, I .I I 1.ir- q At 1 t;i IFr 1, ii i 't I I r, -lI inoa ri ilsi t -olia ii tlr0glp i~ l ir t I I -.1 ~ it ii \ oIIIII r hI.II 11 , III\ I II ,x1,11 A I hi i Ii, r ,t r ,iire weit I le-tirt((I'l -l iti
,lpi l, o tilirii \%iiel I I , in i .i I i la trge : m lvI- Iv,' vi 11i %i- .1li" . rt' p i it , , I l i t \'n -pg\- ti p Itwer
to Fefin r,('till -r-- l i0 tI ,I I,- '1 ii t ii n t 11 a v reIi illv ,I i h ,o i ii mi ii II ,' i i i it q's. te \, ttinti.'I tw 2I 1i''iIl 1itdii o1 th ttrult r I~ I)f i tlit~n I;, ii I, u I witi tit )t. i 'ii ,tI I Iin A tf' WlsingeMor reeitva are itniiir f jiii 0 tri it lr t- \-I -w-ia r(l s iit t i .,I lt Il v i.al -n sptial' axisoptical I "rIssn tteato s I-er., 1-tidon 1 1 -1 1i an 1 lie ei1', or ,11t ii\e u rt ruinh d f thei thr a-ttj1 x~i.
oa, av Ien ddd o h rpetit iI'. htr sii,'l a, fri-giii: Anothenis r ittrrti on of lte (1) whlich iSF siar tore
1.IntiY-rocion setrd It(ri. 1 arale pt soi t all r,ld ,it~n i,,i th tlslltlintirrl on representheligi t t .a i il g tis iitnel lisir w i si p ica l itiivii l it if 'li nc leais ;I1( thle i e iri r liwr se c-n
as arius -I)sigal ra isfo fi-ti, 21) rafn~ hV tifi itli lit i til hoohr axis, Ht wosgi repreisetstietglirl t II la li )i it xielt tutoria lilpae iat art itt iileenl'it 1110l.bt ~n l:iilfit
tptiea provcdii, e ss)(it n i gIvfo t eal Iy no'alii it ar Irlit-i i i' (5 i ter rt a it i- It' Iiss 1' lur i siit ( i onsve ()tf
cI I)(waI sysit n itis a re sit - rs mailsi i I' h e l to~ ai I q-r c t si cii \ ,li lI .li eases- it i it I litgil ist leoll(.
'ni at e ttn ane brug p t 'Itwo m s i let i 1 or % i it ire t ir I) hoiti I %%I , I t- lii I icI\i.c t rwo l, ii lei tit c trlatinials suh as eiiivolutir iirpti elat\lc iii l ite the wht taes p re omlreltan ti hy ic ,, orr 1ti ()about
II w rlon signals. a tiu lrsis litI r -ii tat it he a lt re rsi-I at sglii pt%, i I oIlcd lto tc.tw W glitte hndwdth rod ct ll' iv rde of t m llin. lit tw itiii axes fiitton( li e rIe.T h W~ttiri'r p i t in
opticalprocessing~~pi'tii-ti oprtos wioso n - sg n itt( cl l ie Al"w tlspivetu and ti f oli o at ion,nal hivelicl)addd o tlt rtperoiv, uc as1'e- nhalersl it -i)st iin ofithering itt i siiar the
qunth Yviat ipc a itrm .I\aibe sili l I li AF, i;l i t, yte iittlel ii laldwt Ii rctpr 's-nIItt-girittIFIirl, a dgh lifia d0152n , ' shut well i tetli(1 ol( 'ii s iills fil 'V'llr ' or ill t r li e c-ittltuiuiloti Iie-Dstigna2iel it tlorsiml i truneaion li itt' flT e t'iiidiiitiial lmttt oand Iftitcv A-tt vxvvIey itt Aiiiirii- a on spo1W-ran r tindit'i i .ri'ree atcl lo lhts.mdt
3152evn APPLcED po ein is Vo ven 21. No v? I akil. ar Ssptcmbe 1982o' ,,nl lcas l erseso
38)
11. Formulation the Iliagittitlt (.1 the AlF is (tiriplett'l inlsvrsitive tot
filet -vncrilI/vd( spot'e fretiit'iv Cf 1F 'irt'~iti ii tlr.
of -I)sigalsiiirteliit'dby liei ItissigalBudt h heAl'mii the 'A!)art'spet'ial lassitt othe (;S*isde(letIe(l h Ythe triple integral (listrihiitioni(' nht Iht*lfiitiii(iaih ~
'i Ii = 1. isilti -1 :s puJressedt either Ii ttriiis of (te AF" is
X. Un it 'l i0e~ ." .111 = , _I .I \ -1 : I~ I:~h Ie I: Iti"n
where or inl ternis (dl the WI I ;i-
where 1(x) is the 1I-1) signal, the represe-nts oinplexconjug-ate, and lIP.:) is an trliitrarv ketio'l hint-timi. he~itrePart icuilar kernel tmict ionis le;iil to( i itftereiit spaiteIfrequienicy represettat ions oif It t. ),Two ut the most I5 i1' I Ne-e.:~l eif d. 0121
imiportant GSF1 representations oif thle 1I-1) signal are tIlic it scialed 2 11 l'oiirir t ranni of the kernel ltimetioll
'AFl an hIW.1. 'I'll( kvrtiel tiliti lii b xii ) rt'presenits a1 2-I) ini-TheIA is dlefinied as puilse rcsliii-e tilter ao tjog (on thle sjt:i-e- reqiitnt'Y de-
A-to I ep:P 4= t ?d hi iiln t . i it IlW I ad int, Ah I re rela;;'d li 'v
An alternat e retpresent at ionl oft th I iA lii' ( ' x pressed( as a 2 -1) 'iItevr I Il I lIt( A F".inl ternis of I te IFiitrer ItransftrmIn iffI x A:ll all -pass liliiiIg fuinetion
Jii~~ i)I i'We-' iIt itIrontsliirni tIt he I' tuntu- ion ('into thle WI), Anl im-
The AF tan now lie e'xpressed iii tt'rins of thie Fouurie'r pitlsive kernel fitictiispectra as 1i. I u sl%-iii I
Aii' (I 7r u tx~ie pie j r Wt, i tratisfitrminu ti Cet 4l' fitiction C into the AV, Ot herf ilter hint tills haiv ;iliteareIl i t'e literatuire. For
where vvamupli' tilit' alt ass, htil tr
The WID, another (5F represenitatittn, is ilefilietl as t ronsfturms thle ( ;SF iisiriltii C inito an energy dis-riliiition fiitietin. 'This fitt'r himteloin \w-Its first pro-
~~~trei~~~) ise ibit H~i .5d, ~'~~ i B iek. " Ilii entergy (list riltittionl flinti inhas i, ritirt it's simiilair to thle WDI). A\nothetr filter
Similar fto lie AF", thle WDI can he hmiiu frtomtl I hetPool fuitit ioit is ai I aiis',iai kirtuilrier spect ra as W11e I:I~w - i
The two rep~resentaititons art' nott independenttt'it I n f 'tIrg Ilsrlttio ttiih'siintht i hitii(lie Al"ant ile WD) are related bty a staled 2-) F"outrier tepiauii'si ti tiiaIiti-ia atn itlle
trnfom WI) will alwaYs lhe real andi liot-nu'gttive. Fuirt her, thistransform:~p tYIi' itt ( 1 list riliit ion is then tonsistent with
Mui,t I (f)) I~txi-i- t:W uiet 1lt'isetht'rg's uiii-trtoiiity pirinctiplte.
Thel WI ) of aI I -I ) signal tanl lie ittrpretedl as thle dis-ribIt it in oft s igna vsIito'rgy ver spacet-i andi stpatial I tn - 1ll. Two- Dimensional Filtering of 1- D Signals
(Ititntv. 'liis a 2 -I) oflisp1la 1v I)ftIt lvt WI ) rei1)r(,sent Is t he 'Thet (S" dlist rihiit itl tan it', initerprteteth lereftoreenergy cotitt (if the waveformi ltocaliz'et l i space atui either as a gverali/i'i WDI (MI) itr as a gencralizetispatial frettuene v. I however, this interprtatilt isnot AF" I( Al". While fit', present tlistiissiini will deal withalways true since thitre are cast's when' thlit WI) is niot I te (W1). wit lititt any loss oif getierality' it tan ie( car-piositive,. One ofl thet most inmportant propetrties (it' flit, ritil over to a distitssiion itt the (GAI. Part ittiar fIl-WI) is that a shift inl titier spaet'tr spatial fretttt'cv tcrinlgof the I-I)sigItl/t'xf leadtl oa lew sigtiil /s(x .of (lie I-1) signal leads fto aI firtjirt iotil shift inl t0' 'fi'( WXI) t e flt'tw sig-nal is relaited fit) the W~ot it/o)
corresponding WI). 'Ihiisis niot trivi fur flit'Al" whitr' thiritugh i litear 2-1) filter. TIhis (;S' filter is the WDIsuch shifts lead toit addltitioinal phase factoir.i, Ini fact, if* ft'e I-I) filfer fhint liton.
