the influence of random phase errors on the angular resolution of synthetic aperture radar systems

The Influence of Random Phase Errors on

the Angular Resolution of Synthetic

Aperture Radar Systems

JEAN A. DEVELET, JR., MEMBER, IEEE

Summary-The influence of random phase errors on the angularresolution of a focused synthetic aperture radar system is treated.The principal measure of performance has been taken as the meanenvelope power at the system output. This system output power isevaluated exactly, although not in closed form, based on the follow-ing assumptions: 1) the real beam pattern is Gaussian; 2) the randomphase error is essentially a geometry-independent ergodic processwith a Gaussian amplitude distribution and zero mean; and 3) therandom phase error has a Gaussian correlation function.

The curves presented in this paper can be used to estimate ex-pected system power response, expected system resolution andeffective aperture length beyond which, in the presence of phaseerror, little gain in resolution is expected.

It was found that multiple sources of error with different correla-tion intervals make explicit solution of the integral equation for sys-tem power response practically impossible. In this situation, areasonable approach is to evaluate the system power response sepa-rately for each error. If one of the errors is clearly dominant, it maybe regarded as bounding achievable performance.

I. INTRODUCTION

P REVIOUS ANALYSES regarding the effects ofrandom phase errors on focused synthetic aper-ture radar systems' are extended for purposes of

radar resolution determination and system design.Greene and Moller' present an approximate (Monte

Carlo) analysis of the expected system response for vari-ous conditions of mean-square phase error and phase-error correlation length for an arbitrarily assumed ex-

ponential correlation function. The range of parametersin this analysis, for the most part, is restricted to in-vestigation of system response in the vicinity of the 3-dbbeamwidth.

This author maintains that the 3-db beamwidth has

questionable significance in resolution of a radar system.When expected beamwidth is used to estimate resolu-

tion,2 then beamwidth at the limits of system dynamicrange (20 to 30 db) is the significant parameter. In

other words, when receiving a strong target at maximum

system input, the main question is how close a target of

minimum system input can be placed and its presencestill be resolved. This latter definition of resolution is

Manuscript received July 5, 1963.The author is with the Aerospace Corporation, El Segundo, Calif.1 C. A. Greene and R. T. MIoller, "The effect of normally dis-

tributed random phase errors on synthetic array gain patterns,"IRE TRANS. ON MILITARY ELECTRONICS, VOl. MIL-6, pp. 130-139;April, 1962.

2 This is a reasonable measure due to its analytical tractability.

obviously the most stringent because, as will be shownlater, the 20- to 30-db beamwidth expands very rapidlywith phase errors, while the 3-db beamwidth is conipara-tively insensitive to these errors.The high sensitivity of the 20- to 30-db beamwidth to

phase errors allows fairly accurate estimates of systemresolution from the expected power output alone. Con-sidering the accuracy of phase-error data, use of sta-tistical decision theory is unnecessary for all practicalpurposes. This is fortunate because an exact analysis ofthe present highly nonilinear problem using statisticaldecision theory would be a formidable undertaking, tosay the least.An omission in previous analyses of synthetic aper-

ture radar systems has been the determination of arraylength for a specified resolution in the presence of ran-dom phase errors. In the author's estimation, this quan-tity is the heart of system design. It is unreasonable tobuild a processor, storage facility, etc., compatible witha 4000-foot array when (due to irreducible random phaseerrors) nearly the same resultant resolution is achievedwith a 2000-foot array.

Considering the foregoing, it is worthwhile to reattackthis problem from a new perspective. It is not antici-pated that the approach in this paper is the finalanswTer, but the results obtained from it are consideredto be of greater utility in system design.

Certain assumptions are made as to the character ofthe system in order to obtain exact mathematical solu-tions. The paramieter calculated is the expected systemaverage power output. Exact curves are presented toallow the system engineer to judge for himself the re-sultant resolution. Since the more logical 20- to 30-dbresolution is highly sensitive to phase errors, the resultsobtained using a simple and somewhat arbitrary resolu-tion estimate based on power response alone are not farfrom those that would be calculated using an exact sta-tistical decision theory approach.

