AD-A155 974"DNA-YPR-84-80

APERTURE ANTENNA EFFECTS AFTER PROPAGATIONTHROUGH STRONGLY DISTURBED RAND)OM MEDIA

Dennis L. KneppMission Research CorporationP.O. Drawer 719Santa Barbara, CA 93102 -0719

1 February 1984

Technical Report

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19, KEY WORDS (hoC,,- no. or, rrver, de if ",•"pe .r. a.nd d"fifv h1 tIOt& tuRmR•I?)

"Aoerture Antennas Random MediaSpace Based Radar Dispersive Media"Signal Scintillation Satellite Communications

20_Ionospheric Propagation Frequency Selective Environment20 ABSTRACT ,(Conetm,, ,n ie,,ere side if n•e yos .• d e .•di "w i v bk,,rl r mrn •mhr)

A-strongly disturbed layer of ionization irregularities that isused as a propagation channel for radio waves can degrade the propagatingwave and thereby affect the resulting measurements at the receivingantenna. The antenna aperture itself also affects measurements of thereceived signal hy its inherent averaging process. Here an analytic solu-tion for the two-position, two-frequoncy mutual coherence function, validin the strong-scatter limit, is used to characterize the propagation

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"20. Abstract (Continued)channel. lhe channel itself consists of a thick slab of anisetropic elec-tron density irregularities that are elongated in the direction parallelto the earth's magnetic field.

- Analytic expressions are obtained that give the 'effect of theaperture antenna on measurements of received power, decorrelation time (or

*" distance), mean time delay, time delay jitter and coherence bandwidth.These quantities are determined as functions of the aperture diameter andof the angle between. the magnetic field and the direction of propagation.It is shown that in strong turbulence aperture averaging can be a signi-ficant factor in reducing the received power by angular scattering loss,increasing the observed signal decorrelation time via aperture averaging,

- -and reducing the time delay ,jitter by suppression of signals received atoff-boresight angles.

*. Result's are presented for two cases. One-way propagationthrough an ionospheric communication channel is considered where bothtransmitter and receiver uti-lize aperture antennas. This result is easilyextended to the case that one of the antennas is an omnidirectional pointsource, corresponding to the usual case of transionospheric satellitecommunication from a small satellite antenna to a large ground basedreceiver. The second case invoives transmission and reception of a radarsignal that travels through a disturbed ionospheric channel to a targitlocated in free space. This case is applicable-to the situation of alarge antenna aboard a space based radar or to the case of a ground baseddefense radar.

S

UNCLASSIFtED

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SUMMARY

I n this report analytic expressions are derived fqr the effect

of aperture antennas on measurements of signals that have propagated

through strong anisotropic turbulence. Results are given for the effect.

of Gaussian apertures on measurements of received signal power, dec~orrela-

tion distance (or time), mean time delay and time delay jitter'. The

geometries considered correspond to the case of a single, one-way propaga-

tion path between two aperture antennas. located in free space and

separated by a layer of turbulence and to the case of a monostatic radar

where radar and target are on opposite sides of a strong scattering layer.

All calculations depend on the analytic solution for the two-

position, two-frequency mutual-coherence function for spherical wave

propagation in an anisotropic layer of electron density irregularities.

The analytic. solution for propagation in a-thick layer is derived here

using the quadratic phase structure-function approximation valid for

strong turbulence. This result is then specialized to a thin phase'-screen

approximation to facilitate analytic calcul-ation of the effects of

apertures.

It is shown that antennas that are larger in diameter than the

dec~rrelation distance that would be measured by an omnidirectional

antenna can experience significant angular scattering loss and exhibit0

increased measurements of decorrelation distance, and decreased measure-

*ments of mean time delay and time delay jitter. Increased measurements of

signal decorrelation distance are c;aused by the -averaging effect of theantenna aperture'that smooths (eliminates) the small stale signal fluctua-

i.

1 .- .-

* 0 "

tions. Decreased measurements of mean time delay and time delay jitter

are caused by the action of the aperture to cut off signal contributions

that originate from off-boresight angles and have therefore exper'enced

more time delay than experienced over more direct signal paths.

Generally, the effects of aperture averaging first begin to be

significant for aperture sizes that approach ten times the decorrelation

distance measured with a point antenna. For even larger apertures, the

aperture has a significant effect on the received signal at the antenna

output that can be computed using the formulas and curves provided herein.

*I.

PREFACE

10,6

The author is indebted to Dr. Robert W. Stagat of Mission

Research Corporation-and to Dr. Leon A. Wittwer of the Defense Nuclear

Agency for their helpful discussions regarding this work.

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TABLE OF CONTENTS

Section Page

SUMMARY 1

PREFACE 3

LIST OF ILLUSTRATIONS 5

1 INTRODUCTION 7

2 FORMULATION 11

Thick Layer Solution 15Thin Phase-Screen Approximation 18

3 APERTURE ANTENNA EFFECTS .20

Aperture Weighting Function 24Case of Transmitting and Receiving Antennas 25Case of Monostatic Radar 26

4 RESULTS 30

Angular Scattering Loss 30.Signal Decorrel-ation Distance 34Mean Time Delay and. Time Delay Jitter .42Coherence Randwidth 47

REFERENCES .50

APPENDIX -: IONIZATION IRREGULARITY DESCRIPTION 53

Phase Standard Deviation 55Theoretical Angle-Of-Arrival Fluctuat'on 55

4 ,

LIST OF ILLUSTRATIONS

Figure Page

1 Propagation of signals through a disturbedtransionospheric channel. 12

2 Receiving antenna aperture centered at originof coordinate system. 21

3 Angular scattering loss for one-way path withtransmit and receive aperture antennas; a) versusrelative antenna size; b) versus inclination angle. 33

4 Angular~scatter loss for monostatic radar;a) versus relative antenna size.; b) versus inclina-tion angle. 35

5 Relative signal decorrelation distance in the x-direction for one-way path with transmit andreceive aperture antennas; a) versus relativeantenna size; b) versus inclination angle. 37

6 Relative signal decorrelation distance in the y-direction for one-way path with transmit *andreceive aperture antennas; a) versus relativeantenna size; b) versus inclination angle. 38

7, Relative signal decorrelation distance in the x-direction for monostatic radar; a) versus relativeantenna size; h) versus inclination angle. 40.

I8 Relative signal decorrelation distance in the y-,direction for monostatic radar; a) versus relativeantenna size; b) versus inclination angle. 41.

9 Relative mean time delay for one-way path withtransmit and receive aperture antennas; a) versusrelative antenna size; h) versus inclination angle. 44 -

10 Relative time delay jitter for one-way path withtransmit and receive aperture antennas; a) versusrelative antenna size;.b)'versus inclination angle. 45.

-7

L_

'rw

LIST OF-ILLUSTRATIONS (Concluded)

Figure e

11 Relative mean time delay for monostatic radar;a) versus relative antenna size; h) versus inclina-tion angle., 48 1

12 Relative time delay jitter for monostatic radar;a) versus relative antenna size; b) versus inclina-tion angle. 49

A-I A single irregularity elongated along the magneticfield line in the y-z plane. 53

6I

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- - - -. 9. - - . ~ - -

SECTION 1

'INTRODUCTION

Large high-gairt anteninas are used in many radar and coimrunica-

tions applications to increase the energy collected, to increase angular

accuiracy, and to provide protection against jamming. If the wavefront at

the antenna aperture experiences scintillation, large apertures can a'ct to

avt-rage the signal and thereby modify the, observed signal properties. in

the liate fifties; Wheelon -(1957,1959) ohtained the well-known aperture

smoothing effect for the' aeasurement of phase fluctuations by a' f4nite

circular aperture. Since then, vari-ous investigators have studied the

aper-ture smoothin 'g effect on iritensity~scintillations in the optics. regime

(Fried, 1967; Tatarskii, 1971; Homstad1 et al., 1974) and also the related

effect of optical beam wave propagation to an infinitesimal receiver (Leeand Harp, 1969~; Ishimaru, 1969). Knepp (1975) obtained results for the

effects of .an antenna aperture on measurements of the in-phase and quad-

rature components dnd fhei~r spatial derivatives, f'or the case of weak

scatteri ng..