ISeptember 198? / Vot 2 1, No 17 / APPLIED OPTICS 3153
39,
Foir t' iiiplt. (t, 1a -1 ) linear space- invariant filtering I .oi C t is retIirest'l ted I as
xhere' I! I is t ii' 'j I;it iail I In fit i>w rt'spmuisv. 'the WI) I oth l itIltitr il sigllal k I is
sihnrl Wi I lk tlhetW I itit' illn, ii t ait r Il n to it 1 i ipt i I'l l IilI Iit'l '111
I i) S . h ie WI l I (httd,4 i Ii iil-- n im th easi, a fi i I - I hilt giuatiith he I ) ofm thke
itslt ir 1 11 1vi-liIk Otal Realizationst h W of 2- .Fl rig L 1";11v 0 Fi ',g~ltering of 1-siglil il IIlesp JJ d ii .,(1I ai s Signa1lslo"o, lt
' I )ill etatn tprsent ii) liltigi! iW)iite Asatn )til irtetjtclraiaiistCte
spatial~~~~~~~~~~ frqec lihin ir gi t, sIleW) tleritigre Ii I it-lw signa is lit large ; ntlnilitr ok tipticalFtiC ~ ~ ~ ~ ~ t th ittl pia u tIMii, ( x . '(i9) 11(is anihr elitns Io t' 2l". I'l'ere art' thlerWDen the
lead I ti, lit sina sx 1. la In' ehrex ti).itg~tll- ntir
Thii qtit nrpee nt Ii -f)iltering tali'oitiit li Weiging nie A tr iII t fi eI lieropt icl rlit itie antie 2-D0
thle signal with a slidling windoiw finiet iol intel'rat ill,, )2 vtihertilt joint Foturie'r transformi htt-lttgriipliv2 'ias well as tither IjethI,(lS2 'iaalait to
I.li Il ~ii vo11) tit it dli lili ' v thit A F. While all thle above impjle-TheGAV ) f, il sillll (x )i liltilt i lls, Im Ill- it Iluifictott hile ilsed tit startinity
poiint fir the 2-) filtering it' 1-I) signals, here we tdiscunss1 in( Wtt I~ti ,~ -~ 12 suni exltitiklunnial stiaceintegrating cohlerent tiptical
'Ihe Iisi ciieen I it'a realizationItiCtthe W2)1iIt~~~~ililliililol of' theseilt' faltiesm opfehrations.ilytilt ie~V s
we obtain at newv versioin ( XlI) called a lisetido-WI). showinlilli Fig. I. Thle fititI ini lilnt' P, isThere are tittier filter opera! iions t i!canl itt i('hrlit'ti
o n eitherf /( otr onl 1"~t) that will leatd (i GWI). =j\ tit[\ _ii +.%If*lh&!% 2k ), 6
tlefiniing the filtering oiperationrs on the 2-D) Foiirit'r In plant' P2, wev havetranshirmneti DX) 1 alistit iiii ept'ntding
ilit''cy oir spacte spatial fr'teuut'V delays, either (W1)
or GAF tan lie gt'reratted. x i'\pj -jk/2fis - N ')t'Idvt'dv< (27tAino it her 2- 1) tIlispla v tif a I -I ) sigialI is t it' sl itIrt -spate
P'tirier whetOm(F).1Ti rnfrmi ivi vire k is it', waive niluier andf/isilhe total it'igth tiristin I~~''i ''i riitriis iti the t.vliitiric-;i lteis.I)iei tltyntraleiwehe
at liirit'r t raisitirn in t'e x dhirecttill miit Fret'snlh(,11i iXpil-j110lt ,0ds\ t 2:01 kernetl t raiishiir ini Itle ithelr dirtectin Thll lil' oveii
~~~''~~m lil ega al i'ixpri-sseti us
where 1k(x,'v) is tiht esliressitti give I cI v Eq. t'2 1 ), lT'e CA.spet! grain Stx.ro') is obltaineidli hN ti g the squiart' ott C~ t i it-i)tut-I21
thle FINiagiittite ofi thle 51' itir all piissillt windoltw Jil- 1
I iwc'ver, tile spiectriigramn tan ailso lie exprt'ssi'd as htt illiilitl Ito aI iodoclt tit Voier traiisnitim 'Thiisin liai' P., \% ehiv thit At-I" itilt ilittII lv aill kladrat it'
3154 APPLIED OPTICS / Vol 2 1. No I7 1 Septetmber 19 82
40)
1.1t(-5 tisitig appriiprialte tSs' fgue' if
tlie nlagttitttile ofi the tltered WDI , filt,.red hi.' a~ I'res-nel kernel. andt th. ImIgiittiie of tilie AF (of
five differenit sptatial friiittehiii5. 11l11t, fil-teredi WD) and it-, Al' holi a larve iinu r of*sidlelts characterisi ik (i lgral jug Iatterns;.
a l'ipure :i s IiiVws t he' o pica iim ciit n Iiiie ionii (if a
thtird replresenitation olf ilie 1 -1) signal, thet Ilihac-zek or energy ilistriut iwi. I lere thle I A ) FouiriertranISform-1 oft hie signial is uised ito illumina1:te tileoriginal 1-1) signal miask turned 90t0. UThe first
_______________________b
b
Fig. 2. Outputs, in plaot-s I', and 1). ot Fig. 1, of a 1 1)I iiiaik con-
tainilg five d ifferent splatial frequieficies are gIsiiin: (a) amoplitude iif Cthe modified WI). tb) am plituode of thle A F,
Fg5 lik'oil-t s.*rtiii Iit tilu-n .40 liti ut nmiak, in plane
P ,IFig. I. I.0 and (Iti i's~ It- t1hilkm titei 1 -1) nmk (-i,n-4. i~~~iiiutiuig usis 1t~I ruii ,- 1, 1 i-u nn ( resnit'it this tilik-itigon 0
a t) ask ominiiinuiigt ),%, . I %ririi c st iiiihiot dc
Fig. 3. Cohterenit ipl icat iniiiletiiuital imi for I hie giiifrjIi Iim If I hie 5Iwctral io-n ent,
energy fir ltibiozek ilistrititiiin. 'I'lie 1 1) spaitil liiirir spiiitrufi
illuminates thle t- - igiiai fiiask Itirnedl WO. 'I'tw two spfiirical lenises
image thle energy dist ribhution.
Fig. 4. Fne rgy (fist rifhutiimn (if ai 1-1) niaisk cion-taining live different spatial frequencies.
I I'eft ) Fig. 6. Coherent optical i IIplerneiitatiiin of thle SF1l'. IHere (li riuning isififl ont tiii wvi-hs tie icomiplex 1 1) Fiiurier specira
.f the signal. (Mliddle) Fig. 7. Ia) Spiectriigramn fi tile ilifil-ueu I reipiency ramp.f lBffifse of thle iruiiulsie niatuire tIf the filtering tisspectrogram is termed local spectra. (1)) Liocal Do)ippler spc tra iiftw he i iifle ided friiiiency rani Mu i Ughi Vig.s. (a) Lical spiect ra. (b)
Local Dopfler spectra of a 1-1) mask containing five different spatial frequencies.
1 September 1982 / Vol, 21, No. 17 / APPLIED OPTICS 3155
Mp 41)
input is at phase mnask. Tlhe t ransilltiminatetl mask is processors all thle above ( 5F can Ile displayevd ats wellinl trrn iniaged using two spherical lenises to plane P,~ as miaiiulatid. Exliirimientil restlts, using coihetIII plale P,0~ hit imagnitude (it' I he ciiergv (list ribul ion is p;ice-ilittegrat ing" opt ical pro('essors, have en pre-recoirded. Figutre .1 slhows fill he eicrg tistriliititiino ita sente&l ( It itr tNypes-, (it opltical procsor ciiii also Ile
signal with Iiive difflerent spatial tr1uiiic.nilitwd In di.splay th. e ( ;S1 'F, es prnu essors v, Ill titiul'There are at mnwir of' cohereiit opt ical filtering op aIpplic it ions whirt' t li revil-t iiie display ill tiiie-Ire-
eratins that can hel lierliortned nil tilt, 1 1) signal. F(iir i t- Y i I ItnrI iat i i )Ii is tl int(e restI,.ex~nile usn1h aearneeta nFg wt Ii work wssupported ill part under at graillt rni
thle exception tit placing at vertical slit nil the( twot i ia-;ks L:j ~reOfc fSinii ~sacuAt)l
canliel realizeld. Figuire 5 sfinws tie results ntl this Id- -069ad;(,ltrc fotheRmAiI)el n
tering" operatinil tw o iiid live spat ill Illitiiiitit5, C'enter F196hi28-80 C'-0095.
With iIld withliit dt' tilt ering ill tilie WI)Iiplane. Lresults depict thle Itiagnitlule il* I he itiiliitll \). Til Referencesnioilicat inn is Iliet addiitinal I'rvsnel filtering (Iti ithle cylindrical lens. Note t ie lnuch redliced ettt t i tLtirvI-, i;
F'iguire Gi depicts thle clivereit optical illlllellleltlltilil Il (~g il ji. (1,11 E,!" t t--op., i'l Iur~I",,,i'i~-n
ol, the '41'I. 'There are at niMuther of wiiidiw tuinct i(Is hpivt t '1 (il F"I" INI t --. ' itriiii, pdIiscussed ill the lit erat u re.l1 Inl iir expewr i tuelit we ised S3 123l
at unitoni gate, aipproximlatedl 1)"v a1 45' slit, ats tile will- :I C E I'tiiii.Alipt lit 5, 1 852196,dow functioin. Th~e rulilinig filter can also het placed ilt I Wi'I'. thu-i- adi J. \t tint-nw. Apid. ()li I5. 30173 0976).
the Fouirier plane ii aItteVct t lle Sl"l. Pilacing the run- 5. J. W, 1ilil '. Nivilmiiid, and E' W liiiApp] Ol. 1.6,nmn tilter in thle Fiitrier iiptivcl planie (Iii the 1 -I) silil ill 1 177)i11Ml e i v lit' .ilailgeouis it' we wishl ii reallize at real-tillii.e.\~i( i .t.Viti~,N.U hg-,iilI N ~ii ~p
acitstotitic (AMI iinpleliietatinn of' (lte SF1'. since ii . i I t)~i ttIKN iS. 7there are pih\sicil limlatioils (it lithe size iii' tile active S. . I, n inS1, iiitfl1l.It~-.ii,area that tihe presenit AO) tranls(iucers will suipport. thit n;l tliiJ ,' X u'Vuu, . 1-:1. (.Spnrifige-n
4* slit will dt-grade the perfirmnlce ii' tle SI I ["-r,,(0siPlacing" the Slit inl lit'e FnUrier plane, renveAs Sil1t' ot the I W Opt, Eni. g. 19I,3:;9 19,101shutrden i the A(0 iliitllatiir. Thle 11Magilittl(Ie 5(qua re 101 :V K, N Sm ius tnd 1) C C .... r. Prt hst i-tlt, r -ig. 120, 4123
olit the SF1' is the spectrogramii. BY placing a siquiare-Ilaw II l;
dlevice inl thle SF1' pI~iie, skiitI i as phoitographtic t'ilill or I I tri It U I. Wulku. ini tF. . r.ili.,\1 I- (lilt. 6, 746;,at gillllil-ciirreC'te~ videoi camler, thle spectrngriu it* I;:-I19'1
the I-1) siginal is gteneratedl. Figure 7taJ shows thie Ii, \k ll. AiI llt11 7,i'sP'.spetrogralil i of iiuille-siiled t'reilincy raip ''ie p.NVii:n li.tyI
hasbvieil termied asteIclseta t h -D)signil. p ' si
It is poissible tii generate spice delay aini spatial t rv I t 1(CtIw il. I \til It 'liv, Ciiii p-t. 7,i' 1 P466 Ts(juelicy delay spectrogralms by iiiagiitli (l 51;irijllg thei I' - A H% 16lik, I'' ip; 11.0i Ii~- llai;'i Haudar2-1) iiuirier trailsirin l itthe SII. liguiire T11 I, iii b)itl tShoNiwsthle (lelav spe rigrain (it' lite dntihle-sidedI trequeilt y IS. N I; I itI ti lull.iii N it i,' A r, i Vik -kiiil 2 1 2 , i I97 I
ramip. For allitipuilsive rutnning htler tlinkctiin liti,, I9 A Wi lii.,, -,-k. tiFi.i Fruui- [iti t'-ru Il1-1It. .;6tIlisI
spiectral hnll lit' terllled ats thle local Donppler specta i 0 N, I; tIP. ii-n.lii\ Priti-p iti in pinrn Aiikttn' iii
'The local lilppler spectrai does inot have lite-I iany idi'0 t~. lilt iA~td-i'. N.\ Yt. rk,
sitleliihes chuirac-terist ic- oI' t he gliibal I )ppler spectruml. 2 1t t . itt--N..i-
hFigire ~siniws ie local t'reqlienc *v and I111 hiical I )iippler 2 t2.pn rsi--- 1921l'"
I'retluleily spe'tra oii' tive. tlittereiit spiatiiil trequlellcy 2: 1. P, Kvii.m. in,. Sm 'lit,, (lHi. tiii-rini i-lg Is.', 1301tiiiasi-peritdit- signalls. Niite tilie excellenit lre(Itenc(y I99resollutioii ol boiitl cal spectrl. 2-1. '1' 1. Kiti-n, (',Iii-ntlktIni (iplil Ir~~-''ln' SOC.
