SPECTRAL CHARACTERISTICS OF EARTH-SPACEPATHS AT 2 AND 30 GHz

Robert A. BaxterDaniel B. Hodge

The Ohio Store University

ElectroScienceLaboratoryDeportment of Electrical Engineering

Columbus, Ohio 43212

Technical Report 78429q-7

August !978

Prepared forNational Aeronautics and Space Administration

GODDARD SPACE FLIGHT CENTERGreenbelt, Maryland 20771

https://ntrs.nasa.gov/search.jsp?R=19790008902 2020-04-21T20:52:55+00:00Z

M

TECHNICAL REPORT STANDARD TITLE PAGE

1. Report No. 12. Government Accession No.i

!±

SPECTRAL CHARACTERISTICS OF EARTH-SPACE

PATHS AT 2 AND 30 FHz

7. Author(s)

Robert A. Baxter & Daniel B. Hodge9. Performing Organization Name and Address

The Ohio State University ElectroScienceLaboratory, Department of ElectricalEngineering, Columbus, Ohio 43212

12. Sponsoring Agency Name and Address

NASA, GSFCGreenbelt, Maryland 20771E. Hirschmann, Code 951, Technical Officer

3. Recipient's Catalog No.

Report Date

....Aug_ust_ 19786. Performing Organization Code

rOL /O_.._--/_L ...................................

110. Work Unit No.

4

!-1_-. Contract or Grant No.

_NAS5-_225_5_ .....................13. Type of Report and Period Covered

Type IITechnical Report

14. Sponsoring Agency Code

15, Supplementary Notes

This material was also used as a thesis submitted toThe Ohio State University

17, Key Words (S_lected by Author(s))

16. Abstract

In recent years, communication system designers have pushedoperating frequencies higher and higher in an effort to meet in-creasing bandwidth requirements for extremely fast data rates.However, these high operating frequencies are accompanied by in-creased signal scintillation, especially at low elevation angleson earth-space paths. In this study, spectral characteristicsof 2 and 30 GHz signals received from the Applications TechnologySatellite-6 (A_S-6) a_e analyzed in detail at elevation angles

ranging from 0v to 44v.

The spectra of the received signals are characterized by slopes

and break frequencies. Statistics of these parameters are pre-

sented as probability density functions. Dependence of the spec-

tral characteristics on elevation angle is investigated. The 2

and 30 GHz spectral shapes are compared through the use of scat-

ter diagrams.

(over)

18. Distribution Statement

19. Security Classif. (of this report)

U 20. SeCurity Classif.-(of--t-_s page-i----[21--l_lo---o-f-l_o-ge-s-_-2,_P_ce _*_--u I 80 I"For sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151.

Th

_=ory..=ory,tion

TABLEOF CONTENTS

Chapter

I

II

III

IV

V

VI

Page

INTRODUCTION.................. I

THEORETICALBACKGROUND............. 3

EXPERIMENTDESCRIPTION............. I0

THEDATA.................... 17

RESULTS..................... 43

SUMMARY..................... 65

APPENDIXA......................... 66

APPENDIXB......................... 73

APPENDIXC......................... 77

APPENDIXD......................... 78

REFERENCES......................... 7q

ACKNOWLEDGMENTS....................... 81

iii

CHAPTER I

INTRODUCTION

In recent years communication system designers have begunto use the electromagnetic spectrum above I0 GHz in an effort tomeet ever increasing bandwidth requirements. Overcrowding of thelower frequency bands have forced designers to make use of pre-viously unused portions of the spectrum. Higher operating fre-quencies are also necessary to support extremely fast data rateswhich accompany recent advances in solid state technology.

At frequencies above 10 GHz, however, the effects of the earth'satmosphere upon communication signals dramatically increase. Atthese frequencies, a communications link designer must not onlyhave a thorough knowledge of communication theory and systems de-sign but must also have a working knowledge of electromagneticwave propagation. The necessity for this additional qualificationis evidenced by the literature. Until just recently, the effectsof the clear atmosphere (no precipitation) on a communications

signal _ere assumed to be negligible. However, it has been demon-strated that clear air propagation effects can severely hampera communications link under certain conditions. In fact, for sig-nal scintillations whose minimum levels approach the system margin,the probability of an error in detection increases significantly.

Atmospheric effects upon a communications link can influencethe amplitude, phase, angle-of-arrival, and polarization of theincident signal. These effects are manifest by amplitude and phasefluctuations, gain degradation, and depolarization. The causesof these effects include molecular absorption, scattering, refrac-tive index inhomogeneities and atmospheric noise. These causesare not, in general, in one-to-one correspondence with the effects.

Molecular absorption at centimeter and millimeter wavelengthsis primarily due to the presence of oxygen and water vapor. Theformer has a magnetic dipole moment, and the latter has an electricdipole moment.

Scattering and absorption in rain at these frequencies ispronounced because the dimensions of the hydrometers are on theorder of or larger than the operating wavelength.

Refractive index irregularities arise from spatial and tem-poral variations in meteorological variables such as temperature,pressure, and water vapor content. These parameters are also in-fluenced by wind velocity and direction.

f

In clear sky situations, atmospheric gas emission is the pri-mary contributor to noise in this part of the spectrum.

The major purpose of studying the effects of the earth's at-mosphere on electromagnetic waves at centimeter and millimeterwavelengths is to provide sufficient information for a communi-cations link designer to most effectively use this portion of thespectrum. Increasing research in this area has introduced newapplications and techniques.

In radio astronomy, for example, the effects of atmosphericpropagation are important for several reasons. Earth-based radioastronomers must know what happens to a signal of extraterrestrialorigin as it passes through the earth's atmosphere. Radio tele-scopes based outside the earth's atmosphere (such as those associatedwith Skylab) can use the knowledge obtained from the earth's at-mosphere to study foreign planetary atmospheres and vice versa.

Recently, remote sensing has developed into a practical method,and in some situations the only method, of measuring meteorologicalvariables at a distance. In particular, the structure constantof the refractive index and wind velocity perpendicular to thesignal path can be calculated from statistics of a received sig-nal.

One phase of exDerimental research on atmospheric propagationeffects has been aimed at gathering long-term statistical dataof a received signal. Statistics such as mean attenuation, fadedistributions, and variance have been presented and analyzed inconsiderable detail. However, most of the analysis is performedin the time domain. The few papers that analyze the power densityspectrum of the received signal estimate a slope and compare itwith the theoretical value.

In this thesis, spectral characteristics of 2 and 30 GHz sig-nals received from the Applications Technology Satellite-6 (ATS-6) are presented. These data were taken under clear air condi-tions. Second-order spectral characteristics such as slope andbreak frequencies are calculated at elevation angles ranging from0.38 to 43.89 degrees. The dependence of the spectral character-istics on elevation angle is investigated.

CHAPTER II

THEORETICAL BACKGROUND

The earth's atmosphere is a random medium; therefore, wave

propagation through the earth's atmosphere is a random process.

As a result, literature dealing with this subject involves sta-

tistics of atmospheric parameters.

Centimeter and millimeter wave propagation is primarily af-

fected by the troposphere. The ionosphere has little effect upon

an electromagnetic wave at this frequency because the operating

frequency is well above the plasma frequency.

The first successful theory of microwave propagation throughatmospheric turbulence was that of Booker and Gordon . Difficul-

ties in their theory gave rise to the^utilization of the statis-tical work of Kolmogoroff and Obukoff _'4,

Modern theories of atmospheric turbulence have physical foun-

dations in the dynamics of fluids. The basic picture is that of

a blob of fluid, a turb, which is put into rotating motion by tur-bulent eddies. The turb dissipates its energy either by perturb-

ing surrounding turbs, which creates smaller eddies, or by gen-

erating heat from frictional interaction with surrounding turbs.

Large volumes of the fluid containing large eddies are as-

sociated with significant kinetic energy whereas viscous frictional

forces are thought to be small u. Thus, the spectrum of atmospheric

turbulence, @(u), as a function of spatial wavenumber, u, is di-

vided into three sections: (I) the energy subrang 9, (2) the in-

ertial subrange, and (3) the dissipation subrange" (see Figure

I). The inner scale of turbulence, _, is associated with the

smallest eddy size, and the outer scale, _o, is associated with

the largest eddy size. The inner and outer scales of turbulence

are related to the spatial wave numbers, u. and u , as u. = 2_/_.

and u^ R 2_/_ o. The inner scale of turbulence is°on thelorder Iof I _m _. The outer scale of turbulence for an earth-space link

can be taken as the effective height of the atmosphere since tur-

bulent eddies larger than the effective height cannot exist. Avalue for the effective_atmospheric height has been obtained by

Devasirvatham and Hodge I as 6 km and will be used throughout this

paper.

The shape of the turbulence spectrum is different in each

of the three subranges. Most papers dealing with this subject

assume that form of the turbulenceaspectrum which is applicablein the intertial subrange given byJ

@(u) = 0.033 C2 -mn u (1)

where C2 is the turbulence structure constant and m = -11/3 for

the Kol_ogoroff spectrum.

u)ENERGY

_1 SUB\ RANG E INERTIAL

I I

u o u i

DISSIPATIONRANGE

Figure I. Spectrum of atmospheric turbulence.

The validity of this assumption is based upon the size ofthe first Fresnel zone with respect to the inner and outer scalesof turbulence, i.e., upper and lower bounds are found such that_i <<J-_-L << _n, _ where _ is the operating wavelength and L isthe effective _ignal path length through the atmosphere.

Since propagation through the atmosphere is a random process,a signal received from a satellite is characterized in terms ofcovariance functions of the amplitude and phase scintillations.Expressions for the spectral density of the amplitude and phasescintillations are obtained by taking the Fourier transform ofthe covariance functions.

Analytical theories start with Maxwell's equations or thewave equation and introduce scattering and absorption character-istics to obtain differential equations from which the covariancefunctions can be calculated. The expressions obtained are quitedifficult to evaluate which leads to various approximate methods.On the other hand, transport or radiative transfer theories dealdirectly with the transport of energy through a medium containingparticles. The basic integrodifferential equation is known asthe equation of transfer.