II. ANALYSIS

A vast amount of literature is available on syntheticaperture radar systems. No duplication or repetition ofthese works will be attempted in this paper. It is as-sumed that the reader is familiar with the contents ofthe April, 1962, issue of the IRE TRANSACTIONS ONMILITARY ELECTRONICS.

58

Develet: Random Phase Errors in Synthetic Aperture Radar

This paper deals only with the fully coherent focusedsystem. A number of assumptions are made to simplifyanalytical study of the system.

1) Sampling rate of the radar is sufficiently high thatno angular ambiguities appear in the data andalso high enough that received signal spectrumsplatter due to very short-term random phasefluctuations can be accommodated. This assump-tion allows filtering of the coherently detected re-ceived pulses to yield a continuous waveform Dop-pler history of a target. Continuous waveformsare much easier to treat mathematically and theiruse results in no loss of generality.

2) Range ambiguities for particular system geom-etries are suppressed by proper signal design. Nofurther consideration of ambiguities will betreated.

3) Linearity of the radar system is assumed so thatsystem output to any arbitrary field of targetscan be obtained by voltage summation of the in-dividual target responses.

4) A single dominant target is being viewed in eachresolution cell.3 Many targets of approximatelythe same return strength within one resolution cellmay be approximately handled by assuming thatthe system output is a Rayleigh distribution withsome mean power level. A small number of targetsof approximately equal strength in one resolutioncell can cause "breakup" of the return due to can-cellation and reinforcement. This is one of thereasons for the different appearance of optical andradar maps.

Consider now Fig. 1, which depicts the system geom-etry while responding to a single point target. It is as-sumed that an observer at 0 fixed relative to the realantenna watches the target, T, go by with velocity, v.Time is referenced to zero as indicated in the figure. Theobserver and his data processor are "matched" to thegeometry of Fig. 1. Any change in the true situationfrom that of Fig. 1 due to improper target (vehicle)motion, clock (oscillator) instabilities or propagation-medium scintillation must be compensated for by eithera change in data processing or by acceptance of thechange as random error, with the resulting system deg-radation. This paper is concerned only with uncompen-sated phase errors and their effects on angular resolution.

Let the radar at time, t, transmit the signal

Vt = Et cos ct (1)

whereEt = transmitted signal amplitude, volt,w = carrier radian frequency, radian/sec.

If a dominant target exists and is surrounded by much smallertargets all within one resolution cell, the effect of these smaller tar-gets may be approximately characterized as random phase variationson the return of the dominant target.

PROPAGATIONVELOCITY,C.

PATH

Fig. 1 System geometry.

Assuming Gaussian real antenna beam patterns, it isa simple matter to show that the return signal from apoint target is given to first order by

Vr = Er exp (-2kt2) cos [wt + a + /3t2 + 4,(t)] (2)where

Er=received signal amplitude, voltsk =beam factor, sec-2X=carrier radian frequency, radian/seca=fixed phase shifts which may exist to and from

target, e.g., 2 oRIc plus propagation and equip-ment biases/= coefficient for change in phase vs time due to

quadratic range change (Fig. 1), sec-2+X(t) = ith ensemble member of an ergodic random

Gaussian phase process of zero mean andGaussian correlation function, radian.

It is not necessary to know the value of a as it will can-cel in later calculations. The terms / and k may be com-puted from simple geometric considerations. The resultsare

27rvl

-XR

k = 2

sec-2

sec-2

(3)

(4)

where

L=length at the target to the 1/e(-4.34 db) pointsof the real antenna one-way power pattern, feet

v = vehicle velocity, feet/secX = carrier wavelength, feetR=range to target at t=0 (see Fig. 1), feet.