*V. Summary and Conclusions 25. 1'. C. t i'u. J. tHOi''I, 1'. N t'iuuri. .ini 1 in1iuit. Al. ()Ill,
A new signal representalt ii has beein int riilutedI that t 9, S9f5 (I111
allows thei siiiillaiieiuts spacie midi slllitial tre(tivelicy 26. 1). .ui-.u.ii m~ill It. V. K. N'iu.a~ kiar Appl. Ot IS, ti673filtering tt i-I ) sit .imls. Th'll ne represeitaiiin, di- 2. N 'I'imu,i J. .1. Huidtiz. iuh 'C I' lii (lili. lxit. 5, 401pteniniig oil t he Itlrtivdlr filter tiiict ii,caii repireseintthe& WDI, th~e AF,. variouhs tilteredl WDiI and( iqiisi-VII) 25. I(-.'i'.Sil'uiliitiinriuuti.22hilict ionls as wellI as d1Iteretit sped ruil represtahooilsshe-1.ll tespetctrigrahi~lte lot-alspec-tra, 111111th lIcal 29. %1. .1 Kku~,ia,ili,,A~l (lilt 19, 192(19 801.1)oppler spect ra. Using thle 2-I ) ilitlire ut* optlical sigiill :ll. m. I.lriu. Prc wlk I 1- 6, ",Ii I 97h1.
3156 APPLIED OPTICS / Vol. 21, No. 17 / I Se~pltbter 1982
~i C6~2 42)
PI., in ItlNAcV I. s I'tIoLiiek Q w I Ll Al I Iic~it i on to MoIL I ,Litil I, itt's
M. Ktwa i T. N. Saatiw~'i I). L.. Schlti Ii
JIj u ll , ,ojik, N4. Y. 100 31
ABIS-1Lr c, vj r i!eS lin 1;11 Ilk aid, I i 1 u-,i k i tudI I foinL
Ithis w~c, %.L ui1y -e a tan l..~xu 'tjt:A1IIts cal I, rer tt~oiZoWith (l2lii .. ''tweell
2 rnt.1x ftt o !110 Ie l.nti~ tCt Uj u; il ii 5711 ~ j) itk: o. I;Ijt IX ' the0
ttai:,Cd Oil ('bt~i 1111is; t ll ndL 1110 ., lt 1llI!I, I nt r ix I : t,'~tut I~ I Ii s ,iI t I it -
the uer 's lait ler is ktiii.A ,- ! i 1 l: 1:1 1t'IMNi IS t1UI~jln!t_\I to wf~ uk', t' i j I 1i 1 :50 1 id: 11 ,, .. 1~ nt , II; Ite tI tp. 0: 1i of st; it I (IfI l'o.eltt t'll oeth II, o nlt ti;.p It li.1 I"- . Which i; !., I I I I~ ... ~ . ..
allite SIk JQLC,: IILol h .n c1tii1j Ill I t !''l 0I tilml:Iltn j~o .(ts to theSa~~~ellitt,12 it o ta nd. _i I .itl I!; Ia a all 1 IIntol!t,lt lotn
1 I ai utl cl 1 (.10 '.et "Aat',; aii'le2 oiil. 'II~ a ilval ;.osIS
in7 ju~ckit Swi tchiil 53tC1 ] t e (C1Iunncit. lol JI I I ,l 11.1 i1l 0ii.IL : .i5 il;tOf It !,IrV e JpIe-
~sytt~ts various rtmltac,:vfs seiks.'s )Ii.t.''lei kfot . All knit i0 orJl c Iol iS are infi-lute.
propusLNI in order to accccq disit flexiblo connoeti- III lllt n It'athlvi s oillm" x (ii ften entvity andk efficient ;hto of sate11lie clilnivis. 1 t 1 5 e:I; firj to';iI it filei .Ito'lite.Illose chainnel t iii7atio jt rotrooxls mty I., lort i I. xw i h x- :oi it' htioflno itto Uiroe twillt catjoi i5. 'llh1 fI I S t 3i : t1.11 5I i i t.2ex'i Ithie convoilttienal elLto~x)Iln, si.tc Is t iw]nts' ICY ii1o lit11; it-, uit Ito to tli. jc!Ili 'di-division imiltiple-xiiq .ioooss lit'I, ol t lilt- dlivi- lj'll i' ','~ ttl.;n' lt t 11i:, i111,0 tlo I iuision nultiple access, 'lktA. '11101e 11t0 VfIt'1tiVo 4 dluii;; 1,111 the vItchi is !st 1, r tlhu. distl-for licavy traffic, but Witht a bors, y natnle, 1;to l't ion. 'I110 1vio~: .;1:1 1"t, in1 a:;)11 kktIAa f ixui a Iloej tion otf ch itiel c~ij dc Ity i, I't -!;tl bet II ills;. III. IxI I o. In ' I I2.!i do, its-ly Wa)stefull. JOr snob a si Itiat toln, iiuLi aoi's 0;ily sI'I IW Itd t inili I0l't llteixoz lu. y Intischmci's, like theo AlOHA 1,rotcxel , the Umt oclitvv, viiasttiwsl the, i 'nl. AsotiiAt~/and tite Collision Waomlution Mott ltlwn; i'i'Al- y;; IIth solilt li'i' Il iIT1'i; lo. I.-:1 investtq'i-clorititi), are intr1Aned. 'lio tird', ixttNjoly iS t-1i tot bro c,is;t ii;3 :itelli ts itt ISLIOA 801 ,the dyxv.101uC a.s tpiim'nt p1 otrke I!;. ThesLe itii ne I JEVI: 79J a: i (i3 hit 71thie jull ml arid resvirvat Intl ;hvs I n ,,,ct Ionl 2. W, Jto:oij llt~'i i,~ il-!A, to]1
lit dus jqr m, aitly-ze ai t,tiititxi '1111V, nil- tl, i uilzi ti)i tt t3.1i ac o ods
tiple access, jretC71Cil . T1he .7klIS, s 011I i Lli;:.' the ltriiltittot of the 50110:1?k in mkilti-sptobtaiti-i:J Us'v Lti!;y uhil idle jptl Ilio d t ilot ins -illi sdi lit( t;.siiilar to thle apprniti us,'] t I ('l :81 ] i t hiana~lysis of Wl~IA. Ilwi ntiiiii invi. in '11C I-I s.!;liiltA ii othen prvjosi.I asu alt wqtiilrw'vnitto1 the lan~d In ii' we sil (I 01 t the i;l; 'hhtL\ jIoti)-
IWt'I. Other ranosit tio '11 tpt\ ioti',o s cant lv- kAl l 't a 1 o1 1kt vIkUII 1111;' th~Ile I',Li itarcia to.foundt, for- cx.ittjle in I KI1:l 781 ands 1I111111 801 . I itieilnqi~lviicntation of tho !,cliemoh for tiult i-slot L'.nsatellites is coils itined. - iOiiiiittO 1iittliI. . I.N
11 -. 0 1 dI kI.' iii Ni1 Iin I Iii1 '1':1. . ' It Iii xi -
is tio ouver a widh, area ithl .t h11 ti sp-it) lc-tn li:iltit 11 ot-11 i . 't-j:.,t )I t il. ,it Iti wi -Iantenrta. A multiple !;lk)L ti'.ii ,tlIttIII 161S I Stij- flI 1(w the lit 1:(,1 a ili::ii~ pi": wi~tnificant el h-ct in lmttkilrj the it:.e of avatlahlie k-it' .lcltii si. uI s'. to ,xi.~ I afroiclicy har-Als on sitil I ito tntntu Ii snili -tkt . 'l;iiilolI t ii'.' I,. !, Iot til to\l, I all
tenna is arrailil to fotin N ltiii-ivt'rlplliti !,l.'t si't l.i-itl Is "11it1 tol tli'' tolaltill) I- noibamt
4 , each servinti a scitiitc z.'onio. 171i -is K, I. .i oe lot' ; Il it It ,l i I to'1:.1111 'lh et___l,__ I I- -' Ii I uonn1 1sitl ii N cI I! :.i('it IV t !o tInot
7hi1 i %uzk wan; 1. 1rt talIly sulj tot teoi by A 't Lulii it .1)oili tI ,I:. Ii~ I I Il'Ill tIti Ilt tgrant ARIUI 81-0109 U., flui,' ats In the. clive (it ,Ilotid sot!; ot
7F.3. 1
0536-1486/82/0000 0303 $00 75 4, 1992 Iff
43)
or dirtot y '11t'A, .uld Is- two groups't lilve 1.'', ;i 'itt a pitxf simi lar to lost's ttI)~ isfavoied slot . achi user (or L' r t it s t-1 t ioil) li,1within a group has tie sanme( favored nlot.