4

Amongthe various approximate methods used, the most commonis the Rytov method. The Rytov method assumesthat the sumofthe perturbations on the wave are much less than the magnitudeof the incident wave and that the dimensions of all refractiveinde_OPerturbations are large comparedto a wavelength. Lee andHarp have shownthat it is possible to relax these conditionsby considering both the effects of these conditions on the mathe-matical expressions and the practical limitations of the measure-ment. Ishimaru** states that the Rytov solution is valid onlywhen the variance of the log-amplitude fluctuation is less than0.2_0.5.

A commonlyused approximation in the development of receivedsignal covariance functions is Taylor's "frozen-in" hypothesis.Temporal variations in the received signal can be caused by re-fractive index changes along the signal path due to advection,convection, turbulent motions, etc. In addition, the motion ofthe atmosphere itself results in a transformation of spatial vari-ations of refractive index into temporal variations. Taylor'shypothesis assumesthat the temporal variations of the refractiveindex are only due to the average wind velocity perpendicular tothe signal path and, thus, can be removedby a simple transformationof the spatial variation, i.e.,8

n(7,t) = n(7-_t) (2)

where n is the refractive index, 7 is the position vector, _ isthe perpendicular wind velocity, and t is the temporal parameter.Thus, the effect of the frozen-in hypothesis on the time-lagged

(temporal) amplitude covariance function, Bx(_), can be expressedmathematically as

Bx(T ) = <x(7,t) x(7,t+_)>

= <x(7,t) x(7-_T,t)>

Bx(T) : (3)

where C is the spatial amplitude covariance function, T is thetime la_, x is the log amplitude fluctuation, and < > denotes en-semble average.

A quite significant assumption used in analyzing the ampli-tude fluctuations of the received signal is that of stationarity.The assumption of stationarity implies that the measurement isindependent of the time it was taken. The validity of this assump-tion is tested in Chapter IV.

Mathematical Formulation of the Spectrumof the Amplitude Fluctuations

Using the Rytov method, it has been shown that 13 the receivedfield can be expressed as

u(7,t) = Uo(7)exp[_b(7,t)] = Uo(-_)exp[x(-_,t)+js(-_,t)] (4)

where Uo(_) is the unperturbed incident field and the real partof _b, x, Is the log-amplitude fluctuation and the imaginary partof _b, s, is the phase fluctuation.

The temporal amplitude covariance function is defined as

BxCT) A= <x(_,t) x(_,t+T)> (5)

where < > denotes ensemble average. Physically, By(T) is a measureof the similarity of two observations of the log-a_plitude fluctu-ations separated in time by T seconds. If Taylor's hypothesisis assumed valid, the temporal amplitude covariance function isidentical to the spatial covariance function (see Equation (3)).

Tatarskii 9 has shown that the spatial amplitude covariancefunction for a plane wave propagating in a medium with a purelyreal refractive index and a Kolmogoroff turbulence spectrum givenin Equation (1) takes the form

Bx(%) = Cx(VT) = 2_ f Fx(U,L)Jo(UW)udu0

where

(6)

= _k2L [I kFx(U,L) L u2L

and where J_(d) is the Bessel function of the first kind, v isthe perpendTcular wind velocity, L is the effective signal pathlength through the atmosphere, and k is the wave number (k=2_/X).

The spectrum of the loq-amplitude fluctuations is defined

in terms of Bx(T) as a Fourier transform relationship, i.e.,

oo

Wx(f) = .[ BX(T)e-J2_fT dT (7)--CO

and since Bx(T ) and Wx(f ) are even

Wx(f ) = 2 f BX(T ) cos(2_f%)dz0

Substituting (6) into (8) yields

Wx(f ) = _ 0.033 C2 k 7/6 L11/6n fo

(8)

where

f0

x F _ sin(z+_2)5 (z+e2)

V

u2Lz = k

I (z+Q2) -11/6

_ - f -2J2_L ff v0

2

L_o k

Note that k and X are the operating wave number and wavelength,respectively, and are not related (directly) to the fluctuation fre-quency, f.

Assuming that D>>I and changing the variable of integrationto x = z_--, Equation (9) becomes

2 k7/6 Lll/6 _-8/3Wx(f ) = _ 0.033 Cn f

0

x FII- s_in Q!(x+!)I (x+1)-11/6 x-1/2 dxo L _2(x+1)

(10)

For small fluctuation frequencies R << 1 and Equation (10) canbe approximated as

7

IWx(f) _ 0.075 <x2> fo ' Q <<I (11)

where <x2> = 0.077 C_ k7/6 L11/6. For large fluctuation frequen-cies Q >> ! and Equation (10) is approximately

2 -8/3Wx(f} % 0.57 <x >_ITo

5/3 f,8/3,Wx(f) = 0.57 <x2> fo Q >>I (12)

Equations (!I) and (12) intersect at a break frequency, fb2'given by

fb2 = 2"14fo - 2.14v (13)

Equation (13) provides a means for senSing the average wind veloc-ity transverse to the signal path.

Figure 2 shows a Bode plot of Wv(f) for a medium character-ized by a real refractive index. ThOs, in a medium with a purely

real refractive index, the spectrum of the log-amplitude fluct #=tions is approximately flat up to fb2 and then rolls off as f-_ J.

Wx(f)

0.075 < x2>

fo

i

I s

1 \f b2

Figure 2. Theoretical spectrum of log-amplitude fluctuationsin a medium characterized by a real refractive index.

8

Ott 14 has considered the spectrum of the log-amplitude fluctu-ations in a medium with a complex refractive index, n=a+jb. Thespatial covariance function in this case is given by

oO

CxlV : S U{T(U)Jo(UVT)du (1.4)o

where @T_U) : 0.033 C2 u-11/3 and C2 is the structure constantof the temperature, TT The spectru_ of the log-amplitude fluctu-

ations in this case is found from Equation (7) for low f as

2 ,v5/3f-SnWx(f) : 0.002598 C2 k2(@-_T)

and exhibits an f-8/3 slope.

(15)

Comparing (11), (12), and (15) an intersection frequency be-tween scattering and absorption regimes is found to be

v [ [aa/aT_2 ]-3/8fbl = _ 0.1928 k_J -_ (16)

Thus, the spectrum of the log-amplitude fluctuations can beseparated into regions dominated by a scattering mechanism andan absorption mechanism. This modified spectrum is shown in Fig-ure 3.

The validity of this mathematical model of the spectrum ofthe log-amplitude fluctuations will be investigated in Chapter V.

Wx(f)

I

ABSORPTION IREGIME I

Figure 3.

-8/3

SCATTERI

Ii REGIME II _I , f

fbl fb2

Spectrum of log-amplitude fluctuations in a mediumcharacterized by a complex refractive index.

9

CHAPTERIII

EXPERIMENTDESCRIPTION

The data presented in this thesis were obtained from the Appli-cations Technology Satellite number6 (ATS-6} during the time periodextending from 78 August to 25 October, 1976. The purpose of theexperiment was to investigate the propagation of millimeter wavesthrough the atmosphere, specifically to determine the effect ofelevation angle on various communication system parameters.

The receiving terminal was capable of measuring four frequen-cies: 30 GHz, 20 GHz, 2.075 GHz, and 360 MHz. However, the 20GHztransmitter failed just prior to the experiment, and the 360MHzdata was unusable becauseof radio interference from localradio and television stations. Although only two frequencies wereusable, their separation provided an excellent basis for data com-parisons.

The ATS-6, a geosynchronous satellite, is used in somefifteendifferent experiments. Prior to t_e experiment reported herein,ATS-6 was located approximately 35 East longitude over Lake Tan-ganika for the Satellite Instructional Television Experiment (SITE)based in India. In August, 1976, the satellite wasmovedto aposition over the U. S. {J<J4U West). This repositioning of thesatellite provided an excellent opportunity for measuring the ef-fect of elevation angle on propagation characteristics.

The satellite was movedan average of one degree per day andapproximately one hour of data was taken at a numberof differentelevation angles except for the very low angles where almost con-tinuous data were taken. The observable elevation angles rangedfrom -0.48 to 44 degrees; however, angles below 0.38 degrees wereconsidered unusable because of the possibility of ground reflec-tions.

The satellite's antenna for 30 GHz transmission is a 0.46

m parabolic reflector with linear polarization. The 2.075 GHzantenna is a 9.1 m parabolic reflector with right-hand circularpolarization.

The receiving terminal is located at the Satellite CommunicationsFacility of the Ohio State University's ElectroScience Laboratory(ESL), 1320 K_nnear Road, Columbus, Ohio (latitude: 40°00'10"N,longitude: 83uO2'30"W). The facility along with the antennas usedin the experiment are shown in Figure 4.

10

or---

or--.0_,_

4---

e'-0

+p-

0r-C

00

04_0r--

GJ

rO

o

or,-I.I-

11

The 30 GHz receiver is shown in block diagram form in Figure5. The antenna is a horn Cassegrainean-fed 4.6 m parabolic re-flector with linear polarization. The beamwidth of this antennaat 30 GHz is 0.2 degrees. The front end (noise figure = 18 dB)consists of a solid state mixer and a stabilized local oscillator

which produces a first intermediate frequency of 1.05 GHz. Thefront end is followed by a tunnel diode amplifier, manually-con-trolled step attenuators, and a Martin-Marietta phase-locked loop(PLL) receiver (modified for increased dynamic range). The re-ceiver bandwidth is 55 Hz, and the signal-to-noise ratio averaged52 dB at the higher elevation angles.

The 2.075 GHz receiver was built at the ElectroScience Labo-

ratory, and the block diagram of it is shown in Figure 6. Theantenna is a horn front-fed, 9.1 m parabolic reflector with linearpolarization. The beamwidth at 2 GHz is 1.3 degrees. The frontend (noise figure = 8.5 dB) consists of a transistor mixer anda stabilized klystron oscillator which produces a first IF of 30MHz. The level of the 30 MHz IF was manually controlled by a stepattenuator. The signal was then fed into the IF portion of a Collins75S-3 receiver to produce a second IF of 455 KHz. The receiverbandwidth is 4.5 KHz, which made phase locking unnecessary. The455 KHz signal was fed into a square-law detector. The averagesignal-to-noise ratio was approximately 52 dB.

The outputs of the receivers were recorded on strip chartsand input to a digital data acquisition system.

Table I gives the link calculations for an elevation of 44degrees.