For convenience, / and k are used in the following anal-ysis to minimize symbology.The synthetic aperture radar system correlates the

received signal, given by (2), with a delayed in-phaseand quadrature replica and presents an average poweroutput equal to the sum of the squares of the in-phaseand quadrature correlator responses. Fig. 2 depicts thisprocess. The limits of integration are shown to be in-finite in Fig. 2. In a real system, a finite integrationwould be accomplished. Since mathematical tractability

59

IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS March

Vr

2 -2k(t ) SIN[A t

Fig. 2 System

is sought and, further, since, with the assumed Gaussianweighting, there is little difference in the correlator out-put between reasonably large finite and infinite limits,the analysis to follow uses infinite limits.

transfer function.

Having dispensed with all the assumptions and sy-s-tem descriptions, injection of the voltage given by (2)into the system of Fig. 2 with the reference functions of(5) and (6) gives the following average power output:

P0= : i [exp {-2k [X2 + yl + + (Y A)C]}x Cos [2zAo(x - y) + 5(x) - ki(y)]dxdy. (7)

As depicted, the correlators are "matched" filters forthe signal of (2) and, as such, maximize the signal-to-noise ratio in the output for a time shift A = 0. However,there are occasions when all the data received from atarget on its pass through the real beam cannot andshould not be processed. These situations occur when1) processing time is precious and resolutions corre-sponding to real beam length L are not needed, 2) lackof phase coherence across length L yields a point ofdiminishing returns and processing more data gives littleincrease in system resolution and 3) vehicle constraintshave forced the use of a smaller real antenna than de-sirable from a resolution point of view.As a consequence of the above, the reference functions

chosen for analysis are generalized to the following:

Erexp [-2k ( )]cos [cut + F( \2 + Py](S

- exp [-2k ( )i] sin [wc + 3(t - A)2 + Y] (6)

where -q = (by definition) the fraction of the data utilizedin processing,

0< < 1.

As depicted in (5) and (6), , may be greater than 1 andnot necessarily restricted to the range 0< q<1. How-ever, in the presence of noise the signal-to-noise ratio atthe correlator outputs will drop off for 7j> t.4 Since forevery q > 1 there is a 7q < 1 with the same signal-to-noiseratio, it is obvious from an engineering point of viewthat the range of 7q resulting in the least data processingwill always be used; ergo, 0 <.t <.1.

The bar over (7) is the ensemble average over 4i. Sincethe average of a sum is the sUmIl of the averages and,further, since sin [4i(x) -4 (y) ] - and q5;(x) -O (y) _0,expansion of the cosine function in (7) yields

rx rx~ r - /X_Z\\2

JP xJOx exp ci-2k XI +Y2+(y )]-00~~~ ~~00 ) ]}

X cos [2A,A(x - y)] cos [0i(x) - qi(y)]dxdy. (8)

Since Oj(t) has been assumed to be Gaussian, it is recog-ized5 that

- [&i(x) - oi(Y)]2}cos [sSi(x) - i(y) ] = exp 2 J (9)

Simplifying (9) we get

cos [oi(x) - oi(y)] = exp {- [R(0) - R(x - y)] } (10)

where

R(x-y) =correlation function of the random phaseperturbations, radian2.

Let us assume now that R(x-y) is Gaussian in na-

ture; thus

R(x Y) = o2 exp [-( ) radian2 (1 1)

4 This is because only a matched filter, q = 1, can yield the largestsignal-to-noise ratio for A =0.

J. A. Develet, Jr., "A threshold criterion for phase-lock demodu-lation," PROC. IEEE, vol. 51, pp. 349-356; February, 1963. Correc-tion: PROC. IEEE, vol. 51, p. 580; April, 1963.

60


whereJ2==mean-square value of phase error, radian2T = time it takes the correlation function of phase to

drop to 1/e of its zero argument value, sec.