Scintillation or rap Id vari~ations in the amplitude, phase, and

angle-of'-arrival of. a propagating wave is oftenobservedi over satellite,

links at VHF and 1JHF . Scin~tillation can' also be observed at frequencies

as high as the- Gilz range (Pope and Fritz, 1971; Skinner et al., t971;

Taur. 1976) 2nd is occasionally si~ver'e, oven at L-band (Frernouw et al.,

1978). However, aperture averaging is not generally important at

satellite frejuencies .(lJHF-GHz) under ambient or naturally perturbed

ionospheric conditions.

'7

T- .

Worst case or Rayleigh amplitude scintillation is likely to 7. -occur if the ionosphere is highly disturbed, as for example by high alti-

tude nuclear explosions (Arendt and Soicher, 1964; King and Fleming, 1980)

or by chemical releases (Davis et al., 1974; Wolcott et al., 1978).

Increased electron concentrations and the irregular structure of the Joni-

zation can ,lead to intense Rayleigh signal scintillation at frequencies as

high as the 7-8 GHz SHF band (Knepp, 1977). Consequently, the eftects of

scintillation are important to' any UHF ,through SHF communications or radar

system that must operate through an ionospheric channel and that may have

to operate in highly disturbed propagation environments.

For cases of severe scintillation where the signal varies across

the area of the receiving aperture, the effect of the aperture can be

significant. It is well known that an aperture antenna acts to cut off

energy that is incident at off-boresight angles where the antenna gain is

reduced. This effect may also be viewed as-the result of averaging or

coherent processing of the electromagnetic field intercepted at the

aperture location. In this report the effects of aperture averaging are

antalytically calculated for the case where Gaussian antenna beams are used

by transmitter and receiver. Results are presented for two different

physical situations. One-way propagation through an ionospheric communi-

cation channel is considered.wh~re both transmitter and receiver utilize

aperture actennas. This result is easily'extended to the~case that one of

-- the--antennas is a point source, corresponding io the usual case of trans-

ionospheric satellite communications. The second antenna geometry corre-

spords to transmission and reception of a radar signal that travels

through an ionospheric channel to a target located in free space. This

corresponds to the situation of a long-range ground or space based radar.

It is shown that 'aperture averaging can greatly affect measurements of

scattering loss, signal decorrelation time, mean time delay and time delay

jitter. Simple analytic expressions are given for all these quantities In

terms of the geometry of the propagatio, path and the severity and struc- g

ture of the-lonization irregularities.

.8'

,~ .o-AV.

S . .-.. , .,'' ; ,,.-F , , . ,r , .",.'""'.''.' • .,....E. ;•t .,-, --h,-..-' -. ,,r,-..' ..t r?....-. ,-.;,'•-''....''':''... '

To obtain results applicable to a general geometry of the line-

of-sight relative to the field aligned ionization structure, it is neces-

sary to obtain an analytic soluticn for the two-position, two-frequency,

mutual coherence function (MCF) for spherical wave propagation through a

thick layer of anisotropic electron density irregularities. It is assumed

here that strong scattering conditions prevailand that the quadratic

approximation to the phase structure-function is valid.

This approximation was used by Sreenivasiah et al., (1976) and I.

by Sreenivasiah and Ishimaru (1979) for the cases of plane wave and beam

wave propagation in homogeneous turbulence. More recently the two-posi-

tion, two-frequency mutual coherence function was obtained for spherical

wave propagation using the extended Huygens-Fresnel principle (Fante,

1981). Although the quadratic structure-function approximation can some-

times lead to difficulties (Wandzura, 1980) it is appropriate for the

two-frequency mutual coherence function but not for calculation of higher

moments of the field (Fante, 1980). Fante (1981) discusses the'accuracy

of the quadratic structure-function approximation for the case of atmo-

spheric turbulence with a Kolmogor6v power spectrum. He has found that

the accuracy is a function o the irregularity power spectrum and of the

strength of the tu'rbulence ( rivate Communication, 1982), with accuracy

increasing for stronger scat ering'

This aspect of the work here is a generalization of an earlier

calculation (Knepp. 1983(b)) valid only for isotropic turbulence. 'As in

the former study, the results herm are specializedto the case of a thin

phase-screen approximztion o the thick scattering medium. With this

simplification, analytic re ults are obtained for the received impulse

response function to a tran mitted power delta function. Then results may -

- t

9 j-

_7.

-,,

easily be determined for the mean time delay and time delay jitter for

strong, anisotropic turbulence fn the thin phase-screen approximation.

Results for these quantities in tiis approximation have previously been

shown (Knepp, 1983(b)) to closely approximate the results for a thick

scattering layer for the case of isotropic turbulence.

The effects of Gaussian antennas on measurements of received

power, decorrelation time (or distance), mean'time delay, time delay

jitter and coherence bandwidth are determined. It is shown that aperture

averaging can reduce observed signal power, increase observed decorrela- -

tion time and be a significant factor in reduced observed time-of-arrival

jitter at the antenna output.

t.

4

-I,

10r•

'.-

SECTI,. 2

FORMULATION

In this section the solution for the two-position, two-frequency

mutual coherence function is presented for transmitter and receiver

located on opposite sides of a thick layer of anisotropic electron density

irregularities. In the next section this result is utilized to 'determine

the effect of transmitting and receiving antennas on measurements of the

received signal.

Consider a monochromatic spherical wave E(i,z,w,t) which origi-

nates from a transmitter located at (OO,-zt) and propagates in free

space in the positive z direction where it is~incidenton an ionization

irregularity layer which extends from 0 < z < L. After emerging from the

layer at z = L, the wave then propagates in free space to a receiver

located at (O,O,z,). This thick layer geometry is. shown in Figure 1.

As the wave propagates, its phase substantially behaves as (-i<k>z+iwt) so

write 4

*'E(T,z,wt) U(-6,z,w) exp{i(t-;f<k(z")>dz'} (I)

where <k(z)> is the mean wave number given by

In.1 /2(2

<k(z)> - (1 - k//k 2 ) /k ()

where c is the speed of light in a ,vacuum, Ne is the mean ionization

density, nc is the critical electron density and is related'to the claso

sical electron radius r by n = w/(X 2re) (r u 2.82x10"-1 m))e c e e

10%

Z,

Z,

-ARGEr AN1TNNA

Figure 1. Propagation of signals through a disturbedtransionospheric channel.

In the case that the transmitted waveform is no longer a mono-

chromatic wave, hut can be expressed as a waveform modulated on. a carrier,

the'two-position, two-frequency MCF r is of -interest. is important

for the calculation of pulse, propagation in a random medium and It serves

'as a hasis from which to calculate the important power impulse response

'function and Its moments. Under the Markov approximation,. r Is found to

satisfy. (Yeh and Liu, 1977; Knepp, 1983(b)).

-k 2+ 1 k 72 V b

_____ d dd d s~Y S s dJL3z 2(k 2-k 2 /4)s d

LM1. A(C,n) + 4 A(O) r o 0 (3)

12,

, ,

/-.7-

"where Vs and Vd are the gradient operators in the sum and difference

coordinate system. Here r = <U(x 1 ,y 1 ,z,w 1 )U*(x 2 ,Y 2 ,Z,w 2 )% which is

written as r(C,n,z,wd) after the sum and difference transformation. The

standard transformations

X = (X +X2 )/2 = X

Y= (y,+y2 )/2 n yy y2

k =(k +k2)/2 k =k -k5 1 2 d 1

are used to obtain Equation 3. In addition, it has been ascumed that the

frequencies of interest are much greater than the plasma frequency. The

quantity ks is the wavenumber at the carrier frequency, ks = W0/C.