Vylicii a lisL'r has; a jacket readly f or to ias- + (I6)-iimsin(i .t., rte tit it a pocket arriivc!; at the 1' f- i 4K 2i- ) ~ 61'Ad-Qf-ti0-ill0o (10!,) )sit ion in the bufhi, ho I I G; 1 1 (h)looks at the Ilext N slot s and tit eJlesU Cos siunl Mt~l or~v scii lmulat ion Shown 'in NI ) idix A wein the favutixi Slot Lof his, tJIou( Wilth prLutti1;l ity 4tlJ1a and in anty othse r. tular sLot s tiil the i;LxL 2N -1TN slots with i'roleiility 1, Th1us; (s) = laALI e +2ab+ IN- 2) b I eN t7)
a + b (N-1) 1 Il
let uts explain tis pr(_V;;; ii:; toq a tantte, *2 S' -sy
which has the ntiio s I to N. One of rt t-.o iittiers 13 is) = (-) e 1; 'IF (y) +
appears with proixbility a,* tite oth,,is app;* 'r witpti"Al ity b. fluriitj busy i,-t ±;k '.tion there ateSSaI*- pXACkets in thle bulf Ir, as asj s a kickL't at eIk)L is tianusittud, thle ioulet te is turnk~1 Wtit he:_nce N - -
the IK)L packet traI)SISSlOrt IS S~'hX01ed accxordinj )2ab-+ (14-2) b I e N G (8)to die result Of the 1okilQ1te. IS1riri Idle 1.'r-iedS G=wfien there is to jackets itt the 1to hfr, as s. .u;; as Na packet arrives at the bufter, the roulette schie- 1-dules thte 1hicket Lxausarissjen, Fig. 3. -
If a [.icket encouniters a colsion, which is U 'Kacknow( 1tied af oter a round,-trip t trvt delay, it F (y) U (y u '4
-jotits the buOttec at UIe ilRI-Of -011 - 1. Lt 11" (101) -tion in the, saiite way as; a packet It~xuly qenetatI.
'Vie queue itt ltird 'I wit fe(Rxlhtk is ot R~'it ill 1'ig.4. -r 9711e nlWSSd.3t arival rate inclix3rngi collided pAic- 0 1-i y: T (9)ikets is denoted by A
y' T3. Anal Iystis witt
Lt Di ibe tlte expect-cd packet dulay due torte uset. s buffer. 'The &xpocted packet delay ini U Wx '(0
the synatrn is th~en Cliven by {0 x'C
D 2P + D, + E (DI + 2P') (2) to;L.S (61 ) , 7) s cui eai ily bl atm tie
where irst tt set~ id itattots of jacket service tisr's,
F tihe avezaoqe n;utilkr of retranudssiou ,3'2P' round-trip delay. 3.2 'Th- Av,rae t ro PtIdt=;.,-1,I
3.1 21ie Averaao racket Waitin__Tinv i; v'n~e.s ' tI lt~tO't loilnr-
'To findl the value of D' , we us-e an appr.oach ColdtonL VI1i'' t)t' 010 I SO eithr u'r oiut all-simi lar to the one used by (1 1 771 it isP ~ ';itrt o',th y;t; irujl nanalysis of 'iTtIA. We conisider tw.) kind;; of Sor- G, tite chiiol t tattic:
vice tins! distritutien function iBlx and 13(x in G (1a sinc;ie-server (11-11 noWith PiFs:;oti arri vt]s at 1 + E 5XG packets per socoitd 13( it; tie disftribution of eItt,
apacket whtich initiates a busy piz'icxI with fi rst The e, it\'' a" G \T. itt'! Lil th a haors
andl scood nJtisots L) and b All bsutrl;ieut [tic- t;todNte1'Istiitythtauktikets in rite sanic busy' jVricAi lve ser;vice ttitus tlatlt-ill. to- illlye, ;I o; lt , a t t q lye;; lydrawn jndevjvidontlIy froms the di sti i bution SW Ix I\ T, aod It T . INo to; IP to 1-1'hwith fi rot and secndO litits t b Ianti bt 2' et, N I ''~ ~ " iLo the nlrttber of picket;; in rite
1Y(Il 2-l tl hlly'Ind t" l, a 1Ii'k'
quieue and in set-vuce) at time t. M2 no definettiratdI t iit'(i tit t' htCthe trans forns: dId' w0, of t'tt str:, it ~~i t. O
11 0 n 1 ;t n~c . ntt p tiie ;of t-Iet'ieIt
P(~~~~~~)~ I :S Zt P1 2 lst iritN12())It'qts I;tiv'
E1 P" -(2
(a) C- dI3 W(4) W
1' PttlA) %ucrce'si~trut ~ is(ninaflVl(
Ii It ) f 0t (.+x I~ (, J ! .u c tr nss l t l t ria n t:'t::t o : n in a o n fa o r
44)
4 el ch [cl'; I1,o tie iii rIn nt.1-I~oe ises I o s!
a l + W1 - (14) 1~i t'i1 aitl yt1- 'I iv,;tc
1 1 1
-~~~~ L ~ -it-a M 1%t -1 II. 111 1 cm 1 kl1 -I t. ; r
i-i Im I .' LA t' tI au t0I IO IWk
-a t4 Io In
Iril iI UIp :Kit ,i -l IAI . h. , i; i 1 1 co. ,s 1. s 1 111 t ,1
Iu t Lo t;e Iot )il tt ttii11,ts tu iA ti. Ii lk* 'I I , K, ~h A I.I. 1 !
uplink~~l A tA d'ttitkA, tiih t =aoi~ pakA 14)- 14IjlItN S.I.
.1)lktt 2ik~t5 1 . o ju;;A (Ise. Cc~ to Itv 4Ai (1. II
acal ML I I-)L 11) ( t I I J!i I ait tI 4 JI rt
rtjtx a h -11. 11 11 II, i (wt;1 I), ___ k t 111 11 1.11 ILI:-!'I:b' in lii. it atl Ii- l I tak. 10;0;,:~ Il aliI
ran1td 11151; I H\;~- llll wt u ticr 111 ~ 11 Aq:,1', Wi 'I i. I L Ji t-it i-
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In th~is [iiq'r wot ailaly:e a lali..r,; 'l1i ri- 'U r0iz It l-Sic i 1:i 3. , Tv; W, !!,4-'tPIpl accUss [pr0oucll Th1e anial,7:;S i 5 is hoo d 0:1 i.;t i :;l .na~ of tire !;elirt r 1:1 it j -sictobtaiin thei bu!,y aridl idle jqir-itx iit ral-kt 015 ci;t.tl itssimilar tUr the i 'rcchus ill 11llM "81 inl tie 2analysis of l111itA. Ilhe uisfiril rmt ~ 'M,%\ is 2.': an 'tl\rovlthen pri ose as an liiacvt'.'lit- toi I, '1A rarsi 1:i 0r, We wi 1t ('iii. th )iIudii i 'j!-t ri to-'I U .N. Other rariditizNI 11PA I rLot0)IS cali lw.' o)i IhrI a ,j eul Ii r ul c'Oi:vte c !( it ii:. L arIjIa(, tofot-ind , for ex~ciple in [11.1:l 781 ai [i'illiZ 001 . I -e vicod.Lrqlcr,sntaticron ef till, ichcrnw_ fur inul ti-shutA Lv~allsatellites is corrsido'r'xh. W r 'ni t i eln n..;
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49j
A RESERVATIONS SCHEIIE OF MULT IPLE ACCESSFOR LOCAL NETWORKS
by
T. N. Saadawi A. EphremidesElectrical Engineering Dept. Electrical Enginering Dept.CCNY University of MarylandNew York, NY College Park, MD 20742
Abstract
A reservations scheme for multiple access suited for local
networks of broadcast channels is considered and anal yzed, T he
analysis is based on two imbedded Markov chain models, one for
the individual user and one for the entire set of users. The
average queue size as well as the average delay time are
obtained. The proposed scheme adapts automatically to varying
traffic conditions.
5(i)
I. Int rod uc t Ion
The problem of ef f ic ient transmission of voice and
computer data through shared channels remains a challenging
and, in many ways, open problem in the communications area.
In particular the problem of optimum or "good" design of
multiple access protocols for radio channels continues to
receive wide attention. There is a special need for protocols
that can be analyzed so that their performance can be predicted
and evaluated before implementation.
As is widely known by now, multiple access protocols may be
partitioned into three main categories. The first is the class
of fixed allocation protocols, like frequency division multiple
access (FDMA) or time division multiple access (TDMA). These
are known to be effective onIlv for heavy, non-bursty traffic.
Otherwise they are known to be extremely wasteful of channel
capacity since dedicated allocation of resources is ill-suited
to low duty-cycle users. The second class of protocols consists
of contention-based random access schemes, such as the ALOHA
protocol and its many variants. Their main disadvantages are
known to be an inherent instability and a low channel
utilization rate. The third broad category of protocols
includes dynamic assignment schemes, and, in particular,
schemes based on reservations. Few of the proposed reservation
schemes have been analytically studied. Although comprehensive
reviews of multiple access protocols abound in the literature
2
[1-3] we briefly recall some .pecific protocols selectively
j to create the context for our :4tudy. In [4) a protocol is
described where the channel is divided into two subchannels;
one operated in slotted ALOHA mode for reservation requests,
and the other operated in a dedicated mode for data packets.
Reservation schemes such as this one are particularly
attractive when a significant part of channel traffic consists
of multi-packet messages, became the successful transmission
of a single reservation mini-packet is sufficient to reserve
as many slots as are necessary for the entire message.
In [5] Crowther et al. introduced the Reservation-ALH1A
idea. In this scheme, time is slotted and the slots are
grouped together in fixed length frames. Slots are periodically
assigned to users as in TDMA, but unlike TIMA these are
temporary assignments. When any particular slot is idle in
one frame, the slot is opened up for contention transmission
in the following frame. A user who successfully transmits in
such a contention slot will then be granted ownership of the
slot as long as he has packets to transmit. This scheme was
Ilater analyzed by Lain [6].