Substituting (11) into (10) and (10) into (8), we finallyobtain

P0o= exp (- 2)ff-oo oo

*[exp {-2k [x2 + y2+ ( )

+ ( A)2] 2 exp [

X cos [2 A/(x - y)]dxdy.

Our task now is to evaluate the interesting integr.(12). Expanding the exp{oa2exp [-(X-y)2lr2]} tin an infinite series, which is valid for all values ofexponent, (12) becomes

Po = exp) (-s2) Ej=o g ! _0 _0

*[exp {-2k[x2+ y2 + )

y A ) ] jd Y) }2 ]

*-cos [2 A,o(x -y) I dxdy.

where

Po' = normalized system power response

x= distance along vehicle track, feetxo = (2/7r)(XR)/Leff, -4.34 db (1/e) synthetic aper-

ture response width, under no phase-error con-ditions, feet

Leff = L(2/(1 + I/1,2))iI2, by definition, the effectivesynthetic aperture length, feet

x,=0vT, the distance to 1/e point of the phase-errorcorrelation function, feet

-= (by definition) the fractionprocessed. See (5) and (6).

(12) Two key quantities defined above

al of (15) and (16).7term 2mRthe x0 =\-L-) feet

L 2 1/2Laf = L

of available data

bear repeating as

(15)

(16)feet.

Note that, for small -q, (14) reduces identically to thetwo-way power pattern of the real beam. This is ex-pected since, for small 71, little processing of the receiveddata is performed.

For the case of considerable beam sharpening, xIL(13) -*0, the exact equation, (14), reduces to

jr2J exp

00

Po' = exp (- a2) Ej=O

-j4 +

LL(L(,,)f)j_ j _~~~~1/2

j ! _ ( L,,

Expanding the exponent in (13) and performing theexcruciating double integration with the help of integraltables6 give the following exact result for Po

Po' = exp (_-52) Ej=O

J-2iexpF- (4 !F + ,- 1(T4,L X IL (,,)j f.....rri± X21L (L:ff )J

6 W. Grobner and N. Hofreiter, "Integraltafel, Zweiter Teil,Bestimmte Integral," Springer-Verlag, Vienna and Innsbruck,Austria, p. 143, 1958; Eqs. (la), (lb).

2

(Xq2 + 1)2[

(14)

7Note, in (14)-(19), that the- time variable along the track hasbeen replaced by the more conventional distance variable.

(17)

1 964 61


Note that in (17) the power response of the systemdepends only on 1) o, the rms magnitude of the randomphase error, 2) xc/Leff, the ratio of the correlation dis-tance of phase errors to effective array length, and3) xlxo, the normalized distance from maximum output.Note also, as xc/Leff >X (complete correlation acrossthe array), (17) becomes

PO' = exp [-4(-)] (18)

and is completely uninfluenced by the rms magnitudeof the phase error. This is expected. Eq. (18) demon-

strates the significance of x0 as the width to the 1/e(-4.34 db) power response with no phase error.

Figs. 3(a)-(n) are the results of IB1\I 7090 computa-tions utilizing (17) for various values of xc/Leff and oA.To generalize these results to the case of arbitrary beamsharpening (14), simply add the following decibel cor-rection to the ordinates:

80 logio (e) /x2=-.2 db. (19)

8 Note that with Gaussian weighting no sidelobes exist. There-fore, if sidelobe suppression is important, Gaussian amplitude weight-ing is indicated.

-300 ,

-40re -

r-

e I,

I

111I40 5

0 2.0 3.0 4CDOSTANCE ALONG VEHICLE TRACK, X/X0

(b)

In

et1

tO

I

tE

-40-- - ~ ~

-5 0_; _ _ __ _

60) 1.0 2.0 3.0 4.0

DISTANCE ALONG VEHICLE TRACK, X/X0

(d)

N0 2.0 3.0DISTANCE ALONG VEHICLE TRACK, XIX.