The function A(3) is the integral of the autocorrelation func-

tion of electý.on density fluctuations, BI, in the direction of prlpa-

gation

A(-5,:p)= B•(7 1-ý 2 ,z')dz' (4)

where p1-P2 " (x 1-x2,y 1 -y 2 ) and E AN /•e > so thate e

2wf If exp[iKj.,-(3--P2 )] *y(K.,Kz=O)d2 K. (5)

where i•i tche power spectrum of electron density fluctuations. Equa-

tions 4 and 5 depend upon the validity of the Markov approximation where

it is assumed that the electron density fluctuations are delta-correlated

in the direction of propagation (Fante, 1975). That is 8( I-.2,zI-z2)

13

The unknown two-frequency MCF may be written as r = I' 1ro (Knepp,

1983(b)) where r 0 is the exact free space solution in the parabolic

approximation. Substitution of r.= r1 r0 into Equation 3 and neglect of

near-zone terms yields a differential equation with terms containing thefactors arl/aX and arl/aY. Since the boundary condition at z = 0,

rl(ý,n,z=O,wd) = 1, is independent of XVand Y, the derivatives with

respect to these quantities may be neglected. The change of variables z'= Z + zt, 0 = C/z', = z', and the additional substitution r 1 r2 r3

where

r e - A t p kA (6)

and the replacements

2 kd/0- 7a) '21 d 2

-- kd/k (7b)4 Ps

(abA2)1/!2 a1z' (a

(1 a2p (8b),=(a bA lZ2) z lz a

enable one to write the equation for r2 asA

-_ * + a. - 2 +A2c2)vZI 2 - 0 (9)

14

iI

where A(C,n) has been expanded in a truncated Tayior series

A(c,n) = A0 t A2C2 + A2 A2n 2 (10)

(see appendix). This is the quadratic phase structure function approxima-

tion. The difference wave number kd has also been neglected with

respect to ks. The effect of this assumption is to restrict the valid-

ity of the solution to. a small range of wavelengths centered about ks.

Thick Layer Solution

An analytic solution of the form

r2 f(v) exp{-g(v)t 2 -h(v)£ 2 } (II)

may be substituted into Equation 9 to obtain three equations,. the first

consisting of terms that are factors of )j , the second consisting of

factors of £2, and the third independent of P and c. These equations may

be solved exactly for r 2 , and the results substituted into Equation 3 to

yiel.d

' (O,€,z'=L+zt , ) w =

F exp{-(A-Bt)8' - (A-A~t')€} 2(12)

where

e .1 2__• 8(L+zt)F x exp 4• 1 (a!

(iaz cosh -sa L + sinh $aL)(taa z co~sh A•aIL + sinh &.a I L

t t 13)

c+15

2A = !iztaa 2 (14)

4

2 2 22

nh--L (16)(L+z) aa2B t=(S

(151 + Oa z tanh OalL -tt t = ~(16) .$alzt + tanh $aiL (7

,'

V .1 + Aaa 1zttanh t~a 1Lt' = l~t(17)Aa3alzt + tanh A~ajL

82 = -i4 (18)

In transcribing Equation 6 for r 3, the relationship between thecoefficient Ao and the phase varianFe a given by Equation A-6a isutilized. Equation A-5 gives the expression for the quantity A in termsof the geometry of the line-of-sight relative to the magnetic field"di rect-ion.

To complete the solution,' it is necessary to solve for r, in theregion L <. z . zr (soe Figure 1). Equation 12'serves as. the boundary 4condit.ion at z' * L + zt. Since the region' >' L + zt corresponds tofree space with no ionization, Equation 3 is app:-opriate after the r,substitution is included, inserti on of the z, 0, and + substitutions, and '

deletion of the last term involing the function A that is zero in free Jspace.

16 ' .

v,

The Fourier transform

r.(e,.,z,,d) =z I ei(K e+K•t*)

(19) _

x r (KO,K0,z zwd) dKodK€(9

dJ

may be substituted into the suitably modified Equation 3 to obtain the

algebraic equation

kar_ + i d I (K2 + K). - (20•Z' 2 k2 z2 I

az 2% k

Equation 20 may he solved, and the boundary condition Equation

12 applied at z' L + zt with the result

rl(Ko,KO,,zt+zr,wd) = r.(Ko,KO,L+zt,wd)

(21)

xexp [-iy(K~ + K~)

where

y I'kd (Zr()L)

2 ks2 (t+zt)(zt+Zr)

The final result may then be obtained by taking the Fourier transform of

Equation 12 to obtain rI(Ko,Kl,zt+Zr,wd) and then taking the inverse

Fourier transform of Equation 21., -The required integrals are easily

found,. and the result may be written as

17

Frri(,nz+z ) = F d

[(l+i4y(A-Bt)) (1+i4y(A-ABt'))}]/"

.-

Cex (ABt)/(zt+zr) 2 n2(A-ABt')/(zt+Z )2(2x ex - +;.4y(A-Bt) - 1+i4y(A-ABt') (3 .

Equation 23 is the result for r, after propagation through a thick layer.

characterized by anisotropic electron density irregularities. The full ,

solution for the two-position, two-frequency MCF is obtained by multi- .

plication by r0, the free space MCF. Since ro is not affected by the

random layer, it may be ignored here.p

Thin Phase-Screen Approximation

Much simplification is possible if the thick scattering layer is

replaced by an equivalent thin phase-screen of infinitesimal thickness and

the same overall phase variance. To the. first order, in the thin phase-

screen approximation

cosh AaL 1 (24a)

sinh AaaL =aaL (24b) .

tanh Aa•1L Aaa IL (24c)

Utili ing these approximations as well as Equaiions 8aý-c and Equation 18

for a , a2, and a, the two-p osition, two-frequency MCF is found in the

thin hase-screen limit as

18

S"

r1(C,n,z +Z ,/2t rwd)

x x % expI o~ :~:201t " ± d 1 i i w d .'-.

2 2 2/ (X2/A2 J

Sr e2 0 (26)

0 zt1ioA 2 .

W' = - (zt+zr (27,)

SZZt rA 2

the strong scattering limit, the phase structure-function can be expanded '"in the form of Equation 10. For the anisotropic Gaussian power spectrum "•-•i

ofitrs n hs r r2 where ro is the axial scale""

0 2 2

size of the elongated ir'regularities as discussed in the appendix. The"":parameter 20 is a measure of the decorrelation distance of the complex in

electric field as seasured in the plane of the receiver; w is theax.a-'-c--.

coherence handwidth and will be shown to be inversely proportional to the

time delay jitter; <Yý is given in the appendix as the phase standard

deviation imposed on the wave by the phase-screen..

19

, S.'

.L

SECTION 3

APERTURE ANTENNA EFFECTS

In this Section, appropriate expressions are derived to obtain

the effect of the antenna aperture on measurement4 of the properties of

the received signal during strong scintillation conditions. The first

geometry considered is that of a spherical wave that propagates through

the disturbed layer to a receiving aperture antenna.. Results for this

case are then easily extended to the case of a transmitting and receiving

antenna and, then to the case of a monostatic radar geometry.

Consider the geometry shown in Figure 2 where the aperture is

located in the receiver plane, z = Zr. If the incident field in the -AL

plane of the antenna is U(-,z w,), then the complex voltage envelope at

the antenna output may be expressed at (Price, Chesnut and Burns, 1912)

v Izr - f J(o'.zr w) A(V-W ) d2p' (28)

where U is the solution to the parabolic wave equation at location •'

in the receiver plane for a monochromatic signal of frequency 4W, "P is*0

the location of the center of the aperture antevna and Ar is the 'complex ._

antenna weighting function.