Transmitter Power (dBm)Spacecraft antenna gain (dB)Spacecraft system loss (dB)

Table ILink calculations (elevation angle = 44 o )

2.075 GHz

43.039.5

- 0.4

Free space path loss (dB)

Gas Losses (H20 and 02)(dB )

Ground antenna gain (dB)Polarization loss (dB)iGround waveguide loss (dB)Signal level at front end CdBm)

Receiver noise temperature (OK)Receiver bandwidth (Hz)Receiver noise level (dBm)

Signal-to-noise ratio (dB)

-189.9- 0.02

40.0- 3.0

m

- 70.8

38O04500-126.3

55.5

30 GHz

33.039.0

- 1.0

57.8

- 1.5- 89.0

1800055

-138.7

51.2

12

30 GHz RECEIVER BLOCK DIAGRAMJ

4.6M 1.05GHz

,'N_ ] MIXER "_ DIODE

SWITCHED

/GHz I ATTENUATORS

L Lil STALO I

l

FRONT END

MIXER

rii _co÷MULTIPLIERS

SYNTHESIZER

60MHz IOMHz

MIXER

70.0 MHz

H PHASE iETECTOR

fI0.0 MHz

-----_ I0.0 MHz

10.0025 MHz-----,J,- 70.0 MHz

MARTIN ,MARIETTA

PLL RECEIVER

AMP+

M IXER

f10.0025 MHz

I---- "--

IIIII]

2.5 KHz

1_ 1

1

STRIP CHARTRECORDER

I

I OUTPUT

O-5V

TO DIGITALDATA

RECORDINGSYSTEM

Figure 5. 30 GHz receiver block diagram.

9.1m

'/

2GHz RECEIVER BLOCK DIAGRAM

_I TRANSISTORMIXER i

2045 MH_ COUPLER20, dB

fo

30 M Hz <_ _.IF

SWITCHED

ATTENUATOR

KLYSTRON

OSCI LLATOR

" SRL 7G l

2045 MHz

I DYMEC ZER[_ POWER 1SYNCHRONI SUPPLY

3t MHz

REFERENCE

f if ref

COLLINS

75S -3RECIEVER 455 K Hz__

I I ____ 6.783333 MHzX3 X2 REFERENCE

f ref

MULTIPLIERS

fo =

N=

12Nfref + fif ref

HARMONIC NUMBER =10

Figure 6. GHz receiver block diagram.

SQUARE ILAW

DETECTOR TO DIGITALDATA

RECORDINGSYSTEM

I

STRIP CHART

RECORDER

The data acquisition system is shown in block diagram form

in Figure 7. The output of the receivers ranged from 0 to 5 volts.

These analog signal levels were multiplexed and fed into an 8 bit

analog-to-digital converter (A/D). A controller for the A/D sampled

the data at 10 samples per second and 200 samples per second.

The 10 Hz rate was always used, while the 200 Hz rate was recorded

for approximately five minutes at each elevation angle.

The main controller for the data acquisition system was an

HP 2115A computer. It took care of sampling, formating, recording

on ? track magnetic tape, and keeping a log of operations such

as attenuator settings, recording times, etc. The 7 track tapeswere later converted to g tracks so that the data could be analyzed

using the ESL Datacraft 6024 Computer.

The analog signal levels from each receiver and wind speedswere also recorded on strip charts.

15

Gt LOW-PASSANALO FILTERSSIGNALS"FROMRECEIVERS

(o-sv)

____ A/D CONTROLLER

[._J MUX SAMPLING RATES:

I"-7_/_I'-"l ,o.z200 Hz

I NTERFACE [_ .....C IRCUITS F1

(

AMPEXTAPE DECK

( 7 TRACK )

©©

HP 2115ACOMPUTER

I ii i

LOGGER

Figure 7. Digital data acquisition system.

CHAPTERIV

THEDATA

As mentioned previously, two sampling rates were used: I0Hz and 200 Hz. However, the data taken at the 200 Hz rate werenot used in the work that follows because spectral frequencies above5 Hz were not significant and, thus, were for the most part con-taminated by system noise (noise in the spectra is discussed inthe last section of this chapter).

Figures 8 throug h 12 shRwsome_ypical _ata samples at ele-vation angles of 0.38_, 4.95_, 18.11v, 33.65_, and 43.44v. Eachwaveform shownconsists of 2048 data points taken at the I0 Hzsampling rate (1 data record = 2048 data points). There is ob-viously a significant effect of elevation angle on the receivedsignals. This effect h_s been studied in the time domainbyDevasirvatham and Hodge . In this study, the effect of elevationangle on the signal spectra is considered.

A. Stationarity

If the signal is "wide sense stationary ''15 its mean, m, isindependent of time and its autocorrelation function, R(T), isdependent only on lag time, z, i.e., it is not dependent on abso-lute time. In analyzing a data record, the mean of all 2048 pointswas calculated; thus, the mean is forced to be independent of timefor any given data record (compensation processing for a nonsta-tionary mean is discussed in the next section).

The autocorrelation function is defined in general as 21

TI

R(T) : lim _T F f(t)f(t+T)dt (!7)T÷_ 2T '

It is more convenient, however, to calculate an autocovariance coef-ficient which is related to the signal waveform, f(t), by

p(T) : T+oolim_T T_._T[f(t)-m]o2 [f(t+_)-m]dt (18)

where m is the mean and o the standard deviation. For a periodic,sampled waveform, Equation (18) can be approximated as

17

In,-w>

0

DO Ld

0 Wn-

il

N

_i 1-W _

Od

t I

COOl dNEI

O0o

CO

r _

bJc_Or)

_w

I---

INQ

0

0_-i

Ol 0

[80} dNEI

OO

f_,,,-4

o

II

I1)

4-_

%

or--

°r,._

Q)

*r--

0

(13

Q..

0!,,,i m

'0.0o

EL, = 4.95 °

2 GHz RECEIVER

I I I I I ,I

31&. 13 88.27 tO2,1&O 138.53 170.87 20_. 80

TIME (SEC)

0

amC__w(=,

r,

,mr"

50 GHz RECEIVER

¢)

10,00I I I I I I

31_.13 68,27 I02. _0 138.53 170,67 201&. 80

TIME (SEC)

Figure 9. Received signals at el. = 4.Q5 °.

19

...,_

,'1'1

¢r"

t,O.O0

EL. = 18.tl °

2GHz RECEIVER

I I I I I I

3_.13 88.27 102,_0 136.53 t70,67 20_, 80TIME ($EC)

i.,$i

OD

_'0°O0

30GHz RECEIVER

I I I I I I

3¼.t3 88.27 t02._O t38.53 t70.87 20¼.80TIME (SEC]

Figure 10. Received signals at el. = 18.110 .

20

CE

7'1

EL = 33.65 °

2 GHz RECEIVER

i t m i 10.00 3_4.13 68.27 102. u,O 136.53 170.67 ?.rJg. 6rJ

TIME (3Eti

-7 :30 GHz RECEIVER

el._

cI;

"'I 1 I I [ I IIO.O0 3Li. 13 68.27 I02. LiO 136.53 i70.87 20q,80

TIME (SEC)

Figure 11. Received signals at el. = 33.650 .

21

_.=ql

oo-r'_

O_

rr

EL, = 43.44 °

2GHz RECEIVER

C3

" I I I t I ItO.O0 3t&.13 88.27 I02._0 138.53 170.87 20_&.80

TIME (SEC)

nm

O_

! i

0.00

30 GHz RECEIVER

! I I I I I

3tl. 13 88._7 102. _,0 138.53 170.87 20_. 80

TIME (SEC)

Figure 12. Received signals at el. = 43.440 .

22

i N [f ti)-m][f!ti+j)-m]PP(TiJ) -N i_l. o

with increasing accuracy as N÷_.

If the autocorrelation function is dependent only on lag timethen the time interval that is to be examined for stationaritycan be divided into sections and the similarity of the autocorrela-tion function, and similarly the autocovariance coefficient, willdetermine the relative stationarity of the waveform.

The data samples shown in Figures 8-17 were examined for sta-tionarity using the method described above. An additional 2048samples immediately following the samples shown in Figures 8-12were used so that the signal stationarity could be examined overthe length of one data record. The first 2048 samples will bereferred to as the left half and the second 2048 samples will bereferred to as the right half. The autocovariance coefficientswere calculated for both halves at each of the five sample ele-vation angles and are shown in Figures 13-17. For a numericalcomparison, the squared differences between the left and righthalves were defined as

N

Squared i!I [pL(ti} - PR(ti)]2

Difference = I_ loqlo N (dBl

i 1[p: +pR ti)]120)

where Pl is the autocovariance coefficient function for the first?048 points and PR is the autocovariance coefficient for the second2048 points.

The squared differences for each of the five samples areshown in Table II along with the signal variances. Note thatthe stationarity of the sampled waveform improves steadily withincreasing elevation angle. The nonstationarity at low elevationangles is probably due to large signal variances caused by strongfluctuations; as mentioned in Chapter II, signal (log-amplitude)variances greater than 0.2 to 0.5 are considered to be caused

by strong fluctuations. The signal _ariances are shown in Figure16 as a function of elevation angle . The curves represent re-gressive fits of the data to a spherical-earth model for the sig-nal path length. Ishimaru's strong fluctuation criterion cor-

responds to -7 dB in Figure 16. Thus, only the 30 GHz data under4 elevation angle is in the strong fluctuation regime accordingto this criterion.

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Table IIAutocovari ance coefficient compari son

2 GHz Receiver

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0.384.95

18.1133.6543.44

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increasin_ lag time except for the right halves at el. = 33.65 °and 43.44 v. Their behavior is probably caused by short periodsof no bit changes in the sampled waveform.

B. Processin 9 Techniques

A Fast Fourier Transform (FFT) technique was used to approxi-mate the power density spectrum of the sampled signals (the FFTactually produces a periodigram). The periodigram was producedin two steps: CI) the FFT was used to obtain the Fourier Trans-form and i_s conjugate, F(m) and F*(m), from the time-domain wave-form, f(t)'; (2) then the conjugate pairs were multiplied andnormalized by the d.c. component of the power. CThe spectralcomponents were normalized to the power in the d.c. componentrather than the mean power so that the dependence of the spectralslope on variance was not removed.) The result, S(m), was a per-iodigram which approximates the power density spectrum with in-creasing accuracy as the sampling rate increases. This two-stepprocess is shown schematically in Figure 19.

tNote that f(t) is in units of volts; thus, f(t) represents theamplitude fluctuation and not the log-amplitude fluctuations.However, the amplitude fluctuations and the log-amplitude fluctua-tions are similar if the fluctuations are small.