(e) (f)Fig. 3(a-in) System power response vs distance along vehicle track.

()0 3.0 40DISTANCE ALONG VEHICLE TRACK, X/x"

(a)

2.0 0ISTANCE ALONG VEHICLE TRACK, X /X.

(C)

62

5 ------ L eff=00

- 60, l -60L-0

5.0


-60 Llill zr * 5 DEG4GX~~~10DE

O21. .030 4 0 DEG

OISTACE ALNG VHICLETRACK X/X

30~~~~g

(j)

(h) (i)

(k) (1)

Q 1.0 2.0 3.0 4.0DISTANCE ALONG VEHICLE TRACK, X/xO

( m)

DISTANCE ALONG VEHICLE TRACK, X)X.

(n)Fig. 3 (Cont'd)

1964 63

TFLoff

-20 -I ___D _ ___ ;_

-650

ICC E IEA5.0


180 CEG 120 DEG 100

~~~~~~~~~~~~~~~~~~~40

500ff~~~~~~~~~0 E°I c: _

R~108 -106 f=

& 00 ft5DE0 o 18x16f

zI t X:01f

to°L0

EFFECTIVE ARRAY LENGTH, Le!f, ft

(a)

to3

102 _0t'

_

100a1

0 5000

15,000 20,000

180 DEG 120 DEG 100 DEG 080 DEGi I~~~~~~

10°0

0

-I ~~~~~~XC500)0 ft .-

.- -- ' ~~~~~~~~~Ro=. Sx 106 fti

o< | I | - ~~~~~~~~~~~~~~~180DEG

10l21 120 DEG ':

x ' < ~~~~~~~~~~~~~~~~100DEG

Tflo,Y

fN | ~~~~~~~~~~~~20DEG

c 1 ~~~~~ ~ ~ ~~~~~~40DEG

° 5 ~~~~~ ~ ~ ~~~~~~~60DEG

10 ,000EFFECTIVE ARRAY LENGTH, L,ff, ft

(d)(c)

XC-10,000) f

Ro= 1.8 x 10" f

X fJ ~~~~~~~~~~~~~~~~160DEG

\, ~~~~~~ ~ ~ ~~~~~~~~120DEGi

+>_ ~~~~~~~~~~~~~100DEG

_ ~~~~~~~~ ~~~80DEG

5 DEG20 DEG

40 DEG

Of 10 2

I 1-

_ ri

_C

m_

CfIO. lo

eIl

O

5000 00o00

EFFECTIVE ARRAY LENGTH, Leff ft

(e)

Fig. 4(a-e)-Twxenty-decibel half beamwidth vs array length.

Figs. 4(a)-(e) are interesting and can assist in assess-

ing the effective array length that one should use in thepresence of phase errors to achieve a specified 20-db syn-

thetic beamwidth. The figures were derived from IBAI7090 computations utilizing (17) and, as such, assume

considerable beam sharpening, i.e., neglect of the lastterm in the exponent of (14). These curves are normal-ized and can be used for anv range R or syTstem wave-

length X for which the assumptioin of considerable beamsharpening holds. For the special case of R=Ro= 1.8

X106, feet, and X=Xo=0.1, feet, the 20-db half beaim-

width appears directlyT as the ordinate.Note the rapid expansion with phase error of the 20-

db beamwidth as contrasted to the 3-db heamwidth. A

reasonable definition of resolution is that it is the half

beamwidth to the point equal in decibels to the dynamicrange of the system. It is felt that this is a good engi-neering criterion considering 1) the sensitivity of the

dynamic-range beamwidth to a and x,, and 2) the lack

of precise knowledge of o- and xE. For N strong targets

64

0 OO ft

to_ 1.8 x 106 ftItt1 0.1 ft

5000 10,000EFFECTIVE ARRAY LENGTH, Leff, ft

(b)

15,000 20,000

5,000 20,000

to

DEG 80 DEG

5000

5000

5,000

IEEE TRANSACTIONS ON AEROSPACE AND NAVIGATIONAL ELECTRONICS

in the vicinity and for short phase-error correlation dis-tances, it is suggested that a safety factor of 10 log1o (N)be added to the dynamic range. [See Fig. 3(a).]

III. CONCLUSIONS

Certain assunmptions regarding the physical systemand random phase errors have yielded an exact resultthat predicts the expected power response for a syn-thetic aperture radar system.

Complete freedom to choose the fraction of the re-ceived data to be processed has been left to the systemdesigner by his choice of ij.The assumed Gaussian weighting not only yielded a

mathematically tractable system but, in addition, gavean expected response with no sidelobes.The definition of resolution suggested by the author is

the half beamwidth to the dynamic range point on the

expected power response curve. A figure of 20 db waschosen for illustration in Fig. 4(a)-(e). The definition ofresolution as the dynamic-range half beamwidth is con-sistent with 1) the high sensitivity of resolution to o-