Equation 28 gives the Lutput voltage from an aperture antenna,

pointed in the z-direction for a transmitted monochromatic waveform of

radian frequency w. If the transmitted waveform is a pulse waveform

modulated on a carrier, the transmitted signal can be expressed as

s(T) " Re {m(r) exp(Iw T)} (29)

20 .

, * . 0

..-

z

Figure .2. Receiving antenna aperture centrýed at origin ofcoordinate system.

where w i is the carrier radian frequency and m(r) is the modulation

waveform. After passage through a layer of irregularities, the received

signal may be written (Knepp, 1983(a)) in ,the absence of an atttenna, as

r(•.•Zr,•) = Re{e(P,ZrT)exp'(iwor)} (30)

where

r) %O(OZr)-_ f MW•U65.'rW+(ýo)exp(i urldf (31) ,.,',

The quantity e is called the complex envelope of the received waveform. ZT.

M(w) is the Fourier transform of the transmitted modulation waveform and

is given by

M(W) =" m( -t)exp(-i wT)dr -(Z).""

219Q

- w • • / '1 I

4..

In order to obtain the effect of the ant-,,,ad on thp received -.

time-domain waveform, the received complex envelope U in Equation 31 is

replaced by the expression, for v giver, by Equation 28. Thus the

received time-domain signal at the antenna output may be written as

e(_ýo,Z ,T) f !_ 1 M(W)U(-'"z r'w+W-0

x A( -a-o)exp (i WT)dp'dw (33)

In the case that the receiving aperture is an omnidirectional point

antenna, A* is a delta function and the received complex envelope given

by Equation 33 is identical, to the result given by Equation 31..

For the case that the transmitted signal power has a delta

function behavior in delay, m(.r)m* (T) = 6T(), the correlation function

of the received power can be expressed as

<e(T ,Z T)e*(YZ r ,T)> = G(5 -Y T)1 r. 2 1 2

2

(2w)" r S(Kt)IAr(kgl exp[ii.(pj1 o2 )]d 2 K (34)

where S(K,,r) is the generalized power spectrum (Knepp and Wittwer, 1984)

defined as the Fourier trdnsform of the MCF,

SIK,•T) = f j r(-d,wd) exp (-iK.-d+tWdT)dWO 2Pd (35)

and A (K). is the Fourier transform of the weighting function of ther

receiving aperture

Ar 2 A rP--) exp(*l1(.P) d 0 (36)4w4

22

The vector K is given es k sinO(cosý if + sine J) for the geometry

shown in Figure 2.

To obtain Equation 34, use is made of the fact that the MCF

depends on only the differences in position and frequency. A detailed

derivation similar to the above appears in Ishimaru (1978). Note that the

dependence of the power impulse response function, G, on the receiver

distance Zr has been omitted for notational convenience.

Equation 34 can be written in another very useful form with the

substitution

r(Kwd) = - r(dwd)exp(_il.P)d 2 P (37)

where., r(K,wd) is the Fourier transform of the MCF. If r is used in

Equation 34 one obtains

G(fý 1-52 9,) (21r) f r(K ,wd)fAr(K)I exp[iK.-(PjP 2)+iWdT]dwdd K (38)Ir

It 'will be seen that Equation 38 is preferable for the evaluation of the

* effects of apertures, on mean time del'ay and time delay ji'tter.

For later usage, it is noted that.the function r(K,w d) may beobtained directly from Equation 25 as

r(K.wd) d exp o

K2L2 / wd\ K2 12 / aaexp - 1+ (39)2 1+

4 / 44 7h,-./

.,.

Aperture Weighting' Function

Let the antenna possess a Gaussian beam pattern with gainfuncti on

G(O) = GO exp (_62/02) (40)

where the receiver beamwidth Or is related to the effective aperturediameter D by 02 = (X/D) 2 /(41n2) and X is the wavelength at the

rcarrier frequency. With this choice of beamwidth G(1/2D) = G /2 so that,

0 7this Gaussian aperture has approximately 'the same 3 dB beamwidth as a

uniformly illuminated circular antenna. The antenna gain 'pattern is the

square of the transform of the aperture response function defined earlier.

Thus

A(K) = G1/ 2 exp (-02/202) (41)

r 0 r

The Fourier transform relationship of Equation 36 may be used and the

small angle approximation invoked to obtain the aperture weighting

function and its Fourier transform as

Ar(') 2nk2 92 G1/ 2 exp (-k 2 p2 02/2) (42)r 00 rand

A.r(K) G1/2 exp (-K2/2k 28r) (43)r o r

Before proceeding, it is useful to note that the effect of an

aperture antenna is immediately apparent from the form of Equation 38.

With a point or omnidirectional antenna, the ,beamwidth is large in

Equation 43 and A r(K) is constant. Therefore. with a point 'antenna, the

aperture angular response 'function is not important in Equation 38. It is

apparent that the receiving aperture acts to modify the value of r such

that

ra(K wd) rr(K,wd)IAr(K)1 (44)

"24

where the subscript a on r denotes that this expression contains the"P, effect of the receiving aperture. The subscript r on r on the right

hand side of Equation 44 refers to the case here that the receiver is

located at .z r This equation is an explicit expression of the well knownU result that an apertu're modifies the angular spectrum of the received

signal.

"Case of Transmitting and Receiving Antennas

-I.

* Equations 34 and 38 give the received power impulse response

function for the case of an omnidirectio; ' spherical wave transmitter anda receiving aperture antenna with aperture angular response funciion

Ar(K). A transmitting aperture antenna is easily included in Equations

34 and 38 since its only effect is to reduce the region of the

phase-screen that is illuminated. This is accomplished by includingan additional antenna angular response function At(KZr/Zt) where A t(K)

is the response function of the transmitter with beamwidth 8t and where"the factor Zr/Z transforms the antenna angle utilized by the trans-

mitter to the angle observed by the receiver. This transformation is

equivalent to the observation that a ray transmitted from location -zt

at the angle 8 and that scatters from the thin layer is observed by the0. receiver at angle 0' - zte/Zro For this case one may write

' ,S 2r (K.wd r i~w~j~rX) (k /Z(45)tr d r (wd!rK t rA(K/t)

"where the subscript tr refers to the fact that this expression Includes

the effects of both transmitting and receiving apertures. Since the

one-way propagation path here again utilizes a. receiver, at Zr, r on the

right hand siae again appears with, the subscript r.

2S

For the later calculations of scattering loss, mean time delay

and time delay jitter, Equation 38 is evaluated at Pl-P2 = 0 where

Equation 45 is substituted for the factors r(i~wd)IAr(K) - That is,

for the one-way propagation path with both transmit and receive apertures

Gt(0,) = (2w) 3 f r (Kwd)exp(i wt) d2 Kdwd (46)

tr (,)tr 'ZWd d T) d2wd(6

i For evaluation of the effect of an aperture antenna on the measured signal

decorrelation distance, the necessary function is the two-position, single

"frequency MCF given by the Fourier transform of Equation 37 as

",0) = r (K,0)exp(iKZ-.d)d 2 K (47)- .d'' f.tr '

-.- The use of rtr includes the effect'of both transmitting and receiving

"apertures.

"* Case Of Monostatic Radar-a.

"Now consider the case- that the two-antenna geometry above is

replaced by a monostatic radar geometry. In the radar case, the solution

- is facilitated by considering two one-way propagation paths. Let the

radar antenna be located at (O,O,-Zt) and also let the' antenna angular

' response function be given by At -with beamwidth (t on both trans-

mission and r.ception, First consider the one-way propagation path from

the transmitter antenna to the target located.at (O,O,zr). For this

path, the effect of the aperture antenna ,is obtainid from Equation 45 with,

the omission of Ar, since the point target accepts energy incident from

all directions. That is,,1

r l (K wd rrwd)A / (48)a26

where the subscript ml refers to the first one-way propagation path for

the monostatic radar geometry.