3O

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I F'w (oo)

Figure 19. Generation of a periodigram.

The highest spectral component that can be produced is the

folding frequency, fH' which is half the sampling frequency, fs'i.e., fH = fs/2"

The lowest component is determined by the number of data points,

N, and is found from the relation _ = CfJ2)/(N/?) = fs/N. Forexample, a periodigram produced from dat_ sampled at a rate ofI0 Hz which uses 2048 points will contain spectral components atf = nf /N = 0.00488n Hz, where n=1,2,3,...,I024. PeriodigramspPoduce_ from the typical samples are shown in Figures 20-24.

Two time-domain processing techniques were used to make theperiodigram produced by the FFT more accurately approximate thetrue power density spectrum of the received signal: (I) slope com-pensation and (2) cosine tapering. The purpose of slope compen-sation is to remove sloping trends of the mean signal level. Theeffect of slope compensation on the periodigrams is that of re-ducing aliasing and Gibb's phenomena. Cosine tapering is a win-dowing technique the purpose of which is to reduce sidelobe levelsin the spectrum.

Slope Compensation Algorithm

Sloping trends in the mean signal level are compensated byfirst considering M points (M=IO0 for results that follow) on eachend of the sampled waveform, see Figure ?5. Averages of these

end samples are calculated, mI and m_, and a straight line is drawnthrough these averages from P_. to P ^. Each sample point isthen level adjusted accordingm_o them_ffset level of the line through

P _ and P ^. The slope compensation subroutine is given in Ap-pendix a. m×

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Cosine Taper Algorithm

16 The window function, fcT(t), for the cosine taper is givenby

fcT(t) = ½(I + cos _t)fSAMp(t), T<t<T (21)

or in discrete form as

fcT(ti) -12 1+ cos_ _ fSAMp(ti), i=0,1,2,...,M

(22)

where t. is the time corresponding to the ith sample point andM is th_ number of points to be tapered on each end of the wave-form (the value of M was chosen as 20). The cosine taper subrou-tine is given in Appendix A.

If fSAMp(t) is a pulse of width _T then the Fourier transform

of fcT(t) is

FCT(m) = _sinTm (231m(_2_T2mPl

Note that FCT(m) is proportional to -3 as _r_.

37

Figure 26 shows periodigrams produced from each of these proc-essing techniques. In analyzing the data, both processing tech-niques were used to produce periodigrams; however, only 40 points(20 points on each end of a 2048-point data sample) were tapered.

A third processing technique, filter compensation, was per-formed in the frequency domain. Referring to Figure 7 in ChapterIII, note that the signals are low-pass-filtered before they aresampled. To compensate for the roll-off of this filter, each spec-tral component in a periodigram is increased by a value determinedby the characteristic curve of the filter. Figure 27 shows the_0 GHz filter characteristic curve and the effect of filter com-pensation on the spectral components. The effect of filter com-pensation on the 2 GHz spectrum is quite similar to that shownin Figure _7.

C. Noise

It was desirable to determine the noise level with respectto the mean signal level so that the spectral shade could be ac-curately analyzed and interpreted. The noise present in the powerspectrum can be classified into two types: receiver noise and bitnoise. Even though these types of noise are caused by differentmechanisms, their effect on the power density spectrum cannot beseparated. In this section, two methods are used to determinethe noise level in the periodigrams.

Method I. Consider the _ampled waveform on a day when thesignal was quiet (el. = 43.44 ) as shown in Figure 28. A one-bitchange on the A/D converter corresponds to 0.1.3 dB and is discern-able in the figure. Note that the A/D converter was toggling onebit above and below the mean signal level (0 dB). This randomtoggling could have been caused by receiver noise, bit noise,or the atmosphere. However, since a change of only one bit isobserved, the receiver noise must be equal to or less than thebit noise.

From the above argument, a waveform which is constant exceptfor a single one-bit toggle should produce a spectrum whose maxi-mum component is a close approximation to the noise level in anygiven periodigram due to bit toggling. This level was calculatedas -81.5 dB below the dc power level.

Method II. Assume that the receiver noise is flat uD to 5Hz and drops to zero above 5 Hz. Then with a system margin ofm? dB and spectral lines spaced O.NO_A Hz apart, the noise levelis calculated as

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A periodigram of the sampled waveform shown in Figure 28 isshown in Figure 29. A linear regression line drawn through thespectral components intersects the ordinate at -76 dB.

Considering the noise levels given above, a threshold levelat which the noise "corrupts" the spectrum was first set at -80dB. However, after examining the results of slope and break fre-quency calculations (discussed in the next chapter), this "noisecutoff level" was raised to -70 dB below the dc power level,which corresponds to the peaks of the spectral components in Fig-ure 59.

41

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a2

CHAPTER V

RESULTS

In this chapter the spectral characteristics of the experi-mental data are investigated. A comparison is made between theexperimental results and the theoretical results discussed in Chap-ter II. The spectra are characterized by either one or two slopeswhich are obtained by a linear regressive (least squares) fit.Probability density functions are given for slopes and, in thecase of two-slope fits, break frequencies. The dependence of thespectral shapes and break frequencies on elevation angle (pathlength) is also investigated.

A. Approach

The basic approach was to process the time-domain digitalwaveform using the techniques described in Chapter IV to obtaina reasonably accurate power density spectrum.

The next step was to fit the spectra to a shape similar tothat shown in Figure 3. Note that the lower break frequency inFigure 3 is dependent on the rate of change of the real and imag-inary parts of the refractive index with respect to temperature,_a/_T and _b/3T respectively. The values for these quantitiesare calculated in Appendix B using oxygen and water vapor as theabsorptive gases. These values were used in Equation (16) to com-pute the lower break frequency, fhl- The results of these calcu-lations for 50% humidity at an el_ation angle of 20 o are given

in Table III. The values of fbl given in Table III ar@mseveralorders of magnitude lower than those calculated by Ott _ usingoxygen as the only absorptive gas; the reason for this discrepancyis unknown at this time.

Table IIILower Break Frequency (el. = 20 o )

Frequency (GHz) @aI@TIBbI@T

6.80 x ].N_ IK

fbl (Hz)

4.71 x In -I02.0o x !0 -Q

The above calculation indicates that the lower break frequencyshould not be seen in the periodiqrams. Therefore, a maximum oftwo slopes were fit to the spectra.

43

Two slopes were fit to the spectra by an iterative procedurebased on the minimumof the root-mean-squared (r.m.s.) errors ofeach regression. The first fit, characterized by a slope, AI,and intercept, P_, was obtained by setting the break frequency(the frequency a{ which the two slopes intersect), fb' equal tothe eleventh harmonic*, performing a linear regression on the firsteleven harmonics and then performing a second linear regression,characterized by a slope, A2, constrained about the point at whichthe first regression curve intersected a vertical line drawn throughthe break frequency, see Figure 30.

s(f)

s(o)

Pi

Pc

R

• IA2

•1- SII Ij I

LOGto fb LOGjo fc LOGio f

Figure 30. Geometry of two-slope fit.

Each successive fit was obtained by incrementing the harmonicnumber and fitting two more slopes. After obtaining the sloDesfor the maximum harmonic number, r.m.s, errors at each break fre-Quency were compared. The slopes obtained with the minimum r.m.s.error were considered the best fit, and the harmonic frequencyat which the best fit was found was taken as the break frequency.

*The eleventh harmonic (0.0_37 Hz) was chosen as the first usablebreak frequency to ensure that at least ten points were used forthe first regression.

44

By visually examining several spectra, the maximumharmonicnumberfor the break frequencv search was set at 250, which cor-responds to f =I.22 Hz. The break frequency harmonic was notallowed to com Xcloser than ten harmonics within the harmonic cor-

responding to the noise cutoff frequency. The break frequencysearch algorithm alonq with the noise cutoff algorithm is givenin ApDendix A.

After obtaining two slopes and a break frequency, a noisecutoff level adjustment was made to prevent the slopes from beingcontaminated by the noise in the spectrum discussed in ChapterIV. From the results presented in the last section of ChapterIV, the level at which the noise corrupted the spectrum was -70dB. Therefore, the levels at each point on the regression linesfor the spectral slopes were compared with the noise cutoff level.The frequency atwhich the calculated slope lines intersected thenoise cutoff level line (point S in Figure 30) was termed the noisecutoff frequency, f . Once f was determined, the entire breakfrequency search procedure wa_ repeated; however, only the harmonic

numbers below that corresponding to fc were used in the regressions.

Figures 31 through 35 show two-slope fits as calculated fromtypical samples. In Figures 32a, 33a, and 34a it is obvious thatthe two slopes do not accurately represent the spectrum. The prob-lem is most probably in the error minimization criterion and inallowing the break frequency to come too close to the noise cutofffrequency.

Because of the problems with the break frequency search routine,the results were visually classified into four types: (!) A1 > A2,(2) AI < A?, (3) AI _ A2, and (4) other unacceptable cases dueto difficulties in the computer algorithm. The type (3) caseswere recalculated using a single linear regression curve fit (thesingle slope regression algorithm is given in Appendix A). Onlytype (I) cases were used for break frequency statistics. Slopestatistics were found from the second slope of type (I) cases,and the first slope of the type (2) and (3) cases. The secondslopes of the type (2) cases were assumed to be caused by noisein the spectrum and, thus, were not used. All type (4) cases wereconsidered unusable. The statistics of the data sets are givenin Table IV.

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-.+ + +*++.+ ++ ++*+++ ++,._t*_*++ *C*+% *

4-++++ +

+

ii ,,i iI

FREQUENCY (HZ)

('a)

63

C:)OJI

nr"

ILl

-¢.E3C3

(.3t-_

-¢

E30d(D

ILl i

OD

,.r,.r't

C3

C3O0i

E3I=:3

I

275 - 17= 55

30GHZ RECEIVERDATA TYPE= VOLT

AVERAGE TYPE= SLOPE

COSINE TAPER

NOISE CUTOFF LEVEL= -70,0

FB= 0,18

RI= -3,96

R2= -10.72

÷ + 4- + ÷4- "4- .I.,._ ,.b,. 4- + + 4-

+ - ++ +_+"+ ++ .4- 4- - +++_---_, + ++_ "

+ 4*--.r.,,,+.,,c ++ + +_+ +++

+

[02 _ _ G_÷_o-' _ _ G_5o ° _ _ _FREQUENCY(HZ)

(b)

Figure 34. Two-slope fit at el. = 33.65 °.