In [71 the Round Robin rtervation scheme was introduced
Iin which a slotted channel with fixed length frame structure
is assumed. Each user is permanently as;igned one slot perIframe, and unowned slots (which are present when the number
of users is smaller than the number of slots per frame) are
made available as needed to users requesting them. Users are
3
permitted to transmit il slot,; owned by terminals which current Iy
have no packets to transmit; ';uch inactive terminals can va:',iIV
regalin owners hi p of t he ir s lo t s whel 11 nee d Cd ( 1 o W owi 1 g a 0 o
frame delay).
In [81 a reservation schL me is proposed in which channel
time is divided again into frames, each of which consists of
reservation time slots, preassigned (PA) time slots alld
reservation access (RA) time slots. In the reservation time
slots each user sends a reserva t ion message (consisting of the number of
packets in his buffer minus one). Each user is assigned one
PA slot permanently and a number of RA slots matching the amount
requested in the reservation message.
Wieselthier and Ephremides [21 introduced the interleaved
Frame Flush-Out (IFFO) protocols, in which a long propagation
delay Is assumed (satellite channels). Slots are reserved
by the different users, but, because of the dclay, there may
be slots that will go unused unless they are opened to
contention. The performance of the protocols was eva1alta ed by
computation and simulation.
In this paper, we consider and analyze a simle reser-
vation scheme essentially identical to that proposed by RotLhauser
and Wild [16] and studied by Bux [171 and similar to the one
proposed by Spaniol [181 and Mark [191. This protocol has high
adaptivity properties and is eminently suited to low propagation
delay environments, namely, local network applications. The usual
assumptions are made about user statistics, channel properties,
etc. as will be described in the sequel. Our analysis is new and
is based on a modeling idea Introduced In 191, that consists of
two coupled Markov chains, one that describes the status of the i ler content,,
4
53)
of a typical user (the User I,,rkov Chain) and one that describes
the status of all the users of tie cha~nnel (the Sy'stm Mar'rkv
Chii ii). Such an approach pre i ts us to account for the interaction
between the users, quite similarly to the case studied in 9].
In section II, we describe the channel frame structure and
the user model. In section III we describe the systtu:; Markov
chain, and in section IV the user Markov chain. Delay is
calculated in section V. Finally, numerical solutions and
simulation results are presented in section VI. Two Appendices
provide the analysis details.
II. The Model
In this section, a description of the reservation scheme
is presented. This is done by first describing the channel
time frame structure and then the user model.
a) The Frame Structure
The frame structure is shown in Fig. (l.a), where the
first slot in the frame is the reservation slot which is
subdivided into M mini-slots, (M is the number of users in the
system). Each user is permanen-tly assigned one mini-slot in
the reservation slot, in which lie sends a request whenever
he has a packet ready to transmit. In particular user i is
assigned the ith mini-slot in every reservation slot. T.
equalize delays among the different users this assignment may
be cyclically rotated so that no individual user has a
permanent advantage over the ithers.
j
54)
Mini-slo ts are ve ry slhort S sin11ce thL I ne!edI t LJU Au aumdt L
Ctle tra nsminis s ion of a s in glIe -Li e iniid icaiti iig the nieed to
reserve a slut inl the next. f riie. Thlt lengrth o10f thle 1-Iill fra-me
is a random variab le arid depends, Oil the, number Of user s
r e que sting t r ans m is s ion a t th. bi- eg inin i i f thle f rai:. 1%( or
example , as del)ic ted ini Fi g. I , Users inmber 2 and niumbeur 4 a re
requesting transmission. Sinick ulser I hJs nothing9 to tranismit,
user 2 knows hie will be thle f irSt to transmit Lifter thle
reservation slot is over. Also, since user 3 did not request
transmission, user 4 kniows hie wvill transmnit after user 2, and
then the new f r ame will1 start . ThuLs , L. I e f r ir I, e 11g t 11 e2qu1l iIS
to three sl1o t s. Note that t he frame length cannot be less r hanl
one slot and cannot be more thain (M+l1) sIls ( corcespotidiing to the
case thiat all, l users are requoestine trn1isEc) 'e aIssumle thatl
messages consist of single packets each of which f'its eXactlY inl
one slot . Also we assume that tile channeil i s miini tored by all users
withiout delav or noise. Note alISo, that a uer Ltransmiits Onlly anc
D a ck et r)e r f ra me r eP.a rd Ie -;s of thle niu mb er o f p)a c ke ts i n i is buLif fer.
b) Thev Ise r Mo del1
The system consists of M teriniaIs , each o f wh ich has anl
infinite size buffer. Thlle a rr ival p)r o c ess at ea chl I e thM
terminals is assumed to be a Bernoull Ii process witLh ra te c
Thle user may be in one of two stLateS, anl idle state, i f li is
buff er i s e inp t y o r anl a C t i v e !; t al L L- , i f h i S b U ff e r- i S ii on1 - 12i IN y
As shown in Fib. I .b, whenever the user has a packet inl
his buffer , hie sends a r eserV t ic on reques-'t at tilie beginlning, Of
the f rame, ad at t lie S iul C vi I Lhe pac ke t at the head of
line (iOL) is moved to the tI:mmitter and waits there until
it is transmitted in its assi t ed slot.
We need the following de i nil io n
Let
A s t ad a at t C p r 0 ab iii o f lv ing i
packets ifi the buffer at the be, ginning
o f a f r ai II e.
10 prob at) i lity a a uI f 1- 1 ci 1 .p lt
e cprobabi ei 1 oof a s er Leini idle.
1 -T; 0 1)r obab i Ii Iy o f a ts e r bt!i ii a c te
Iuch a model produces a non-s t;ndard queue in system t ilter-
I acting The service t inC of a packet in one qnen
depends on the status of the L iter queues. If we let n k rcpreselt
the contents of the input buffer of tLe k- user, then a
complete description of all M users requires the spec iicat ion
of the joint probability distribution L (n ,L 0, . , n ) The
determination of these probabilities is very complex, if iLot
impossible, and demands the solution of a lar ge number of sets
of equations. Fayolle and Iasnogorodski [10] showed the
limitation of this direct approach. They considered the simlple
example of two expo nential seCIrver S the service rate of each of
which depended on the status oI the other server's queue.
Using the theory of analytic continuation they reduced the
I I7
6
determination of the joint prob Pi1 i ity distrihution o the tW)
queues to a Riemann-lilbert prob 'em and were able to obtain a
very complex closed form for the solution. Furtherm:ore extension
of their analysis to the case of more than two users is not Ieasible.
Leibowitz [I1 presented an apprL, ilatC :ethod of treat lig mu ltiquece
systems. lie studied the case of : queues with a single server, in
which the queues are served in cyclic order, 1ie ade the aSsulMption
that the distribution of the quete size of one queue is invariant if
it is the same for all queues during one cycle.
11ashida [ 12, 13] used Leibo'.,it;t's approxirat ion in tie a ivlysis
o f mu I t i queue s ys t e s A lso in S ) , the -uthors u s Jed i t ow i t ' s
approximat ion alon g with an indepeindence ass Umpt ion.
We consider that the ma in contribUl ion of our paper is the proposed :%atheatical
model for the analysis of interactLing buffered termina Is. The ;:odel,
as we mentioned earlier, is basi'a I y a cohbinat iOn of two , arkov
chains imbedded at the beginning of each frame; oce -arkov chain
that describes the state of the user, ref red to as the user Markov
chain and another Markov chain that describes the state of all the
users in the system, refered to a.; the system, Markov chain.
8
j LI 1 :I L;:;111ned VieW Of t heLA iiA IreservatiOn Slot
!44
f rame lenhtll h new framie=3 slots
(a) Typical frame structure for the reservation
S c Ie -
(b The User Model
Fig. I
-- ~
11 'The Sy'stem Mirkov Chain
The state of the syste:n is , ,scribed by the number of users
req u e s t iig t r a 1 si J.;i on a t tihe b . 11 ining of the frame. F'rst, we
need the olIo wing definitions: Yet
j n ml 1;1b e r of et s ers r c(I t ug tr 1 L i s 11 ;;;1 s to the
beg iin i n f a t ravi.
f rame I eng th when users -equest t! rLL 1 r :1:i ol a t the
beg inn ing of tl.e frame
S1 = +j
steady t C ) r 0 l) t 1 f i L, CV ,;,0
o r o h) a b 1 t i d I ',I SC C Fc (2 11 L' 1 .transmission at itL he,'ianziiig ,fle ti-ra:.,
o . ,' 'robabi[try an idl. ,user seneat! i.t : at iii;;:, ,,le
pac ket d u r ing a f ri:':e o f 1c: + I h . I ) i IL I ,
i I - (I - L) j I
F ' pr o1)a b i I ty f o a L) I ,,'ly k; Ic al er 1o- a t
t lie )eg i nl i o f a Ir a :1,.: .' en e rs a c t i v
0
T "t e s y s t L il ark r ha c i a in i i I Iu 1 a t ra L c d i n F i 2 .a w Ii 1 e
its transition 1)ro ibilit ics are )htaine d in tppendix A.
IV The User )rk V v 1b,1 in
The state of the user is de ribed by die V .,er of packets
in the buffer at the beginn11in11 g o t.! tr I c, lt'
n = number of packets ill buffer ot t L h bC,inning of
a f atle
1= steady state probab 1) ity of n packets in the buffer
at tie be, inn ing of t C r a ie
I
Let 7-(z) be the generating function1 Of 7; n e
n=O 1n
In Fig. 2.b, we show the user Markov chain. Note that only
one packet can leave the buffer 1,er frame. We show the derivation
of tile transition probabilities in Appendix B.
60)
A. (n)
Fig. 2.a System Markov Chain
n n+
B (n) <
B n(u
B 0(n)
00
n+1 B (n)
11 n>M+ 1
B *n
Fig. 2.b User Markov Chain
61)
V Delay Analysis
The average total delay per packet, D, is the sume of the
average waiting time in the buff.cr, W, plus the average time from
the beginning of the frame during which the packet is scheduled for
transmission until the packet is transilitted in its assigned slot.
We refer to the latter as the service time, S.
Hence,
D = W + S
Using Little's result, we can write W as;
where, Q is the average queue size.
Assuming the cyclic assignmnit discussed earlier in section II,
the average service time equals to one plus half the average number
of requesting users at the beginning of the frame, i.e.