and xc and 2) the inaccurate knowledge of these param-eters.

It is felt that an exact decision theory attack on thisproblem would not be fruitful because of its mathe-matical formidability and the inaccurate available dataon a. and xc.

Eqs. (11) and (12) indicate that multiple sources oferror with different correlation intervals make explicitsolution of the integral equation for system power re-sponse a formidable undertaking. In this situation, areasonable approach is to evaluate the system power re-sponse separately for each error. If one of the errors isclearly dominant, it may be regarded as boundingachievable performance.

Binary Quantization of Signal Amplitudes:

Efect for Radar Angular Accuracy

DUANE H. COOPER

Summary-An investigation made by P. Swerling 141 determin-ing the limits that receiver noise imposes upon the accuracy withwhich the angular position of a target may be estimated by a pulsedsearch radar, is extended to determine additional accuracy limitationsimposed by the quantization of signal strengths prior to angle esti-mation. This analysis follows that of Swerling in form. Theresults are presented in graphs. For many cases of practical interest,the additional limitation in accuracy may be neglected.

INTRODUCTIONINARY QUANTIZATION is intended to mean aprocess in which a continuous signal is convertedinto one with a rectangular waveform which has

a fixed magnitude during times for which the originalsignal exceeds some preassigned value, and a zero mag-nitude otherwise.1 There are at least two instances in

Manuscript received July 18, 1963. The research reported in thispaper was made possible by support extended to the CoordinatedScience Laboratory, University of Illinois, jointly by the Depart-ment of the Army (Signal Corps), Department of the Navy (ONR),and the Departmernt of the Air Force (OSR), under Signal CorpsContract DA-36-039-SC-56695.

The author is with the University of Illinois, Urbana, Ill.I Quantization into more than two levels, or that quantization

which preserves only the polarity information of an oscillatory sig-nal, is beyond the scope of the present consideration.

which it is desirable to know the effects of imposingbinary quantization as an early step in the process ofinformnation extraction. These instances will be men-tioned in connection with the quantization of the signalfrom a pulsed search radar.

For the first instance, binary quantization may betaken as an extreme example of simple amplitude satu-rationi, such as may occur in video amplifiers or incathode-ray display devices. As has been shown by J. I.Marcum [1], if binary quantization does occur, thenthe imposition of any other monotonic-increasing non-linearity prior to quantization causes no further loss ofinformation in the quantized signal. It is in this sensethat binary quantization may be taken as an extremeexample of such a nonlinearity.

For the second instance, one may consider the mech-anization of the radar observation process. This usuallyinvolves the storage of a representation of the radarsignal in order that certain computations, as entailed inthe observation process, may be performed. The re-quired storage capacity may be greatly reduced if itshould prove that a binary quantized version of the

651964

the influence of random phase errors on the angular resolution of synthetic aperture radar systems

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