Now consider the second one-way propagation path from the target

I back to the receiver. This is the propagation geometry initially

considered in this report except for the interchange of receiver and

transmitter location. For this one-way propagation path, f' is replaced

by rt appropriate to the one-way propagation' path from zr to -zt

I Thus

IA

r 2 (Kwd) = rt(Kiwd)IAt(K) 2 (49)

* Note that the occurrence of At here as well 'as in Equation 48

. signifies the use of the same aperture antenna (with beamwidth Ot) for

reception as well as transmission. The only effect of the reversal of the

direction of propagation is an interchange of Zr and zt in thecalculation of the MCF. This interchange has the effect of replacing t

0

in Equation 39 with the term I' = Zt~o/Zr"

In the case of the monostatic radar geometry under considera-

tion. the principle of reciprocity states, that the propagating signal

takes the same path over each of the one-way propagation paths. This is

true provided that the random medium is "frozen" for the duration of the

* signal traversal.

The power impulse response function for the signal received at

the target due to a transmitted delta function of power from the radar is

given by Equation 3$ as

Gi(O, r), (21) 3 f (K. d)exp(i dT)d2Kd *d (50)'

- 27

If a delta function of power is transmitted from the target, the

power impulse response function at the radar receiver is given by a A

similar expression

G= (2,T) (2w) m2 (K,wd)exp(iWdT)d2Kdwd (51)

The net power impulse response function after transmission of a delta

function from the antenna, scattering from the target, and propagation

over both of the one-way paths is given as the convolution (Ishimaru,

1978)C.

G (O,T)= fG(Ot')G2(Or-t')dt' (52)

where the subscript m refers to the monostatic radar geometry. Using

the convolution theorem, it is easy to show that

Gm(O,T) (2w)7 m rl(Kwd)d 2 K r 2 (K', d')d2K' exp(iWdT)dwd (53)

This equation is the equivalent of Equation 46 for the case of the one-way

propagation path.

The above considerations apply to the calculation of tht power

impulse response function Gm(O,,r) for the dase'of monostatic radar.

For this geometry the signal decorrelation distance is easily obtained as Q

a simplification of the previous development for the one-way propagation .

path. For a monostatic radar, the decorrelation distance is obtained by

transmission of a signal to the target and observation of the scattered

signal at two spatial locations in the plane of the receiver. The point .4

target acts as a. transmitter of 'a spherical wave which is then observed, by

28

%4

the radar receiver. For a small target and frozen irregularities, there

is no dependence of signal decorrelation distance on the one-way propaga-

tion path from the transmitter to the tarr-t. The necessary two-position,

single frequency MCF is given in a form similar to Equation 47 as

• ,4

r(-d,O) = I r Km2 (,O)exp(iK-")d)d 2K (54)

where rm2 is given by Equation 49 and includes the effect of the

receiving radar antenna,. It will be shown that evaluation of the analytic

form of the expression r(Td,O) immediately yields the signal decorrela-

tion distance in this strong scatter calculation.

2'

.;p

*2;3

29 -

-. .. . . . . . ... . ... .. . . -.• . • . ,t.--'.-"--•••=' • - ••• "•

L.

SECTION 4

RESULTS

In this section the derivations presented in Section 3 are

utilized to obtain analytic expressions for several properties of the

received signal as measured at the output of the appropriate aperture

antenna.

Angular Scattering Loss

An antenna aperture acts to coherently collect the energy inci-

dent upon the antenna and to deliver it to the receiver. In the transmit-

ting mode a directive antenna is designed to transmit energy only over a

selected angular region. In te, receiving mode this same directive

antenna accepts energy only from a narrow range of angles. Thus, relative

to an omnidirectional point antenna, a directive antenna will experience 4what is referred to as angular scattering loss when the signal at the

aperture exhibits scintillation. This angular scattering Toss arises from

angle-of-arrival jitter present in the incident wavefront that may cause.

energy to arrive at the antenna propagating at angles greater than those

accepted,by the receiving aperture.

A different but equivalent way to view the effect of a receiving

antenna aperture is as a coherent integrator of the signal arriving on the

aperture face. If the antenna is pointing towards the source of an undis-

turbed plane wave, then the antenna output is maximjm. If there are fluc-

tuations in, the signal phase across the aperture or the incident wavefrontis' tilted relative to the antenna boresight ,direction, the coherent inte-

grator output is decreased. '

.30

o-.

J

It is convenient to compute the angular scattering loss as the

ratio of the total power received with an antenna (or antennas) to that

received with an omnidirectional antenna that experiences no loss. In

both cases the transmitted signal power is taken as a delta function in

delay. The angular scattering loss may be written as

limrt 8 r+ f G(O,.)dT

loss = t' r(55)f G(O,T)d.

where the power impulse response function G is given by Equation 46 for

the one-way ,path with both transmit amd receive antennas and by Equation

52 for the monostatic radar case.

The integral of Equation 46 with respect to T is easily

evaluated using the identity f exp(iWdT)dT = 2 116 ( d). This leaves a two-ý

dimensional integral of r(K,O) over angle which involves easily found

integrals of' Gaussian functions. The result may be written as

.2a29x + ýaex')[ ae;'2I/2loss = I + + 2Ory + (56)

where cox and a0y are the Standard deviations of the theoretical

angle-of-arrival fluctuations in the x- and y-directions over the propaga-

tion path from -zt to Zr. The quantities cox' and a8y' are

the same standard deviations calculated for the one-way propagation path.

from Zr to -zt (see appendix). The quantities Ot and Or

are the 3 dB beamwidths of the transmitting and receiving antennas,

respectively.

31 ''

. 4

The expression for angular scattering loss given by Equation 56

contains the effects of variations of the properties of the propagation

medium and of the geometry of the propagation path. To show this result

pictorially, let the antenna beamwidths be identical and let the equiva-

lent phase-screen be' located midway between transmitter and receiver. In

this case V' = I. To simplify matters further, write the antenna beam-0 0

width in terms of the diameter as discussed* earlier. Write the angle-of-

arrival jitter as a function of the signal decorrelation distance using

the expressions given in the appendix. These substitutions give

2a2 /02 = 0.28 D2/Z2 and 2a2 I/e2 = 0.28 D2/(.o2/AZ). In order tooyo 0 0(O3/ 20

separate the geometric effects of inclination angle variations from 4

aperture averaging effects, define Lp as the lecorrelation distance

for propagation parallel to the magnetic field line. The quantity lp .

is invariant with respect to changes in the inclination angle and is given

from Equation 26 and A-6b by Xp on

the axial ratio q which is here taken as 15 as suggested by Wittwer

(1979).

Figures 3(a-b) show the angular scattering loss in decibels as a

function of the relative antenna size. D/10 and of the inclination

angle. In Figure 3(a)'the inclination angle between the magneti fielo

and the direction of propagation is shown for values of 0, 15% 30% 450

-and 90% In Figure 3(b) the relative antenna size takes.the val es of 1,

3, 10, 30 and 100. It is seen in both figures that only when th antenna

diameter is large with respect to the decorrelation distince is he

angular scattering loss significant. Small inclination angles case

Increased values of the observed phase standard deviation If a d thus 9

give 'increased angular scattering and increased scattering loss is shown.

An Increase in the inclination angle calOses a. decrease in If a d an

effective increase in the decorrelation distance Ao; therefore, the

amount of angular scattering loss decreases.

3?

6...

-Aft

35 -'U

En~ 25-

-3 20-

0

wto too 1o0oRELATIVE ANTENNA SIZE - D1

(a)

40 vi I

.35-

DA 100-5

C,25-

-,% ..0

0

".550 10 20 30 40 50 60 70 80 90

INCLINATION ANGLE (DECREES)

(b) .. '

Figure 3. Angular scattering loss. tor one-way path with transmit andreceive aperture-antennas; a.) versus relative antenna'size;b) versus inclination angle.