S=1+
VI Numerical Solutions
Equations (A-l) through (A-5) provide the values of the quantities
that were needed in order to solve the state transition equations
for the system probabilities pn" However, some of these quantities
are not solely expressed in terms of the parameters M and o, but
also in terms of F, i.e., the prohability of one packet in the buffer
at the beginning of a frame given user is active, which is a function
of I and i0, But, v0 and I both depend in turn on p 1. Thus, in
total, we have a set of simultaneous coupled non-I inear equat ions in
TO 0 i1 and ' {pln In Fig. 3, we show the flow chart for the numerical
procedure used to solve these equations. We start with an initial
value for F, then solve (M+l) linear simultaneous equations in (pn}.
We then determine O and -, and use Eq. (B.1) and Wegestein 's iteration
1I1
scheme [14] to obtain the new value of F. Two to five iterations
were required for convergence to a solution for F within a 10-7
tolerance in the values of F and 1 p ).
For M=4, Fig. 4 shows the probabilities n O (the prob. of an
empty buffer), p0 (the prob. of zero users requesting transmission)
and pM (the prob. of M users requesting transmission) versus channel
throughput. Notice that as throughput increases both 0 and P0
decrease while PM increases. Figs. 5 and 6 show the average waiting,
service, and total times versus throughput for M=4 and 8 respectively.
Throughput is defined as X=Mo. Such a definition presupposes steady-
state stability so that the average input rate is equal to the average
transmission rate. Although detailed stability analysis is not
considered here, it is fairly obvious that since there are no wasted
slots (due to contention or idleness) except for one reservation slot
in each frame and since all users are assumed similar, any value of
A less than M--- produces a stable equilibrium.
II
I
I
I
1
I II I I I I I I III I I I ... . .
63)
Li Start With initialvalue for F
Of +I)Li iicar EqIs. System Markov chain
New I.-dU -sing Wcen'elstein 'S Determine FItera in Scheme
1~~~I Toleranceag Deay
Fig.g 3 r F0o Cha t t, Derin Average Delays
for the Reserva tion Scheme
II
.1|
--- '.2 ii*
I' /4-
CA.
16I
/
1- °
// -
,
7,
I0
L .0'
//
kE
~16
I.'
n
I -
1 0
V1
0
('3
\l~~ ~ <II
I
I H.12 - -.I is
II?
W 1!
-C-
3-.
$1
I
The reservation scheme has been i; imulated on the computer. ILL'
j simulation program was written in FORTRAN, it was not necessary
to use a language specifically designed for simulation. The numerical
jsolution for the reservation scheme required the implicit assumption
that some processes are independent when in fact they are not.
i The simulation produced agreement with the numerical results within
5% for the average queue size. Table I shows the comparison between
numerical and simulated results for average waiting, service and
I total delays for four users. The maximum difference in the average
waiting time is approximately 6% and in the average service time
is 5%.
Throughput D W S
.10199 A 2.2016 1.1448 1.0568
S 2.1687 1.1414 1.0560
j .30597 A 2.812 1.5905 1.2215
S 2.7613 1.5419 1.2194
.50995 A 3.9797 2.4602 1.5195
S 3.8001 2.3691 1.5194
.71393 A 6.8565 4.7208 2.1357
S 6.7625 4.5175 2.245
A: Analytical results
S: Simulated results
Table I . Comparison of Numerical and StimulatedResults for the Average Delays M = 4
1
19
VI LU I s i :i
in this papur, we hvu intr C.cd aai .iCvl1.'cd . ru~,urva-oiu
scbe::.u, in w1hich rescrx';,tion ;ac ta arc s. r,.:c ,: :t' . uf:
over a fraction of the ::aij c --::, . :.c .';cr,,' '.a L :..,, iLrVLcu
an11d t tal ti::e5 ! avL b L obtai .< . ab'.'i::;.
co::1 na t 1o1 o two i::,bedUd Ma v tL,!u 1c 1:t v
for th itc bt cn t, U;cr. 'ch .) C1,jt2 L2,)CI-en
i n.plIuaeatud by the .Orc in t " ann,'iz :- ,. t!, a
finite nu:iiber of u : fer-d ucer ' in ,
the ,aalysin o"f t .. reurivatiano :. ,:c: !:. e: ]rcasc' t c .i , t : a '.c r or
a Lana o: Co1li :i ?,cac~tia:. .".'.,,iL.:.; St 1 .::,-i. Ur e
a II II~C .:nl I L IIrc a .. , ,. =
I
Ar'PiNI: IX A
System Markov Chain for tie Reservat ion Schv,.L
Let
K = number of active u.;ers out of the j requesting users
The distinction between the numbL,r of active users and the nu:aber of
users requesting transmission is lecessitated by the ls:uUption that
if there is only one packet in the buffer, it is automatically moved
to the t ransini t ter box at the beginning of the frame and hence the
corresponding user is considered idle at the beginning of that
frame. Thus
%-K = number of idle users at the beginning of a frame
Recall that F is defined as;
F = Probability of one packet only in buffer at tle
beginning of a frai:ie given user is active
i( r\ 11 -: O (A.
Let
CK j) = Prob. [K activ(, users out of j requesting users
a t tihe beg inn i ng of a f rame]
( (1-F) K (F A. 2)
In order to use the ah1,e binomial distribution, it is
assumed that the event that a user is active is independent fr-om
the state of the other users. Similar assuMption has been stated
and justified in [ 131 for- thle ca,- Of ai single server orvlu N lines f:
cyclic order.
Let
q ~ 1 r [ J o I the M K id lu users geeaepackvet; inl
aI I rame L,, , i th t ) users are request ing
Iid K o f tit, , ., i c t iv I
S - (I-c I - (A.3)
2 3
Now we can write h(i,K j) , t lie joint p rob a b ii ty dist ribution
of i and K conditioned with resi,',t to j, aS fo1lows:
h(i , )APr[if out of M-K id>, users; generate packets duriag a
fraite and K out of the j requesting users are
active given j u,;ers are requesting]
i - i K) cM- [i-.[ 1-o.( ) (1 - ) K F)j-
i K C KK (Sq(i ,K, j)CK(3) (A. 4)
Referring to Fig. (2.a) we can write A. (n) , the transition
probability from state j to state n, as;
m in (j , n)A.(n) = Pr[ (n-K) arrivals and K 1sers active given j
S K=0
LIs e rs a -or e (IqueLs tin g
li .iin (Q , n ) Ii( -K j I<j <M<MK=O15i 3 h=-KKI ) Ij.,U5<>
SM-n (A. 5
Of course then we have
Hp = F A. (n)p. , O<n <zHP2 2 0 < n <..
1
A P P E 1) 1.
The LUsser Mtarkuv Chain fr ' tC KUS urv1 L ion S eC1:1e
In this Appendix, we determioc tie transition probanil ti es
for the user Markov chain shown i !ig. 2.b. Alo, expres sions
for 0 the probability of an eru t. buffur and q, the bufL er size
are obtained. First, we need the following definitions; L et
g (ij+ 1) 1 Probability that i out of j+I possible packets
arrive during a frac given j requesting users at
the beginning of t1ie frame
j +1
j=0
H( z) A it(z,j l)p ( B.2)j=O
Now, referring to Fig. 2.b, w. h ve;
Bn (n) = U Prob. Lone packu'e leaves and no packets arrive
given j reljuesting users at the beginning
of a frai-,cj
M= ['U g(0,j+l)pj]/([ pn = , . .(B.,3)
Notice that the initial value of the summation is j 1, since at the
beginning of the frame we have at "east one user requesting transmission.
For the same reason, we normalize by dividing by (1-P 0 ).
Similarly,
V (n) = Pr. [one packet Ie;Ives, and one arrives during an
frame given users requesting at the beginning
of the framej
?3
= (B.4)
g (0,j+ 1)p / 0( B- 4
[ 0 j+ /-pj =o J
Note that for n 0, we can have a ;!.ax imu of (M- I) requesting users
hence, we need to normalize by d iv id ing by the probab ilt y of having
at most (M-1) requesting users.
Similarly,
B .(n) = E Pr. [one packet leaves, i+I arrive given jn-d -J
~requesting aIsers]
M
= [n g(i+l ,j+I)p 1/(l-Po) n B 2,3j=i
i = 1l,min(n-I ,M) (B.5)
~andS- 1
eneB 0 (n) = [.n-l g(n,j+l)p. /lpM I ..... ) (B. )Hence,
B n+ (n) .1 +Bn (n) n +E B (n) (n)I n - + B<,
T = (B. 7a)
nB+l) n+l +Bnnn + l Bni n- >
71 = 0i-TT 1i+ Bo0 0 To (B. 7 b
O = B 2 1 r 2 I+B ( ) 1 O7 B 7 )r1 =B 2 (l) 712BI 1 l+BO(1),( ( .c
To obtain IT(z), we need to define the foilowin ; w e n t
M
G(O)AX g(O,j+1)pj (B.8)j=O
G()_C g l~+Ipj Bg. 9)M
I!
73)
Multiplying Eq. (B.7) by z and timming over n, we get;
1(z) E n
n=O
W.... m i u ( n - i , M )Bn+l n+ . n+B n l (n ) TTn1+1
z + ' B 11 (11) 7, n Z' ' " B n- (n r J! 11 '1z u(n-.
n=O n=O nO i= i
+ B 0 (n); 1 y" u(n-1) (B. 10)
n=0J
Using Eq. (B.3), we get
n B (n) Tr z =[G(O)-g(O,1)1Z (z)-' O (B. I1)
n =0 n+1 1C(+)1 (0,l0p 0 ] 0
Using Eq. (B.4), we get;
n I [ c; ( )- (0 ,1.- 1 pZ B ( n) 7o a G ( 1 ) I( 1)P o -(z) + M
n =0 n 1- P00 0 1 -p~ 0( B. 1 2)
j From (B.6) we get;
M ME B 0 (n) zn = : : g(n,j) I)z 0Jn jo/ (l-P10)nl j=l n=l
= 0 [11 (z)G (0)H (z ,M+I) pM+g ,M+I)pM] (. 13)(I-p )
Finally, we have
a, min(n-l,M) MF+B .(n)in n = B .(t)n .zE T n-i n-i t n1- 77 Z
,n=2 i~l i=l n=i+].