4Q

C.

C3C,.I-I

n-

W

I

(J

._I .

I

0

"7

292 - I8= _

2 GHZ RECEIVER

DATA TYPE= VOLT

AVERAGE TYPE= SLOPE

COSINE TAPER

NOISE CUTOFF LEVEL- -70.0

FB= O. 05

At= 6.98

R2= -t9.0 t&

4- .4-

_+ 4-++

i+ .+ .,+.,-++_. +_ +_+ ++ + +

÷ ,+/,4.÷++++'*4,+++++%++ %++++,

•_* ++++ +4-+ +4.4-

1 I I I| Io-_ _ _ _'_769_1o-' _ _ _.d' _ _ _sFREQUENCY {HZ)

(a_

C)

(:3Od.

I

mr"

ELI

--¢

I

0O

292 - IB= B_

30GHZ RECEIVERDATA TYPE_ VOLT

AVERAGE TYPE= SLOPE

COSINE TRPER

NOISE CUTOFF LEVEL = -70.0

FB= 0.06

Rt= t.99

R2= -t2.90

(b)

Figure 35. Two-slope fit at el. = 43.44 °.

5O

Table IVResults of Data Classification

No. of Cases No. of cases

iType at 2 GHz at 30 GHzI

Total

142!Ig]12

q8

471

23853

11762

470

B. Probability Density Functions

In Chapter II it wa_ _hown that the spectral slopes shouldroll off according to f'_/_, which corresponds to -26.7 dB/decade

on the periodigrams. In this section, the distribution of theslopes and break frequencies are described by probability densityfunctions. Figures 36 and 37 show the slope probability densityfunctions. The mean of the 2 GHz data is -23.5 dB/decade and-21.8 dB/decade for the 3n GHz data. The distributions about thesemeans are quite different. The _ GHz data has a very high peakat -25 dB/decade and has a wide range. The 30 GHz data does notcontain a single high peak as seen in the 2 GHz data; in fact,the 30 GHz data exhibits a double peak behavior. This type ofbehavior might be the result of bit noise contamination, i.e.,the density to the left probably corresponds to turbulence-induced spectra while the density to the right might be the resultof spectra contaminated by bit noise.

There are several possible explanations for the average valueof slope to be less steep than the theoretical value. Random bittoggling would cause the spectral slope to approach -20 dB/decadeor flatter (see Appendix C). Aliasing would tend to enhance thehigher frequency components which would also make the slopes lesssteep. Causes of the variations in slope might include spatialand temporal inhomogeneous turbulence, changes in average winddirection, and nonstationarity of the process.

The break frequency probability density functions are shownin Figures 38 and 3g. Note that the bulk of the frequencies isat the lowest value that can be obtained from the break frequencysearch algorithm. This high density toward the lower values sug-gests that there might be a significant number of break frequen-cies just below the minimum value. Also, since the wind speeddata have a fairly symmetric distribution, the break frequencies

2 GHZ RECEIVERDATA TYPE: VOLT

AVERAGE TYPE: :_LOPE

COSINE TAPER

NOISE CUTOFF LEVEL= -70.0

C_

<::3

c_cOu3u_

_--._rt ""

COrmCI1

LLC3

°

r'-_ cu

A• ' ' L ' I ' ' ' _ I ' = ' D ' ' ' ' --I

-80 -70 -60 -50 -q.O -30 -20 -10

SLOPE [DB/DEC:I

Figure 36. Probability density function of spectral slopesfor 2 GHz receiver.

52

30GHZ RECEIVER

ORTA TYPE: VOLT

AVERAGE TTPEz 5LOPE

COSINE TAPER

NOISE CUTOFF LEVEL= -70°0

¢3

&-

cI:o'300_

(_3COrnrr

L_ ,,..m

)--I--

CO

r'_

-70 -60 -50 -LiO -30 -20 -I0SLOPE (DB/DEC)

Figure 37. Probability density function of spectral slopesfor 30 GHz receiver.

53

2 GHZ RECEIVER

DQTR TYPE: VOLT

QVERRGE TYPEs 3L_PE

C_SINE TRPER

NSISE CUTSFF LEVEL= -70.0

t_

Lr_

CIZ

O3

tOu_

(_3

¢fi

CE

LL._

_L4-(.j -

>--

F-- "

CO

ZU3LLJ

_00'

I; ']_' ' J I ' ; J ' ] ' ' ' ; I J _ ' ' I

0.1 0.2 0.3 O. LI 0.5

BRERK FREQUENCY (HZ)

fill I

0.8

Figure 38. Probability density function of spectralbreak frequency for 2 GHz receiver.

54

30GHZ RECEIVERDATA TYPE* VI_LT

AVERAGE TTPEs SLSPE

CS$INE TAPER

NSISE CUTSFF LEVEL= -70.0

C]SCO

03_

COrr_

C]S

b-c_

l'mH

0")Z

C_

' ' ' " '_ , ' _ , i°0.'0 0.5 0.8

' I ' ' ' _ I ' ' ' ' I ' ' ' ' I ' '

0.1 0.2 0.3 0,4BRERK FREQUENCY (HZ)

Figure 3g. Probability density function of spectralbreak frequency for 30 GHz receiver.

55

should also have a fairly symmetric distribution. Thus, someofthe break frequencies might represent second order break frequencies.For example, if the actual break frequency was below the minimum(0.05 Hz), then the break frequency search would attempt to finda break frequency higher than 0.05 Hz.

The range of values of break frequencies is quite reasonablewhencomparedwith break frequencies corresponding to appropriatewind speed data. For example, a break frequency of 0.1 Hz forthe 30 GHz data corresponds to a Berpendicular wind speed of 1.54m/sec at an elevation angle of 20 . For the 2 GHz data, a breakfrequency of 0.] Hz corresponds to a perpendicular wind speed of2.38 m/sec.

Since wind speed was recorded on a strip chart and wind speedand direction was also available from the weather bureau (about4 miles east of the receiving terminal) a comparison between meas-ured wind speeds perpendicular to the signal path and wind speedscalculated from the break frequency data was made. However, thecorrelation was quite low, particularly at the low elevation angles;the wind speeds calculated from break frequencies were mostly greaterthan those measured directly. A poor correlation was not surprisingsince the measured wind speeds were point measurements on the groundand the calculated wind speeds represent an average value alongthe signal path. For this reason, inhomogeneities in the atmos-phere could cause poor correlation between measured and calculatedwind speeds, especially at the low elevation angles.

Co Dependence of Spectral Characteristicson Elevation An_le

According to _ discussion in Chapter II, the spectral slopesshould obey the f- " (-26.7 dB/decade) law. No dependence onelevation angle is found in the weak fluctuation theory. However,

a_ mentioned earlier, the 30 GHz data at low elevation angles (below4 _ does not satisfy the weak fluctuation criterion according toIshimaru. The strong fluctuation theory predicts a spectral slopeproportional to f or 0 dB/decade (see Appendix D). It is notclear at this point how the strong fluctuation theory interfaceswith the weak fluctuation theory.

Figures 40 and 41 show the dependence of spectral slope onelevation angle. The crosses represent maximum and minimum valuesand the circles are the average slopes, in dB/decade, at each ele-vation angle. The dashed line indicates the theoretical -26.7dB/decade slope, and the solid line represents the average valueof all measured slopes. The effective path length scale is basedon a spherical earth model with a 4/3 radius and a homogeneousatmosphere of 6 Km. The effective path lengths ranged from 8.6Km to 318 Km.

56

Oo J-

I

,L.3_LLJ I,-m

rn

_JQ_

J lCO

!

f_I

I

!.

o.

)(

2 GHZ RECEIVERDATA TYPE: V_LT

AVEflAGE TYPE._ 3L_PEC85I NE TAPER

NSISE CUT[IFF LEVEL-- -70,0

tI

' i

EFFECTIVE PRTH LENGTH [KM)200 SO 30 20 15 I0! _ I I I I

' ' ' I ' ' '' I '' ''I'''' I" ' '' I '' ' ' I ' ' ''I ' ' ''I '' ' 'I

0 5 10 15 20 25 30 35 40 45

EL, (DEG)

Figure 40. Dependence of spectral slope on elevationangle for 2 GHz receiver.

57

CD

0

?

c_oJ-

!

C_) co-

CZ_

L_EL_

__]CO

C_

CD

I

(

d cI"

-- MEASURED

THEORETICAL

A VOGEL, ET, AL.® O.S.U.

A)( +¢

' I

30GHZ RECEIVERDRTR TYPE: VOLT

AVERAGE TYPE: $LSPE

COSINE TAPER

N_ISE CUTOFF LEVEL= -70.0

-- m m --_ .

I

EFFECTIVE PRTH LENGTH (KMIaoo so 30 20 Is lol,,,,,,l, L L t i

I ' ' I ' ' ' ' I ' ' ' ' I _ ' ' ' I ' ' ' ' I ' _ ' ' I ' ' ' I ' ' ' '" I

0 5 ID 15 20 25 30 35 L_O u,5

EL. [DEG)

Figure 41. Dependence of spectral slope on elevationangle for 30 GHz receiver.

58

The 2 GHz data has a quite large maximum-to-minimum spread,although the average is quite close (3.2 dB/decade above) to thetheoretical value. The average points at the low elevation anglesand at the high elevation angles are in good agreement with theory.Most average points between the elevation angle extremes are sev-eral dB/decade higher than the theoretical value.

The 30 GHz slopes have a much narrower maxim_-to-minimumspread. Data taken by Vogel, Straiton and Fannin is also shownin Figure 41. Both data sets seem to be in fairly good agreement,with the data taken by Vogel, et al. tending to have less steepslopes at the lower elevation angles. It is unknown whether thefact that the data obtained by Vogel, et al. was measured overwater might have caused less steep slopes at low elevation angles.

Note that _here seems _o be a trend toward steeper slopesfrom el. = 0.38 V to el. = 5_. This trend is also apparent in thedata taken by Vogel, et al. This trend might have something todo with strong fluctuations; however, the slopes do not approach0 dB/decade as predicted by the strong fluctuation theory. Aper-ture averaging might also be an influence of this trend since aper-ture averaging would reduce the small-scale fluctuations whichwould in turn cause steeper slopes.