33

For thecase of a monostatic radar, the power impulse response

function is given by Equation 52 as the convolution of the power impulse

response functions of. each of the one-way propagation paths. A simple

change of variables is utilized to reduce the integral fGfft(O,T)dT to

the product of two integrals of G, and G2 . These integrals are easily

performed using the delta function identity discussed previously with

resulting angular scattering lors

/ x 2a coy2 o2'

loss : + . ____+ _57 ''

where Ot is the radar antenna 3 dB beamwidth. A comparison of this L;:

result to Equation 56 shows that the, addition of a second one-way propaga-

tion path essentially squares the'scattering loss.

Figures 4(a-b) show the angular scattering loss for the case of pa monostatic radar geometry as a function, of the inclination 'angle and of

the relative antenna size D/. Here t is the decorrelation

distance observed for propagation parallel to the magnetic-field along the,

path from (O,O,zr).to (O,O,-zt). The overall behavior of scattering -

loss is quite similar to that exhibited in Figure 3. The major difference

observed is the increased loss ,caused by the additional one-way propaga-

tion path from the target hack to, the radar.

Signal Decorrelation Distance

As another aspect of the aperture aveiaging effect, an aperture

can act to increase the measured decorrelition distance at the antenna

output. This observed increase is caused by the averagi.ng effect of the

coherent integration that smooths the more rapid fluct.ations. To obtain

34 p.

N-p.-,

40-'

> -0

-4o

0400

20

RELATIVE. ANTENNA SIZE, D/L,

(a)

* -.. , 100mt 60

'.4040-'5'

-24

Figure 4.- Angular scatter loss for monostatic radar; a) versusrelati e dntenna size; b) versus inclination angle.

35

the signal decorrelation distance for the one-way propagation path from(OOzt)to (0l,O,zr), it is necessary to perform the integration

specified in Equation 47. Fortunately, r given' by Equation 45 is theLIproduct of simple Gaussian functions. The integral is available instandard tables and the resulting expression contains the factor

(..~2/.2 n/I. ) where the decorrelation distances in the x- andox oy

y-directions (see'Figure A-3) are

2. =£1+ +Ox 06 1 (58)

r

_ 2 1/2.

+82 + B (59)

If small apertures are used so that t he beamwiidths are largewith respect to. the theoretical angle-of -arrival standard deviations, -theparenthetical expressions above are unity leaving I and X. in

Ox OY* agreement with the coefficients of C and n given in Equation 25 with"wd set to zero.

Figures .5 'and 6 show the. relative' signal decorrelation distancesZ A and. Xo /1(2/A) in the x- and y-dlrections for the case ofNidentical transmit- and recei've antennas separated by a centrally locatedscattering layer. The quantities to and t0/A are the values ofdecorrelatlon distance in the x- and y-directlons, respe~ctively, that are

*observed with omnidirpetional antennas. In the figuris the results areagain shown as a function of the ratio of antenna diameter to the

*decorrelatlon distance for propagation parallel to the magnetic fielddirection.

36 .*

jt

40

CA,

~~00

04

RELA~TIVE ANTTENNA* SIZE D/L,

(a)

60 -0

20

03

30

Cl0 10 20 .30 40 s0 60 70 80 90INCLINA TION ANMCLE (DECREES)

Figure S. Relative signal decorrelation distance in the x-directionfor one-way path with transmit and receive apertureantennas; a) versus relative antenna size; b) versusinclination angle.

37

o°oU

500

40

2,0 p

04

cr4j

o- ,o ,o'• ,o2RELATIVE ANTENNA SIZE - D/4,

(a)

h60

1100

-.j 20

C10

S 10 2.0 30 40 50 60 70 so 9004 3 1 INCLINATION ANGLE (DEXREA'3)

'," (b)

Figure 6.- Relative signal decorrelation distance in the y-directionfor one-way path with transmit and receive apertureantennas; a) versus relative antemna stze; b) versus"inclination angle.

,

Figures 5(a-b) show the relative decorrelation distance in the

x-direction as functions of D/1p and inclination angle *. Apertures

with diameters less than the decorrelation distance do not affect measure-

.ments of the signal decorrelation distance. It is evident that apertures

large with respect to lp can greatly increase the measured decorrela-

tion distance (relative to an omnidirectional antenna) for sufficiently

"""strong angular scattering.

- It is immediately evident that the decorrelation distance is not

"reciprocal. With omnidirectional antennas the decorrelation distances in-I

the two orthogonal directions, A and I are dependent on the pathox oy

geometry as given by Equation 26. These values are not reciprocal;

t and' . are measures of the average distance between fades in the", ox oyreceiver plane and consequently depend on geometry.

I In the case of the monostatic radar, the signal decorrelation

. , distantes may be obtained directly from Equations 58 and 59 by using the

parameters appropriate to only the second one-way propagation path with

transmitter (target) at (O,O,zr). The results are

2a• ,/2+- 6/.

* 0 ( 82

" ' wher . o + (61)-' oy a" @•

Swhere t and t1'/ are the signal decorrelation distances in the x- and- 00

y- directions that correspond to the'one-way propagation path from a

source in the zr plane to a receiver int the -zt plane. Figures

7-8 show tox and toy, for a monostatic radar as a function of

normalized antenna diameter and the angle between the magnetic field and

. the direction of propagation. .The results are similar to those shown in

Figures 5-6.

a 39,

N

'50-

> 40-

00

105*

10Opt

600

Q.. too t o .

REAIEATNN0IE - D1

C4120-

0 1.0 20 30 40 so 60 70 so 90

INCLINATION ANGLE (DECRESS)lp (b)

Figure 7. Relative signal decorrelat~iewdistance in the x-idirectiovifor monostatic radar;, a) versas relative anten'na sire; b)versus Inclination angle.

% P

6q 0-

!,0

> 40

.30 01o

0 20o

T o, ' I 0 ,10''

1 ,0-) 10o 1 102

JEL.4TIVF ANTENNA SIZE - D/lp

(a)

60

~50

1000

Ctoo

020-

0 w . . ',. ...,

20 30 40 50 60 70 so 90

. 0 10

S 00INCLINA TION ANGLE (DEGREES)

Figure 8. Relative signal docorrelation distance in the y-directionfor monostatic radar; a) versus relative antenna size; b)versus inclination angle. ,

.41

Mean Time. Delay and Time Delay Jitter

In the same manner that an aperture antenna with a small beam-

width neglects or averages out energy incident at off-boresight angles to

reduce the received power, it also acts to reduce the measured time delay

and time delay jitter. This reduction occurs because the energy arriving

from directions away from boresight typically travels over longer paths

than the more direct signal. These longer paths require a longer propaga-

"tion time for that portion of the received signal and henceý contribute to

increased <T> and aT values. If this energy is neglectedi by an

aperture antenna, then the signal at the output will be characterized by

smal-ler <T> and OT than that measured-by an omnidirectional (point)

antenna.

Now define, the mean time delay, <T>, as

<T> f G(OT)TIT (62)

I G(OT)dT

and define the time delay jitter. aT, by the second moment

a2 f G(OT)r~dr 2 (63)

II(O,T)dT

where G(OT) is the power impulse response measured at the jutput of the,

receiving antenna due to an, impulse of power originating from the trans-

mitter. .