1 E g(i+l ,j+l)p z [T (z) -71T Po i=l j U
M i- z g(i+l,j+l)pjz [71(z)-nO]
0 j=1 i=
"I
I
1 -N- z [ii(z,j+l)pj-g(] , i+l)pj z-g(0,j+1) p 1 {, (z)-O Ij- l j=I
-I~
'-Po [LTW(z)-,rO] [[ (z)-zG(l)-;(O)j (1.14)
From (B.11) - (B.14) and (13.10) we obtain
X(z) (1. 5)
where,
Mz) = flP- H (z)+P 0 (1 -p,)l(zM+1)PM
(B. 16)
Y(z) = (l-Po)Z -11(z) + po(g(O,1) + zg(1,I)) (B.17)
To obtain 7 0 , we take the limit in Eq. (13.15) as z-1, hencu,
I0 = Y(1) (B. 18)x(l)
wher e
Y Yl)Y(l) 3 ( ( - p (1-() it(]) B 19
(i-PM) (M- PO)II()+- 0 p 0 (I-PM)-(l-Po)p((M+l)o+I)}
(B3.20)
To obtain Q, we differentiate (B.15) and take the limit as z-1;
Thus
= 2i 0 ) (1.21)
2?l
where
X(1 z 2 z -p M M 0l (0) 2 0(-1)l)-(l1--10O) N
[ii ( , M+1) -2 1(I , M+ 1 (B. 22)
and
Y(1) = 1(i) (B. 23)
26
Re fe rences
75)
[I] Tobagi, F., "Multiaccess Pro! tc ols in Packet Communication
Systems" , IEEE Trans. on Colm!li cat ios, Vol. 28, pp. 468-488,
April 1980.
[2] Wieselthier, J.E. and Ephremides, A., "A New Class of Protocols
for Multiple Access in Satellite Networks", lEEk Transact ions
on Automatic Control, October 1980.
[3] Kleinrock, L., Queueing SysLlms, John Wiley, 1976.
[4] Roberts, L.G., "Dynamic Allocation of Satellite Capacity Through
Packet Reservation," AFIPS Conference Proceedings, 1973 National
Computer Conference, 42, 711-716.
[5] Crowther, W., R. Rettberg, D. Walden, S. Ornstein, and F. Heart,
"A System for Broadcast Communications Reservation-ALOHA,"
Proceedings of the Sixth Hawaii International Conference on
System Sciences, University of Hawaii, Honolulu, January 1973.
[6] Lam, S.S., "Packet Broadcast Networks - A Performance Analysis of
the R-ALOHA Protocol", IEEE Transactions on Computers, Vol. C-29,
No. 7, July 1980.
[7] Binder, R., "A dynamic packet switching system for Satellite
broadcasting channels" Proceedings of the IE'EE International
Conference on Communications, Vol. 3, June 1975.
[8] Balagangadhar and R.L. Pickholtz, "Analysis of a Reservation
Multiple Access Technique for Data Transmission Via Satellites",
IEEE Trans. on Communications, vol. COM-27, No. 10, October 1979.
[9] Saadawi, T.N. and A. Ephremides, "Analysis, stability and
Optimization of Slotted ALOH{A with a Finite Number of Buffered
Users", IEEE Trans. on AC, June 1981.
27
7C
[10] Fayolle, G. and lasnogorodski, "Two CoupIh. d Processors: The
Reduction to a Riemann-llill ert Problem" Wahrschvinlichkeits
Theorie, Springer-Verlag, 1979.
[11) Leibowitz, M.A., "An Approximate Method for Treating a Class
of Multiqueue Problems," ilbl Journal of Research and Development,
Vol. 5, No. 3, 204-209, July 1981.
J [12] Hashida, 0. and K. Ohara, "Line Accomodation Capacity of a
Communication Control Unit" Review of the Electrical Communicatiori
Laboratories, Vol. 20, No. 3-4, March-April, 1972.
[13] Hashida, 0., "Analysis of Multiqueue", Review of the Electrical
Comm. Lab., Nippon Telegraph and Telephone Public (Corpration,
Vol. 20, No. 3-4, March-April, 1972.
[14] Lance, G.N., Numerical Methods for High Speed Coiaputurs, Iliffe,
I London, 1960, pp. 134-138.
[15] Saadawi, T.N. and A. Ephremides, "Analysis of tie Tree Algorithm
with a Finite Number of Buffered Users", Proceedin-- ICC, June
1981.
[16] Rothauser, E., and Wild, D., "MLMA: A Collision-free Multiaccess
Method", Proc. IFIP Congress 77, Amsterdam, 1977, pp. 431-436.
[17 Bux, W., "Local-Area Subnetworks = A Performance Comparison",
IEEE Trans. on Comm., Vol. 29, No. 10, October 1981, pp. 1465-1473
[18] Spaniol, 0., "Modeling of Local Computer Networks", Cecpkuter
Networks, Vol. 3, pp. 315-326, 1979.
[191 Mark, J.W., "Global Scheduling Approach to Conflict-Free
Multiaccess via a Data Bus", IFEET Trans. on Comm., Vol. C0,1-26,
No. 9, Sept. '78, pp. 1342-1352.
,i
ii
AN rYIjGOP1I=~ FOR SCEN MATCHIMNG
By
Tarek N. Saadawi and George Eicbrnann
78)
Abstract
A fast aljoritri-i for scene matchincq is studied anal co.-.ared
with ot- er ali ,orithoms. The mreasure of similarity is the s,.-, of-, the
absolute values of the c iffer.e:nces of 1-ma(ee i-ntensit*y, be~tween t-he
te-=I1ate and the scene. Conparison with other -easures ofsilart
is prosented.
79)
(1) Introd-uction
ThQ ne~ed : cr axnalvzini,- the imacies of the earth taken by v~~
sensors on hiobi and-' low altitude aircraft is increasinq at a rapid ce
These im-aoOes have Leo?,-n uSed for several purposes, incluin-: navio ,ati ;nal
quianc, oocar~lcand tc,,x) rapnmc map matchin-, natural resourco -nalysL-,
weather ardct , :n,:nirc-TKnt studio2s.
A co.-rin probleim that arises in these application:; is sce ne 7.atcninc;.
Given a Dictorial d escrioDtion of a recion of sceno, determine whicr.reio
in another imaqe is si;-pilar. The rr-ethod used to solve this -,roblem in its
sinlest form is called template matching. An imrage of toamlate is given,
an, it is desired to determine all locations of t-he template in anothner
ixmaqe. If a sufficient nim~ber of corresnondinc.rgin could be utnac
then the image could be accurtely regist-ered, facilitatitc- location of
correspondina cbjccts.
Including th'e basic tamplate matcninq imcthc stDvra approacocs anc
tcri.niciue-: have beeon used, such as seaunonti-- s.4-H_1aritv cci-,ariscrS,
matchi-ng with invariant miccments, aund matChring OC-' ait td~ 0 etrs . A
ed-.c, as defined h1-ere, is a local chani,c ondietna' n ira17C- ,a~rnanct..
General-',' iimaa-o racino- can be sho-wn in the c2- wn hct a darn
Sen, Preproccssino- Ee -IehnInput trtcn Aerdo
Imaooc"
Preprocessingi is an important procedure in imaie i-,atchinsi. Its basic
obj -,- is correocting two degjraded imag.ies (su~ch as t o iage frcm txwu
di: :re-nco sensots) to (let new imfagjes to niuet cci- cAIn re-qui. ollunts Ldcr\
and proje-ction. r.0eal it includc:- cjecretrical correct ion adintens'-i.ty
corr-ct.;c-n. Geni( xtrical correction is a tasr to chainge( each pixel of'
distorted(. imagte t-rom origiinal coordinates,- ('x, 'to aqtia v2a
corrected c(YCrd mates (u, ,). ["er inten.sity oel c oa a i rn
,it 1011~ UCcal lud ~;r i~- ectasomwar; wid'Ivuse
1E(1,i infonnation is ii-qx)tnti- bu 1(th hunran anc ci 1 Tl
and a consider-aLle azoiunt of restearch h;as L 1- donie in attent1its'L to -xrac
WoeL1( iniformation fran pictures.
in this paper we study an alcioriths; for imiaqle riotchins.. ih j lor'iths.
osraresloss cenm)utation tiincK and achieves a hicih reyistratim( xrorano
In the next section we discuss the algorithm and in Section -ll w analyze
the results. Section Iv conpares the perforrrcrices of the different jrcasures
of similarity for iihaye registration. The conclusion is given in Section V.
:;. i , -fl ,ndM rtin, in (I), .l) x* xc' ,j x'<n,"e £ uy-d c :it Iij!I ) "V ~ itq III ~ I11 fu!- I( It iti I I s1ui 1 .ct1
101in du , k - 111,L42 I-e(is I -'Ati )II, tll',. :L ( s UV c: ItLi I shi, J1,, -it, 6 t",'t i(,):I
a&,. rith'i (f]SDA). nThey used the nom~lizW correlation ct !Ii cI -it cis
the !:k"-sur- of :mtch. The nonrolized co-i-reliation ccx_,fficient is d( finle(I
_____- _____. ___o - i,]* 1 (1-a)
C(i,j){(M' . ) (M ) }
where• = " Si (P , q )
p =1 q:i
M M 'C, : C E S i ] (p,q)"T(p,q)
p-- q: i
M M 2= C { pq (1-b)Pi q=i
M M "c7 M Si 3 (p,q)
M M= T(p,q)
Where T(p,q) denotes the template with diny-nsion MxN pixels, S (p,q) is
MxM subimage of the scene located at (i,j) coordinates with dimension
Lx.L. Hence the search area would be (L - M + 1) x (L - M + 1). The
formula in equation (1) requires a large computation time.
A critical issue is how to choose a thresholding technique for teniinatina
the calculation of C(i,j) early. If the thresholds are too high, we ma-y elimi-
nate the best registration because of the early behavior of the calculation of
C(i,j). On the other hand, if the thresholds are too low, many poor reoistra-
tions will avoid early termination of eqn. (]). The authors in (U) proposed
an empirically derived function as the threshold sequence dependent on the
subset number b.
i : ,: .1 :, !.t.'a tw,,' l:'i, a? , i: m~ : a., I?: < Vi Fe m Ia it,:. p!], : ,
evi::i.! 1,. , 11 i. :: , n as
A , ]) - - " ( ,q) - '(p,q) (2)
I--. c .1 Il i( pr11 iur-s are sLi,,lar witji tii S+DA a]: oi i
we use a stusct of tile tvo data fields (Si j and ') at a ]c ' rLs(W ution
calciate an ustuiiute of A.(i,j). For exan')ie," 1 (iJj would i calcui tu l
using one-sixteenth of the data. This A (i,j) could then be c(vqared to a
threshold. If it passes the threshold, thie nevt estimt A, (i,i) at th,
next high level resolution is calculated using an additional one-sixteenth
of the data. This procedure is continued until all of the data points are
used, giving the best estimate Ab(i,j) or until a threshold is not exceecee,
which aborts the evaluation process.