Immediately following 5o, the slopes show a sudden changetoward less roll off. The data taken by Vogel, et al. also showsthis sudden change, but it is not as pronounced as the O.S.U.data. This sudden change in values seem to suggest a transitionregion between two mechanisms; for example, between strong andweak fluctuations or between angle-of-arrival fluctuations andamplitude fluctuations.

The slope s above 5o are mostly flatter than the theoreticalvalue. The reason for this is again probably due to bit noise.

The 2 GHz break frequencies as a function of elevation angleare shown in Figure 42. The spread is seldom more than 0.05 Hz,and, as pointed out in the p.d.f.'s, most of the break frequenciesare close to the lowest possible frequency which could be obtainedfrom the break frequency search algorithm.

The 30 GHz break frequencies as a function of elevation angleare shown in Figure 43. The spread is much more than the 2 GHzdata, sometimes as much as a factor of 20 greater than the 2 GHzdata. The 30 GHz break frequencies are not as heavily concentratedabout the lowest value as would be expected if the break frequen-cies are indeed related to the perpendicular wind speeds as givenin Equation (13) (Chapter II).

59

2 GHZ RECEIVEROATR TYPE_ V_LT

RVERRGE TYPE; SLOPE

COSINE TRPER

NOISE CUTOFF LEVEL= -70.0

N

>--

0ZL_J

L_ "n-

L_

3_CEcu

C_

EFFECTIVE P_TH LENGTH (KM)so _0, _0 ,I0

c; } I,,, s ,I, ,, =, , = .... _,, 1 I,' '' I .... I, ,, I,, ,,10 5 10 15 20 25 30 35 [_0 u=5

EL. (BEG)

Figure 42. Break frequency as a function of elevationangle for 2 GHz receiver.

6O

30GHZ RECEIVERDATA TYPEs VOLT

AVERAGE TTPEI SLOPE

COSINE TAPER

NOISE CUTOFF LEVEL=

e

-?0.0

LO

C3

I

(9+

÷_P L"__ _ _ -®EFFECTIVE PATH LENGTH (KM)

30 20 15 10200 50

0 5 10 15 20 25 30 35 _0

EL. COEG)

Figure 43. Break frequency as a function of elevation

angle for 30 GHz receiver.

61

D. Comparison of Spectral Characteristicsbetween 2 and 30 GHz Data

To gain a better insight as to the nature of the data, thespectral characteristics at 2 GHz and those at 30 GHz were comparedvisually from scatter plots. Figure 44 shows a scatter plot ofthe 30 GHz slope versus the 2 GHz slope. Theoretically, the spec-tral slopes should be -26.7 dB/decade at both frequencies. Thisscatter plot shows that the slopes are indeed heavily concentratedaround the theoretical value.

Figure 45 shows a scatter plot of the 30 GHz break frequencyversus the 2 GHz break frequency. Theoretically, the ratio ofthe 30 GHz break frequency to the 2 GHz break frequency is 3.87,as indicated by the dashed line. The solid line represents a linearregression through the scatter points. The agreement with theoryis quite close even though the lowest attainable break frequencyis most likely too high.

62

C_

OJ-

Ia_l zQ_

_J

C3

I

COX i-80

• I

• • eee , e_ _e_w ? e e• .. : .. I.. e6"

• • ¢ _ • g

• 0 0 • •

II il IO Ol • • • il

i . O 1111 I _llV I •

•,' .,.. ,: . I

'S |' '.f, •

• J O |• lJ is

I i Ii

18I i

It

-- 26.7

-60 -_0 -20 0

2 GHZ SLOPE

dB/DECADE

Figure 44. Scatter diagram of 30 GHz spectral slopeversus 2 GHz spectral slope.

63

THEORETICAL

REGRESSION

0 0,I 0,2 0.3 0._ 0,5

2 GI-tZ BREFIK FREQUENC"f0.6

Figure 45. Scatter diagram of 30 GHz break frequencyversus 2 GHz break frequency.

64

CHAPTERVI

SUMMARY

The results presented herein characterize spectra of 2 and30 GHzsiRnals rRceived from the ATS-6 at elevation angles rangingfrom 0.38v to 44V. " "

The spectra of the received signals are characterized by atwo-slope linear regressive fit and a break frequency correspondingto the intersection of the two slope lines. Cases in which thetwo slopes were approximately equal were recalculated for a singleslope fit. The results are presented statistically in the formof probability density functions. The dependenceof the spectralcharacteristics on elevation angle is given. Scatter diagramsare used to comparethe 2 and 30 GHzresults.

Average spectral characteristics are in general agreementwith predictions based on turbulence theory; however, variationsof the slopes can be quite large. In fact, temporal variationsof spectral s]opes are v_y similar to variations of the refractiveindex structure constant . Possible causes for variations inslope include spatial and temporal inhomogeneousturbulence, changesin average wind direction, and nonstationarity of the process.

A trend toward steeper slopes appears to be evident at lowelevation angles (below 5U) in the 30 GHzdata. The cause andcertainty of this behavior is unknownat this point and shouldbe carefully examined in future experiments.

Becausethe time-domain waveforms are _ostly low-order bittoggles at elevation angles greater than 25 , the spectral slopesand break frequencies above 25 are probably biased representa-tions of the spectral characteristics.

65

APPENDIXA

COMPUTERALGORITHMS

The following routines are computer algorithms whichthe techniques discussed in this thesis.

Technique Subroutines used

I. Slope compensation MEAN

P. Cosine taper COSTAP

R. Power density spectrum FFT

4. Break frequency search CORRUPT,BREAK

5. Single slope regression ONESLP

6. Regressions LINREG,CLR

implement

Lines

!-40

44-60

64-106

1!I-182

186-213

217,298

66

I

2 £

:',c_

4 C

5 C

E, C

7 C

_ r

9 £

10 C

11 r

1,5

14

15

17 c

1_

lq

20

21

23

24

25

28

2_J

51

_2 c

.55 £

5t¢

55

3_

57

56

_9

4 o

4"1 c

4.'> c

*¢5 £

47 c

4g f

5(I C

b_

5q

b.9

In

2O

30

40

5O

,RUPROUIII'E. FIEANIY,_HAV.tAVI )

£LF*PE COhPII'tSATION

KAV--O:POIhT AVEHAGE

VAV=I;r;L(}PI:. CO_PE_,I_ATION

Y -- TIMF-DOMAIN SAMPLFD W/_VEFoRIv_

H'AV -- AVERAGE TYPF

AV1 -- PIEAIkl OF" SAMPLED WAvEFf_RIV_

n II_ENS I0|.,Iy (20_,8)

qATA NENU/IOO/

AVI.-O.

IF(KAV.EQ.{j) GO I0 30

CURREC T FOR SHIFTS OF ME'AN LFVEL [3Y ,.£AI_PLING NFN['_ POINTS

pRoM TIME-I_or,IAIN WAVFFORI,_

AVP.=N,

MP 01N T-- 2 (lU,8-1klE NO

O0 I0 I=I,NENr)

AVI=AVI÷Y{ I)

K--_PO I t',IT+ I

AVP=AV_÷Y (I_)

AVI=AVI/NEND

Av2=Av2/NENO

AVF}IF= (AV2-AVI ) / t2{!R,8-t,tEI_[})

/_VCON=AVI-AV[)IF*(NE#fl)/2,)()0 20 I=1_2D48

Y ( I )=Y ( I )-AVcON- (AVI)IF_, I )

AV_I= (AVI÷AV2)/;?.

RETURN

CAI. CULATE PIEAN OF SAI_PI ET) W/_v[Fr)I:tM AN[.) _uI]TRr. CT

PROM Y

I_O 40 I=I,204(I

AvI=AvI+Y ( I )

I)0 50 I=1,204P,

Y (I )=Y ( I )-AVI

PETURN

rNn

sUr_ROUT INE COsTAP (y.M, t,l)

cOSINE TAPER

TIME-DOMAIN SAMPLE{) WAVEFOf,_I"_

P'i=hiUMIF_,EROF F:,Af'_PL.EF_IO F'_F T/_pFlil:l) Of,.IEACH E_F[1 oF RECOH|)

N=t,IUMI::IFR OF SAMPLEs

F)IMENSION y(1)

A=DIIt.)

lq) I0 T=l,._

J=N+I- I

"=A*(I'I-I)/PI

67

56

57

58 10

59

60

61 C67. C

6.'; r

6q

65 C6_. F

67 C

61_ C69 C

70 C71 C

72. C

73 C7q

15

76

77

7( I.

7q

80

8182

838q

8586

87

88

89

911.

9P.,..)&

9_

'95

969798

'99

100101

10210/

iOb

106107

108

109

110

C

5C

C

C

c

lu

r

C

Cc

c

r

C=.5_(1.+EOS(R)!

Y(t]=Y([]_Cy{,I)=Y(J)*C

RETUR[JENn

_URROUTINE FFT(X_Y_S)

p_OUCF PFIRIhOIGRAM FROM SAMPLLI} WAvEFOR_

ilSING FFl (FORTI

x -- ARRAy OF HARMONIC FREQI.IFNCIFS

(MULTIPLES OF TLF)

y o- TIME-DOmAIN SAMPLED WAVFFUR_,

-- ARRAY OF SPECTRAL CoMPF_ENIS

cOMPLEX H

CO_MON/ARAyS3/RI_O_8)tSS(512]

_I_ENSION X( J 024) _Y(_O_8} _S(! _2_)

co_MOrJ/SLII/NFRE(_, _iHATF;q TLF

COPIMON/sET3/NDTYP eMATCH _NTAP, KAV, CUTOFt AVI t _Pl_

nATA _,¢PNTRI_OqAI,Nt_F/lO24/q_/II/

TLF=FLOAI(NRATEI/FLoATINpNTS)

RET UP SINE TARLE

CAIL FORIIB,MtSS,O,IFFRR)

DO 5 I=I,NPNTS

9{II=C_PLX(Y(I),O,)

CONTINUF

FA%l FOUHIFH TRANSFORM SAMPLFI) whVEFOH_

CAI.L FhRI (U,M,SS,-2 _ IFERR)

IFIIFEIIR.N[.O)NEHR=YEPII3HERROH IN FORT,13_O)