To compute <T> and aT for the one-way propagat-ion path 1

with transmit andreceive aperture antennas.. it is convenient to utilize

Equation 46 for the power impulse response function. The integrals of -

and T2 give first and second derivatives of delta functions which yield

first and second'derivatives of rtr(K.wd). These derivatives are ,

evaluated at zPro wd as a result of the T inteOgration. The remain- ,

ing integratlonwith respect to 1" is easily performed. The results may

he written as

42

m -m - ., ,

/d

= ~ + 2 0 2 02 o2w' 2 2

r t

(64)

+ A2( + 2OY 720YA~-'

02 22r t

2 2 - 2 -2

1 (65)

Figure 9 shows the effect of aperture antennas on the observed

mean time delay by showing the mean time delay normalized by the value

observed with omnidirectional antennas, (1+&2)/(2,w'). In the figure,•

transmit and receive apertures are identical and the equivalent phase-

screen is assumed to be located midway between the antennas. Figure 9(a)

shows the relative mean time delay as a function of D/tp for values of

the inclination angle of 0*, 150, 300, 450 and 900. It is seen that large

apertures can have a signifidant effect in reducing the observed mean time

delay, particularly when the direction of propagation is close to the

magnetic field direction. In Figure 9(b) the abscissa and the parametric.

quantity are interchanged relative to Figure 9(a).

Figures to(a-b) show the aeffect of aperture antennas'ea the

observed time delay jitter or. Here OT is normalized to its valuefor omnidirectional transmit and receive antennas [((+A4')/(2w'2),]./2.

It is further assumed that the ratio a 1/W0 is small, as is always the

case for GHz and higher frequencies. At ,P - 0* the antenna aperture has

identical effects on 07 and, <T>. At other values of the inclination

43 -

i *°

107

160

10

~109

S10-

*too cot 102RELATIVE A NTENNA SIZE - /I

(a)

3

0 to 20 0 40 0 G 70910 INCLINA TION ANGLE (DEGRE'ES)

Figure 9. Relative mean time delay fori. -a ahwihtasi nreceive aperture antennas; 4) versus relative ante .nna size;*b) versus Inclination angle.

44.-

d4

190

#~~45'

16-1 300

105

I loof

*-.. 10

RELATIVE ANTENNA SIZE - D/L,(a)

3:

110

E-.

C4~

10IQ-

1010 0 30 40 5 60 0 80 0

C44

Figre 0. Reltie tmedelay jitrfor one-way path with transmit

and receivre aperture antennas; a) versus relative antennasize; b) versus. Inclination angl Ie.

45,

angle 4, the results shown in Figure 10 are similar but not identical to

those shown in Figure 9. The decreased measurements of <T> and aT

are the result of the action of the antenna to exclude signal energy

incident from off-axis directions that arrives later than energy that

propagates over the direct path.

For the general case that the power impulse response function is

given as the convolution of two other functions as by Equation 52, it is .

easy to perform the required, T integrations to obtain

<Tm> - <TI> + <T 2 >; (66)

2i =2T + 02 (67)TM Ti T2

These results state that the mean time delay and time delay variance are

the sums of the values obtained over eachof the two one-way propagation

paths. The required one-way values may be obtained from Equations 64-65

with attention to the direction of propagation and antenna placement. The

final results are

a; e °x- 2, 20 + y8<T +,, f +J +

et t

]!.202 20 .2

2ýx 2o2t 72't0

I

46.'

N.

iFigures 11-12 show the values of mean time delay and time delay

jitter for an aperture antenna normalized to tieir values for an omni-

directional antenna. Again the factor crl/wo is assumed small for

this monostatic radar case. These normalized results are very similar in sform to those presented in Figures 9-10 and again show the dramatic effect.

of a large aperture antenna that operates in strong scatt*.ring conditions.

Coherence Bandwidth

The coherence bandwidth is appropriately chosen as the inverse

of the time delay jitter exhibited by the received waveform due to a

transmitted power that is a delta function in delay (Knepp, 1983(b)). In.-

the cases considered here where 0./wo is small, the effects of

antenna apertures on the measured coherence bandwidth are easily obtained

from Equations 65 and 69. The results for the one-way propagation path

with transmit and receive antennas and for the monostatic radar case are

given as the reciprocal of the values shown in Figures 10 and 12,

respecti vely.

47.0

47 '"

, , jt b0

t o- i°-

"4100

100 10po- 1W2RELATIVE ANTENNA SIZE D1/p

(a)too .... . . '. . . .

too

, t-o-M

0 10 20 30 40 50 60 70 '80 90INCLINATION ANGLE (DEGREES)'

(b)Figure 11. Relative mean time delay-for monostatic radar; a) versus'relative antenna size; b) versus inclination angle.

48

• .%Y

10

1"0 ,

,o2E-.

t o - 4 . - .

to- TI 100 to12

I C"I 10 2

RELATIVE ANTENNA SIZE - D/l,

(a)

10-

S- , •,:-V::.-q 0

.... Io

107-0

0 20 , 30 40 50' 60 70 80 90INCLINATION ANGLE (DEGREES)

(b)-

Figure 12. Relative time delay jitter for monostatic radar; a) verusrelative antenna size; b) versus inclination angle.

"49__ *.* * .v.. q. ~ * ~ ~. - - V. ..! .~. '

• , 9

REFERENCES

Arendt, P. R., and H. Soicher, "Effects of Arctic Nuclear Explosionson Satellite Radio Conmunication," Proc. IEEE, Vol. 52, No. 6, pp.672-676, June 1964.

Briggs, B. H., and I. A. Parkin,, "On the Variation of Radio Star andSatellite Scintillations with Zenith Angle," J. Atmos. Terr. Phys., MVol. 25, pp. 339-365, 1963.

Davis, T. N., G. J. Romick, E. M. Westcott, R. A. Jeffries, D. M.Kerr, and H. M. Peek, "Observations of the Development of Stri'ations Lin Large Barium Clouds," Planet. Space Science, Vol. 22. p. 67,1974.

Fante, R. L., "Two-position, Two-frequency Mutual-coherence Functionin Turbulence,.' J. Opt. Soc. Am., Vol. 71, No. 12, pp. 1446-1451,December 1981.

Fante, R. L., "Electromagnetic Beam Propagation in Turbulent Media,"Proc. IEEE, Vol. 63, No. 12, pp. 1669-1692, December 1975.

Fante, R. L., "Some Physical Insights into Beam Propagation inStrong Turbulence," Radio Science, Vol. 15, No. 4, pp. 751-762,July-August 1980.

Fremouw, E. J., R. L. Leadabrand, R. C. Livingston, M. 0. Cousins,C. L. Rino, B. C. Fair and R. A. Long, "Early Results from the DNAWideband Satellite Experiment-Complex Signal Scintillation," RadioScience, Vol. 13, No. 1,,pp. 167-187, January-February 1978.

Fried, 0. L., "Aperture Averaging of Scintillation,* J. Opt. Soc.Amer., Vol. 57, pp. 169-180, February 1967.

Homstad, G. E., J. W. Strohbehn, R. H. Berger, and J. M. Henegan,"Aperture Averaging Effects for Weak Scintillations,* J. Opt. Soc.Amer., Vol. 64, pp. 162-165, February 1974.

BIshimaru, A., Wave Propagation and Scattering in Random Media, NewYork. Academic Press, 1978.

Ishlmaru, A., "Fluctuations'of a Beam Wave Propagating Through ALocally HomogeneousMedium," Radio Science, Vol. 4, pp. 295-305,April 1969.

50

'I.

REFERENCES (Continued)

King, M. A., and P. B. Fleming, "An Overview' of the Effects ofNuclear Weapons on Communications Capabilities," Signal, pp. 59-66,January 1980.

Knepp, D. L., "Antenna Aperture Effects on Measurements of Propaga-tion Through Turbulence," IEEE Trans. Antennas Propagat.. Vol. AP-23, pp. 682-687, September 1975.

Knepp, D. L., Multiple Phase-Screen Propagation Analysis for DefenseSatellite Communications System, DNA 4424T, MRC-R-332, MissionResearch Corporation, September 1977.

Knepp, D. L., The Effects of Aperture Antennas After SignalPropagation Through Anisotropic Ionized Media, Contract DNA001-81-C-0006, MRC-R-744, Mission Research'Corporation, March"1983(a).