The samping rule of the data set is shown in Fig. 1. 'Ihe coarse
spacino is taken in a pseudo-random mamer. Each pass req-uires tat one
additional -i:.:e] be chosen from each block. TIhe choice of the sr':0inl
:ttithed is iri .xrtant and affects the matchina procedure. Thir7 %il lb
studlied in the follewing section.
The threshold soquence is taken as a constant value. It is shcwn that
the threshold constant is between 30 to 501 of the averaqe value of the,
intensity per pixel in the template imnae.
- - I i IIII I I II I I I I I
A! , tlL Ci 1, 11%,- lk i t ol t~t I~t' 'h
rr i I-' m u kt i oz 1and1 backrc'1v L. y
Til t I t C uns)I s 1 cIharj ic t r " F6
lt0 iriiisity iiistribution has Lxenuanmiz' , UUt( 1 2.8 s;"
~~has (,*t w-GuSsian di st vibut i " ,-r:-h
n 43, t-ho backr.rouid (ot T"iI) has noixc. re' - i JInt~
ouafiftizeui lovels Letwen 0 and 30.
Wjt~j ' "the intensity distribDution can be~ exoresseci as
(x,) = 128 x CI X1>:nI- (x-x k>i j c4- :<3k= k
rect (- 'x rect (vh)'(a
'-hore '"" is brok\en into K blocks (1 = 3)
rect (N.')function is dlef Lnt-d as Whe 'slit'" unction;
and1 tue ausia noise ha-s the forrs;
(t~ . \ X j 4 3x43) )(3h)le'T, Ute backurcundi noise, dist-ribution is
I TN'r{ 15 0 x + 2 -x (RLN (1) 0.5j (30)
N',we expresq the tejIcplate as
+~ 0. 5 -,: 1 l tnl
wnr , , 1 at intoD 3,4, an
Considr ~.n th Ois, thent21
is = 1 (1 4~ 0. 5 :-: f I
The template and scone imracles are shown in ficurines ---a :- 2bD. Then
total and averagle intensity of both the terplate and scune arot :ivon l
Table 1. The ratio of average noise inte-nsity to average sic~nal. initensit:
is very high.
1B) Similar-ity Measure
In rrhbere(Jisteration, the hqxrtant criteria is th ereo:
si;-ilaritv ~t~e the t:xk ir-Ia 10s. The Tleasurll> C,- sin.1arll itv sheep: -,e
abeto providie ca larcea separation lxtenthe i,,ch peDint a:..: ail otho: .,
test locations. Diftferent values for thec rtasure of siLi larit',, are s2.
in Fi.3. We notice that the no-tchina7 points (denoted by + )have iovwer
dlistinct ;,b(i,J) values from those of the non-matche-d points.
C) The Samplincg Rule
'!he ruile to pi1ck up the samulces can af11fect tlie- llctc.ineoejrs
in Fici. 4, four different coarse spacinis , ranr:Lxalv hte hv xe
cons i ccred. Fi(Tures ",.a, 5.b, 5.c show the dependence of the :casure of
sum ilarity At~ji,j) for different (i,j)'son the samoplin,- -x-int-q. Theic
figaues dumnstrate that block 1. convergjes faster. 1P block 1, each
additional pixel is taken as far ais possible from the =iiinal. pi>:el.
D) Thresholdinj Sequence
From Fig. 3, wi notice the clear distinction lx:txtwcn tlie 1 'st nkitch
A 31 923 ACUIITO MAG ANDDATA COMPRES U CIT COL NEW '20ORK D L SCILN E 0 R8 A 09 SR T 9-8 3050D5
i S A 0 R-81-0169.UNC ASFI D FG172N
lli1 28I-
11111.25 11111J14
1 I P f jT1 N 1
IV Perforrunce coiparison of the different criterion function
WeM have cpa~re] the e~rfono nce of different nuasures of sidlarity.
(These results have been shown in Table Ii. It turns out that the absolute
difference measure is faster than other measures.)
These ieasures of similarity are as follows:1. Averaue Absolute Difference Measure (ABS):
M iABS(i,j) 1- T(pq) '
M*" I = 1 10
2. Absolute Difference Measure (AB):
AB(ij) 1Sj(p,q) _ T(p,q)4M p=1 q=l
3. Correlation Coefficient Measure (COE):
M 2202 - a5
COE(i,j) =
fM - 0 4 2M0-515
where oi, 0a, 03, 0 and as as given in equation
4. Normalized Correlation Function (NCF):
M MS S (p,q)T(p,q)
NCF (i, j) p=q=
M M
5. Direct Correlation Function (DCF):
M M
DCF(ij) 1 Z E Sij (p,q) T(p,q)DCFVi j) = .Zjt1 q--
IV Pcrfonuince comarison of the different criteirion function
Wck have cC lpared the perfoni-ice of different nmiasures ol sinularity.
(These results have been shown in Table 11. It turns out that the absolute
difference measure is faster than other masures.)
These neasures of similarity are as follows:
1. Average Absolute Difference Measure (ABS):
AB=ij 1 M M siT p q)
ABS(ij) - S J(p,q) - T(p,q)M =' p T- ci
2. Absolute Difference Measure (AB):
1 E E ""AB(i,j) = 1 q- S 3 (p,q) - T(p,q)
3. Correlation Coefficient Measure (COE):
M202 - 0405
COE(ij) =
Me- 4 2 MJM2 a 3 a o 2
where o0, 0, , 0 and o5 as given in equation
4. Normalized Correlation Function (NCF):
M M ..E E. S 3 (p,q)T(p,q)
NCF(i,j) = p=1 C-I
mM1) E Si (p,q))}2
5. Direct Correlation Function (DCF):
M M
DCF(ij) = 1r S q= (p,q) T(p,q)I q I
F7)
Thuc results of the comparison between the five muasu-es of sinilarity arc
shown in Table II. It turns out that the A13S n asure gives the best match
between the template and the scene. The COE is tine consuming. The
thresholding sequence f(b) has been used as in reference (1), where f(b)
is an enpirically derived function.
f(b) = (b"' - 0.25b - 0.25)/ b(b + 1)
The NCF nimasure is slightly slower than the COE measure. DCF function is
the worst.
.1
88)
V Conclusion
In this paper a fast algorithm for image matching has been studied.
The image similarity measure used is the sum of the absolute intensity
difference between the template and the scene. The thresholding sequence
has been found to be a constant and equals approximately 50% of the
average intensity per pixel.
The performance of the algorithm has been tested by cinputer
generated images (English letters) and Gaussian noise.
Ccararison of the algorithm with other measures of similarity proves
that the average absolute intensity difference is faster than others. We
intend to investigate the algorithm performance with real images and at
real time.
89)
(1) 1). R. Sullivu and J.S. Martin, "A coarse search corrIcation tracker
for inmauu re istration," IFE T-ans. Aerosp. Electron. Sst., vol.
AES-17, pp. 29-34, Jan. 1981.
(2) D.I. Barnea and H.E. Silverman, "A class of algorithns for fast digital
image registration," IEEE Trans. Ccmput. vol. CP-21, pp. 176-186, Feb. '72.
(3 M. Svedlow, C. Mc-illemn and P.E. Anuta, "Image rejistration:similarity
measure and preprocessing method ccparisons," IEEE Trans. Aerosp. Elec-
tron. Syst., Vol. AES-14, pp. 141-149, Jan. 1978.
(4) E.L. Hall, D.L. Davies and ME. Casey, "The selection of critical subsets
for signal image, and scene matching," IEEE Trans. Pattern Analysis and
Machine Intelligence, Vol. PAMI-2, pp. 313-322, July 1980.
(5) R.Y. Wong and E.L. Hall, "Performance coniparison of scene matching
techniques," IEEE Trans. Pattern Analysis and Machine Intelligence,
Vol. PAMI-I, pp. 325-330, July 1979.
(6) R.Y. Wong, "Sequential scene matching using edge features," IEEE Trans.
Aerosp. Electron Syst., Vol. AES-14, January 1978.
(7) H.K. Ramapriyan, "A multilevel approach to sequential detection of
pictorial features," IEEE Trans. Ccuput., Vol. C-25, pp. 66-78, Jan.,'76.
(8) R.Y. Wong, E.L. Hall, and J.D. Rouge, "Eierarchical search for image
matching," in 1976 Proc. IEEE Conf. Decision and Control, Dec. 1976.
90)
Block 1
subset b=1 subset b=2 subset b.3
Er ElFig. 1 Sampling manner of the data set(each nass reauires that one additional pixel be chosen from each block )
91)
0 . .V .. ---7-.. . .. .4
•K - -, B "
-T.- - " .4 - 4- + .,--
* . -"9- -4 . 4 -
-4
- .3
Pig.ld. Templat~e image
92)
.. ..- + + 4 • 4 4- .
+
.-. 4 4 4- + * 4. +
.-4- . ..
. --+.- 9 + . - . .-. - 4 -- 4
-' 4 . -.4.
, -. 4--- + .- - .4
" * . - --t .
... ..... .. . , , .
93,
1ir
Av<; )
+ -,
+ -.. , :
4•, -.", "
* 4 t 4.
4. ++1 + * 4.I- + + * 4 t W.
- -I
- . + i l " _Ii= " i-41
. Q.'.- LA 47 a uv *f
,,CiCK hO SDA+
Fig.3. he satitica beavio ofthe . 0 C~i)
94)
Elc.1 Bb-ck 2Blick 3 Blick A
7Ti~j774 II 1~I
12 * IC 6 I .j~
Fig.4. Sampling manners
95)
z )
-1
7-
3k~ ~7k
-4
F -~
~F-I-V
f\j~1
b
~z~1 -'
CAU$IAN AEC~3.Z
?ig.~a. BOLCK 1,2,3,4 A~S(13,2)
9~)
~
.d~ ~ I
/
/ ~2z~
/
rC'7~C~I If
* F
L
F
I- I-<I
A . . .
~U~TAN *~pcr ''~7~
Fig.5~b. BOLCIC 1,2,3,4, A~s(13,7)
_____ J
97)
7
- ,ASA rE-
Fi.3c B C ,23,,AB(/2
ILt