CALCULATE DC COMPONENT

DA=REAL(H[I))**2

pcD=AVI*AVI+DA

CALCIJLAT_ POWER SPECTRAL COMPONENTS

IN UB BELOW MEAN LFVLL

DO 10 I=_NHF

J=NPNTS+_-I

L=I-I

SW=2.*REALI_(IIIHIJ)}

_{I_)=IO._AIOC;]I)ISW/(]CP)

xC=TLF*L

x(L)=ALOGIo(XC)

coNTINUE

coRREcT SPECTRAL CoMPoNE_ITS FOR

LOW-PASS FILTE R HULL-OPF

cAt. L BWCOR(X*S)

RETURN

ENn

68

111

112 c

115 C

114 C

115 c

II_-, c

117 c

11_ C

119

12(I

121

122

123

1215

125 10

126

1_7

12A _0

130

I_P. r

15_ (-

Ib5

13_ C

157 C

15_ C

159 C

I#O C

141 C

142 C

143 C

144 c

145 C

146 C

147 C

14_

149

150

151

152

153

154

155

156 I0

157

ib8

161

167 C

163

164 C

16b

,_Ur_ROUT I NE CORUPT (CLIF,CI_TOF}

NOISE CUTOFF ADdUSTMENI;

FINO NOISE CUTOFF FRE.OUENCY hNd SEARCH

FOR f_REAK FREOiJENCY ONLY UP TO T_AT FREQUENCY

cLIF -- NOISE cUTOFF FRFoUE_cY (HZ)

cUTOF -- NOISE CUTOFF LEVEL (0_)

CO_MON/SEI_/FB,AI,PltA2#_R

COMMON/AHAySI/X(1N24)_Y|Po4A),S(_024}

COMMON/AHAYS2/XP(ln2_),YP(2OILB).yOUT(_Op4)

nATA NI4FM]/102_/

TLF:IO./PO_.

NO i0 I=I,N_IFMI

IF(YOUT(I).LI.CUIUF)GO TO 20

CLIF=5.

RETURN

CLIF=I*TLF

CAt.L BR[AK(x_s,F_AI_BI,A2,FP,I)

RETURN

FN{_

_;uP,ROuTINE _E_REAK(X,Y,F_,AI,_,A2,Eo,NLNF))

F_RFAK FREQUENCY SEARCH ArlO TWU's[OPF _IT

x -- ARRAY O F HARMONIC FRFo, UE_,)CILS

Y -- ARRAY OF SPEcTI_AL CO,nPONENTS

Fn -- _HEAK FREOUENLY

A1 -- FIRST sLOPE ([}_q/DFC)

BI -- POWE_ LEVEL INTERCEPT (05_)

A_ -- SECOND RLOPF (On/DEC)

ER -- TOTAL RMS ERROR

_E_!D-- ENDING HARmONIc FOR SFARc_

DIMENSION X(I),¥(I)

COMMON/SETI/NFREO. NRATE, TLF

coMMON/AR AYS2/E ( I 0 :>4 ), YP (_04_ ), yOUT ( I 024 )

DATA NMIN/lO/,NMAX/250/

TLF:10./2048,

NF R=NM I I',1IMIN:I

I:O

?-I+1

NFf_=NF[_+I

r,_OI F =NENO-NF

IF(NI)IF,LT,IO)GO To 15

CAI CUI.ATE FIRST SI.OPE AND POI,_ER LEVEL INTERCFPT

CALL LINREG(X.Y_I,NFH,AI.8].rHHI,YOUT)

('AICULATE SEcoN[} SLOPE

cALL CI.R (X, Y ,NFB, NE N{}, AP_. ERRS>. YOIIT )

_;LI'_¢ RMS ERRORs FNU',I HOTH I(_C-,PFSSI(}I_S

F ( I ) =ERRI+I- HR2

69

166167

168 15

169 C170

171

172

115

1 TM _0

175 C176177

178

179lt_O

181

182

18._ C11,14. C

1.85 C186

187 C188 C

189 C190 c

191 c192 c

195 C

17_ C

195 C

19_ c

197 c

1':]8

199

2UO

201 C

202

2O5 C

204

2U5 i o

2.O6

207208 20

209 C

210 C

211 30

212

21.5

21q C

215 C

216 c

217

218 C

219 C

2211 C

TFINFB.GE.NMAX)GO TO 15

GO TO I_

FR=E ( I )

FIND MINIMUM ERROR

t)(} 20 ,l:2,1I

IF(E(J).GT,ERIGO TO 20

IFR--EiJ)

IMIN=J

c OI',IT I Nile

_ECALcULFORE sLOPES FOR MINIMUM F-RROR

NFD=NMIN+IMIN

cAI.L LINREG(X,Y,I,NFB,AI,B1,FRRI,YOUI)

CALL CLR ( X, Y,NFB, NENfl,/_2,ERP2, YOUI )

FB=NFE_*TI.F

FR--ERRI÷ERR2

RETURN

FND

_URROUTINE ONF,_LPi×,S,AI,BI,FR,CI IF,CUIOF )

_INI;LE SLOPF RFGRESSION

X -- ARRAY oF HARMONIC FPE@uENCIES

-- ARRAy OF SPECTRAL CnMP()NFNTS

AI -- SLOPF lOB/DEC)

B1 -- POWER LEVFL INTFRCFPT (F}I_)

ER -- ToTA L RMS ERPOR

CLIF -- HOISF CUTOFF FRFOUFr,,Cy (HZ)

cUTOF -- HOISF CUTf_FF LEVEL (OLd)

DIMENSION X (1), S( 1 ), YOI.IT ( i02_. )DATA NHFf'_I/I 0_._/

TLF:lO./2048.

CALCULATE SL OpF.

cALL LINREG(X,S,1,]O23,AI,Bi,EH,yOUT)

FIND N()ISE CUTOFF FRE(_UENCY

nO 10 I=],NH_M].

IF(YOUT(1).LT.CUTOF)GO T(_ 2.0

CLIF:5.(4,0 TO _0

cLIF=I*TLF

RECALCULATE SLOPE USING _pECTHAi COMPONENTS

F_ELOW _JOISE cuTOFF FREQUFNCY

CALL LINREGIX,S,I,I,_I,E_I,ER,YOIJT)

RETURN

END

RuRROuTINE LINREGIx,Y,NLO,NHT,St_OPE'yI_tT,ER,yOHI )

I.INEAR RE6RFSSION

70

221 C

222 C

223 C

22_ C

225 C

22_ C

227 r

229 C

23O £231

232

23._

235

236

23F C

2.38

2_0

241

2q2 ]

2_3

2_42_5

2_c,

2*7

2_8

2_9 C

250251

252

253 2

25_255

256257

25_ C

259 C

260 C,2(,1

262 C

265 C

26_ £

2b5 C266 C

267 r

268 C

269 C

270 r

271 C

27P C

273

27@

275

X -- ABSCISSA ARRAY

Y -- ORI_INAIE ARRAY

NLO -- SIARIING INDEX OF PrGRFSSInr.I

NHI -- ENI)ING INf)EX OF REGRESSION

£LOPEI -- SLOPE OF y VERSUS X

YINT -- Y INTERCEPT

ER RM S FRRORyOUT -- ARRAy OF SLOPE AND y LNIERcEPT LINE

(yOUT=SLOPE'_X÷YINT)

r_IMENS IOKi X(I) , Y (I ) ,YoUT ( 1021_ )

I"A T A XO/-2./

XSI IN-- f) ,,YS_IM:O.

YY e.Ur_l=O.

Y2_UM=O.

rALCULATE SLopE AND Y INTERCEPT

r_o 1 I=NLO,NIII

_S i IM=XSNM.I-X ( I )-X l)

YSLIM--YSUM+Y(I)

XY_UM=XYSUIh +(X(Ii'XO)*Y(T)

W'2._IJM:X:_SU M+ (X ( ] ) "XO )w=*2

_X:XSUM/INHI-NI.O+])

Fy=YSUr'_/( Nh I -NLO÷I )

E XY = X Y SUI';I/(NHI -NLO÷I )

E X2= X2SUi_i/( NIII-NLO_'I )SLF_PF= (ExY-Ex,=EY) / (Ex2-Ex,_Ex)

Y I_!T=EY-SLOPE.EX

CALCULATE RMS ERHOR

FRRSUM=(I,

r)O 2 I:NLO,NIII

YoUT(I )-'SLoPE*IXIII-XOI+yiNT

FRnSIJM=E|_RSUM÷(IYOIJT(I)-Y(I))**2)

FR=ERRSUM/(NHI-NLO÷I)

{.H=SQRT (ER)

RETURN

END

._uBROuTINE CLR (X, Y,NLO, NHI, SI.OPE, ER, YOuI" )

cONSTRAINED LINEAR REGRESSION

X -- ABSCISSA ARRAY

Y -- ORPlNAIE ARRAY

NLO -- STARTING INOEX

NHI -- ENPlN_ INDEX

9LOPE -- SLOPE of: y VERSIIS X

ER -- R_S FPROR

ynuT -- ARRAy OF SLOPF LINE

PIMENSION X(1),Y(1),YOLITIIOPq)

CO_IMON/SET]/NFREQ,rlI<_TE,TLF

xu=NLO*TLF

71

216277

278

279

280

281 £282

2B_

28_

285 10286

287

288

289

290 £291

293

29q 20295

297

298

299 £

_00 C

Yo=ALOGtO(XO)

YO=YoUT(NLO)

XSUR:O,

Y2sUM=O,

Yy_U_I=O.

CALCULATE SLOPEDO 10 I=NLO,NHI

WstJM=XSUM÷X(I)-XO

_2SUM=X2SUM+(X(II'XO)=*2Xy_url=×YSUI_i_(XII)-_O;_Y(T)

_x=X_UM/(_JHI-NI.n÷I)

EX?=X2$1J_/(NIII-NL 0_1)

£XY=XYRUrl/(NHT-NLO_I)

_LGPE=(EXY.EX*YOI/FX_

CALCULATE P_S FRROR

FRRSUM=_,

nO 20 I=NLO,NHI

yOHT(1)=SLOPE-(XEI)'X0)+Y0

_RRSUf_=EHRSUM+(YOUI(1)'Y(1)}''2

ER=ERRSUM/(NHI-NLO÷I)

FR=SQRI(ER)

RETURN

END

72

APPENDIX B

REFRACTIVITY OF AIR IN THE PRESENCE OFWATER VAPOR AND OXYGEN

Refractivity, N, can be expressed in general as

I II IIN: NO÷ _ sir _ + Nw+ j (_SiFi + Nw) ppm1 1

(25)

where summations are over all spectral lines and

= frequency-independent refractivity (ppm)o = line strength (KHz) I

F = line shape factor (GHz-)

Nw = non-resonant refractivity due to water vapor (ppm).