Knepp, D. L., "Analytic Solution for the Two-Frequency MutualCoherence Function for Spherical Wave Propagation," Radio Science,Vol. 18, No. 4, pp. 535-549, July-August 1983(b).

Knepp,..D L., and L. A. Wittwer, "Simulation of Wide BandwidthSignals That Have Propagated Through Random Media," Radio Science,,

/ (in press), January-February 1984.

Lee, R. W., and J. C. Harp, "Weak Scattering, in Random Media, with- Applications to Remote Probing," Proc. IEEE, Vol. 57, pp. 375-406,

April 1969,Papoulis, A., Probability, Random Variables, and Stochastic

Processes, McGrawill: New York, 1965.

"Pope, J. H., and R. B. Fritz, "High Latitude Scintillation Effects"on Very High Frequency (VHF) and S-band Satellite Transmissions,"Indian J. Pure App. Phys., Vol. 9, pp. 593-600, August 1971.

Price, G. H., W. G. Chesnut, and A. Burns, Monopulse RadarPropagation Through Thick, Structured Ionziation, Special Report 4Contract DASA01-69-C-0126, SRI Project 8047-, Stanford ResearchInstitute, April 1972.

Salpeter, E. E., "Interplanetary Scintillations. I. Theory.*.Astrophys. J., Vol. 147, pp. 433-448, 1967.

I'"

o.1

REFERENCES (Concluded)Skinner, N. J., R. F. Kelleher, J. B. Hacking and C. W. Benson,

"Scintillation Fading of Signals in the SWF Band," Nature (Phys.

Sci.), Vol. 232, pp. 19-21, 5 July 1971.

Sreenivasiah, I., and A. Ishimaru, "Beam Wave Two-frequency Mutual'-coherence Function and Pulse Propagation in Random Media: An Analy-tic Solution," Applied Optics, Vol. 18, No. 10, pp. 1613-1618, May1979.

Sreenivasiah, I., A. Ishimaru, and S. T. Hong, "Two-frequency MutualCoherence Function and Pulse Propagation in a Random'Medium: AnAnalytic Solution to the Plane Wave Case," Radio Science, Vol. 11,No. 10, pp. 775-778, October 1976.

Tatarskii, V. I., The Effects of the Turbulent Atmosphere on WaveSPropagation, translated by Israel Program for Scientific Transla-

-tions, National Technical Information Service, U.S. Department ofCommerce, 1971.

Taur, R. R., "Simultaneous 1.5- and 4-GHz'Ionospheric ScintillationMeasurements," Radio Science, Vol. 11, pp. 1029-1036, December 1976.

Wandzura,,S. A., "Meaning of Quadratic Structure Functions," J.Opt. Soc. Am., Vol. 70, No. 6, pp. 745-747, June 1980.

"Wheelon, A. D., "Relation of Radio Measurements to the Spectrum ofTropospheric Dielectric Fluctuations," J. Appl. Phys., vol. 28, pp.

*• 684-693, June 1957.

Wheelon, A. 0., "Radio-Wave Scattering by Tropospheric Irregu-larities," J. Res. Nat. Bur. Stand., Vol. 63D, pp. 205-233,September 1959.

Wittwer, L. A., Radio Wave Propagation in Structured Ionization forSatellite Applications, DNA 53040, Defense Nuclear Agency, December1979.'

Wolcott, J. H., 0. J. Stmons,'T. E. Eastman, and T. J. Fitzgerald,"Characteristics of Late-Time' Striations Observed During Operation

* STRESS," Effect of the Ionosphere on 'Space and Terrestrial Systems,edited by J. M. Goodman, pp. 602-613, U.S. GOvernment PrintingOffice, 1978.

Yeh,'K. C., and C. H. Liu, "An Investigation of Temporal Moments of"Stochastic Waves," Radio Science, Vol. 12, No, 5, pp. 671-680,September-October 10'r7T

52

"APPENDIX

3IONIZATION IRREGULARITY DESCRIPTION

The ionization irregularities are assumed to be elongated along

the direction of the earth's magnetic field as shown inFigure A-I. *The

electromagnetic wave propagates in the negative z directibn. The magneticfield vector "ies in the y-z plane at an angle of * with respect to the

z axis. The ionization irregularities are assumed rotationally symmetric

... . with autocorrelation function

2

00.B.(rB,s)- = e ep .... ~ (A-i)

• .' •r <Ne >2 XPr (qr0)2)

imWAVE

Sline in the y-" plane.

<.3

I

where s is measured along the magnetic field direction and r is

measured perpendicular to this direction. The quantity q is known as

*the axial ratio (Briggs and Parkin, 1963) and ro is the correlation

distance or irregular'ity scale size.

The s, r coordinate system may be related to the x, y, z

system by the equations s~ (y sin ,+ z cos )2and r2 =x+

(y Cy os 4,-z s in 4,2. Using the above transformation the irregularity

* correlation function may be expressed in the x, y, z. This expression can

easily be integrated according to Equation 6 to obtain the function

A(co) as

/2

02n q r2 2

j.1

xexp (-2-~ ~r2 r,2 (q 2Sin 2 *+COS2J)~

T The above eqiation is essentially Equation 10 inrBriggs and Parkin

(1963). A(cn) is easily expanded in a Taylor series ard, retaining only

terms up to the quadratic, one obtains Equation 10 in the text waiere

-02 rw q r.

A0 - ce (A-3)

<N> 2

e

I2

AZ e (A-4)

4~eo0

54

.1 Th ,rcodnt ytmma erltdt h ,y

A2 -. 1...__ (A-5)22 2q sin24 + cos24

This same Formalism may also be applied to irregularity power spectra that

are non-Gaussian. For the case of a power-law PSD the coefficients A0

and A2 are different than above but behave in essentially the same

manner as a function of the outer scale, size as long as the three-dimen-

sional in-situ electron density PSD falls off at least as rapidly as K-4.

Phase Standard Deviation

For an ionized medium, the phase standard deviation is given as

the integral of the irregularity autocorrelation function along the

direction of propagation (Salpeter, 1967). The result may be written as

. (Knepp, 1983(b))

-2 =1k LA /k 2p (A-6a)

or

02 1w (Xr )2 qArOLae (A-6b)

* ,Theoretical Angle-Of-Arrival Fluctuation

* Consider a plane wave traveling in the negative z-direction and

i incident in the x-y plane at an angle e 'from the z-axis. The electric

, field is given by E(x,y) E exp {lik(slnO x + cos6 z)} where the0

exp(iwt) time dependence is suppressed. The angle-of-arrival measured

along the x-axis is computed as

.4.," I

'- 55"Il

M . . '

-

""1 E(xz) (A-7)"- x ikE ax A7

0

This computation gives sin 6 which, in the small angle limit, is

approximately e.

SIn the case of interest here the lnc-dent field is given as the

solution to the parabolic wave equation andthe angle-of-arrival may be

" measured in the x-direction as,%

SX 1 au(f',z,W) (A-8),x ikIo ax

with a similar expression for 8 . Here the measurement depends on they

frequency. At a single frequency the rms value of Ox is related to

the second derivative of the two-position MCF (Papoulis, 1965, p. 317)

2 1 a r(cn.,d=O)I

ex k 2C I= (A-9)

"." With a similar equation' for oey. For the case of propagation from a

transmitter in the -zt plane to a receiver In the zr plane, oeX

and oey may be obtained from derivatives of Equation 25 in the text, as

""Oex /Tl(k..o) and coy, a7/'/k(10/4)],

For the case of propagation from a transmitter in the zr,

plane to a receiver in the -zt plane, the interchange of trarnsmitter

and receiver may be taken Into account by substituting V' for L where.. ,o 0

-" 0 to ' -'zt /z X This substitution gives the angle-of-arrival fluctuation -

over the reversed path as ox, /(kI) and a , /'[k( ],

56

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.A'

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8-85