The frequency-independent refractivity is given by 19

P. PwNo : 77.67 _+ 71.7 T _ + 3.744 x I05_-_ (ppm)

I-(26)

where P = dry air pressure (mbars)

= water vapor pressure (mbars)w temperature (OK)

Water Vapor. The resonance lines for water vapor are describedby

f [fl - f fl + f ]- (fl+f)2+y _ (GHz-I)(27)

f [ YI Y1 ]F_ =_1-1 (fl-f)2+y_ + (fl+f)2+y_ (GHz-1) (28)

The only water vapor line relevant to this study is characterizedby

S1 = 4.465 x 107 PwT-3"5 e-643/T (kHz)

= 0.0861 (P + 4.8 Pw)T -0"6 (GHz)

fl = 22.235 (GHz)

73

The nonresonant contributions to refractivity are

N' = 2432 P f0"IT-]'2 (ppm) (2g)W W

,, = pwflNw 1.1P "3T-3"5 (ppm) (30)

_. Since there are numerous oxygen lines it is2_onvenientto de_ these lines by a single line characterized by-

f2_f ( P V3002]013.25J_T ) (31)

F_- 2

+'--_-2) -J

1 (1013.25

along with the parameters

S2 = 5.22 kHz

Y9 = 3.92 GHz

f2 = 60.0 GHz

Derivatives of the Refractivity

To compute the lower break frequency of the spectrum of am-plitude fluctuations it is desirable to obtain an expression forthe derivatives of the real and imaginary parts of the refractivitywith respect to temperature.

a) The real part of the refractivityis found from Equation (25) as

2

Re(N) = NO + Z SiF _ + N'w (33)i=l

74

Each of the terms in the above equation is temperature dependent.The derivative of Re(N) with respect to temperature is found as

a Re(N) aNo aSl{)T : {)T + FI-aT-

{)FI {)%+ SI _ + S2 _ + _ (ppm/k)

(34)

where

{)No P Pw Pw: -77.67 71.7 7.488 x 105--=

{)T 7 T(ppm/k)

{)$1 1010 .Se-643/T{)T = -1.563 x 108 PwT-4"5 e-643/T+ 2.87] x PwT-5

(GHz/k)

{)F_ If _{)'{1_ f!+f

2Y1 _ -:i_l'l,/{)TI[(f1+f'--"i2+y_] 2 _ fl -f 2} -!)[(--_l_f) 2+y_] (GHz-l-k

aYI T_I.6{)T = -0.05166 (P+4.8 Pw)

{)F_ (f2-f-a-?--:-1"8x105-_22 )

1__T3 (GHz-l-k-1)

aN'

w PwfO. I T-2.2{)T = -2918 (ppm/k)

b) The imaginary part of the refractivity is found from Equation(25) as

2Im(N) : _ S.F" + N" (ppm) (35)

i=I ] 1 w

75

The derivative of Im(N) with respect to temperature is found as

_sI _F_' _F_ a,"DIm(N) = F_ _ + SI + $2 + w_T _T _ _ aT (ppm/k) (36)

where

PaF" (10!.3.2_)2 -I..8x_.05

aT 'Y2 [1 ,,,---_2 j

(GHz-1,_k -I )

T3 (GHz-l-k-L)

NwaT - -3.85 PPw f1.3 T-4.5 (ppm/k)

76

APPENDIX C

SPECTRAL SLOPE OF A RANDOM TELEGRAPH SIGNAL

It is desirable to find the slope of the power density spec-trum of a random telegraph signal (which is similar to random tog-gling of the least significant bit of an A/D converter).

From Papoulis 15, the autocorrelation function, R, of a randomtelegraph signal is exponentially damped as

R(T) = e-2_I_I (37)

where T is the lag time and _ is the occurrence rate. The powerdensity spectrum, S(m), is the Fourier Transform of R(T),

4X2+m ?

The spectral slope is

dS(m_ _ -8kw

dw (4X2+_)2)2!3Q)

An appropriate value of X can be obtained from Figure 28 ofChapter IV. There are approximately 310 bit changes in the 204.8second sample which gives a value of 110/204.8 = 0.54 for X.

If th_ spectral slope is evaluated in the 0.25 Hz to 0.5 Hzregion, _ mm and the spectral shape is approximately

dS(m) _-8_8___ (40)dm 25m2

Thus, the spectral slope of a waveform consisting of randombit toggling is -20 dB/decade, which is 6.7 dB/decade flatter thanthe spectral slope due to a Kolomogorov turbulence spectrum.

77

APPENDIXD

SPECTRALSLOPEOF STRONGFLUCTUATIONS

Ishimaru11 has considered the effect of strong fluctuationson the mutual coherence function, F. The mutual coherence func-tion is similar to the covariance function defined in Equation(5) except that F is defined in terms of the specific intensity,I, instead of the log amplitude fluctuations, i.e.,

F(x) = .r l(x,_)e -jk's d_ (41)L

where x is the spatial lag distance, _ is the path vector, andthe integration is along the path, L.

If the turbulence is strong, x is large, and Taylor's hypoth-esis is valid then for a spherical wave

l!(x) _ F(x.___._) (42)x 2

where

F(x) = exp{_1.46k2(w)5/3 _ Cn(X2')dx'} ,o

and v is the perpendicular wind speed, T is the lag time and C2is the refractive index structure constant, n

Because Taylor's hypothesis is assumed, the product vT isslowly varying; furthermore, changes in w do not have a signifi-cant effect on F(x). If it is assumed that w _ I0 then F(x) m I,and the Fourier transform of r(x) is simply

S(_) = -_, _> 0 (43)

which is the power density spectrum of the strong fluctuations.Thus, the slope of the strong fluctuations' spectrum is 0 dB/decade.

78

REFERENCES

.

.

o

.

o

.

9.

I0.

I].

D. M. J. Davasirvatham and D. B. Hodge, "Amplitude Scintilla-tions on Earth-space Propagation Paths at 2 and 30 GHz," Re-port 4299-4, March 1977, The Ohio State University Electro-Science Laboratory, Department of Electrical Engineering;prepared under Contract NAS5-22575 for NASA-Goddard SpaceFlight Center.

H. G. Booker and W. E. Gordon, "A Theory of Radio Scatteringin the Troposphere," Proc. I.R.E., vol. 36, April, 1950, pp.401-412.

F. Vlllars and V. F. Weisskopf, "On the Scattering of Radio-waves by Turbulent Fluctuations of the Atmopshere," Proc.I.R.E., vol. _3, October, 195_, pp. 123-]_39.

R. A. Silverman, "Fading of Radio Waves Scattered by DielectricTurbulence," J. of Appl. Phys., vol. 28, April, 1957, pp.506-51!.

A. D. Wheel.n, "Relation of Radio Measurements to the Spectrumof Tropospheric Dielectric Fluctuation," J. of Appl. Phys.,vol. 28, June, 1957, pp. 684-693.

Armel Picquenard, Radio Wave Propagation, John Wiley and Sons:New York and Toronto, 1974, Cho 2.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, R.A. Silverman, Transl., McGraw-Hill: New York, 1961.

Akira Ishimaru, "Temporal Frequency Spectra of Multi-frequencyWaves in Turbulent Atmosphere," I.E.E.E. AP-20, No. I, January,1972, pp. 10-19.

V. I. Tatarski, "The Effects of the Turbulent Atmosphere onWave Propagation," transl, from Russian. Israel Program forScientific Translations: Jerusalem, 1971.

W. Lee Robert and Jeffrey C. Harp, "Weak Scattering in RandomMedia, with Applications to Remote Probing," Proc. I.E.E.E.,vol. _7, No. 4, April, 196q, pp. 375-406.

Akira Ishimaru, "Theory and Application of Wave Propagationand Scatterinq in Random Media," Proc. I.E.E.E., vol. 65,No. 7, July, 1977, pp. I03_-I061.

79

14.

15.

16.

17.

P. A. Mandics, R. W. Lee, and A. T. Waterman,Jr., "Spectraof Short-term Fluctuations of Line-of-Sight Signals: Electro-magnetic and Acoustic," Radio Science, vol. 8, No. 3, March,1973, pp. !85-_01.

Akira Ishimaru, "Fluctuations of a BeamWavePropagating througha Locally HomogeneousMedium," Radio Science, vol. 4, April,!969, pp. 255-305.

R. H. Ott and M. C. Thompson,Jr., "Atmospheric AmplitudeSpectra in an Absorption Region," I.E.E.E. Trans. AntennasPropagat., vol. AP-26, No. 2, March, 1978, pp. 329-332.

A. Papoulis, Probability, RandomVariables_ and StochasticProcesses, McGraw-Hill, New York, 1965, Ch. 9.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra,Dover Publications, Inc., New York, 1958.

Warren D. Brown, "A Model for the Refractive Index StructureConstant at Microwave Frequencies," Sandia Laboratories,New Mexico, February, 1977.

18. W. I. Vogel, A. W. Straiton, and B. M. Fannin, "ATS-6 Ascending:Near Horizon Measurements over Water at 30 GHz," Radio Science,

vol. _2, No. 5, September-October, 1977, pp. 757-765.

19. H. J. Liebe and G. G. Gimmestad, "Calculation of Clear AirEHF Refractivity," Radio Science, vol. 13, No. ?, March-April,_.a7£, pp. 245-2_1.

'_0. R. H. Ott and M. C. Thompson, Jr., "Characteristics of a RadioLink in the 55 to 65 GHz Range," I.E.E.E. Transactions onAnt. and Prop., AP-24, November, 1976, pp. 873-889.

21. A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill: New York, 1962, Ch. 12.

80

ACKNOWLEDGMENTS

The authors wish to express special thanks to Dr. David M.Theobold for his numerousand fruitful ideas, especially those de-veloped in Appendices C and D, given to them throughout the courseof this work.

Thanks also go to Robert C. Taylor for his expert assistanceduring and after the experiment, to D. M. J. Devasirvatham forthe preliminary data analysis which inspired this work, and toDorothy M. McGinty for typing the manuscript.

81