AD-AO810 857 ARMY MISSILE COMMAND REDSTONE ARSENAL AL TECHNOLOGY LAB F/6 17/9MATHEMATICAL ANALYSIS OF POLARIMETRIC PHASE AND AMPLITUDE WIT--ETCCU)

SEP 79 P M ALEXANDER

UNCLASSIFIlED D iSI -T - 7 9N L

NNN! ,1 11112 .240 1111120-

Illil IInl IImL ~

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6 4

MICROCOPY RESOLUTION TEST CH/PTNATIONAL BUREAU OF STANDARDS 1963-1

LEVEI ,q~t TECHNICAL REPORT T-79-92

AMATHEMATICAL ANALYSIS OF POLARIM~ETRIC PHASETi AND AMPLITUDE WITH FREQUENCY AGILITY FOR RAflAR

TARGET ACQUISITION

P.Martin Alexander

Technology Laboratory

2September 1979 C

LL Zj F:?3dfAtcP"' Araelnwa, A~ibmwa 36600

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A Mathematical Analysis of Polarimetric TcnclRe~Phase and Amplitude with Frequency AgilityTehia Q tJfor Radar Target Acquisition*.-VW0NiGOG-IM NE

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LFM Waveform Frequency SpectrumPolarimetric Pseudo-coherentFourier Elliptical polarizationTrihedral Corner Reflector Monostatic RadarDepolarization

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Target models have been used to generate a mathematical analysis of the outputvoltage from a frequency agile radar with polarimetric processing. The purposeof the investigation was to determine the usefulness of such a system for tar-get acquisition. It is shown that frequency agility provides a frequency spec-

* 4 trum at the output which is characteristic of the major reflecting centers £1-luminated by the radar. Polarimetric processing is not required to obtain thisinformation. It is shown that each pair of individual reflectors contributes afrequency to the Fourier transformed siqnal. For a linearly frequency maodulat-

D IJAW, 473 DTOOIOSIUSLT UNCLASSIFIEDSECURIITY CLASSIFICATION or THIS PAGE (Wen Doanti4J

-3?397

SCUMATY CLAWOPICATIO#I OF THIS PAOR(inm Def &aft,*

ed waveform, this frequency depends on the difference in radar range betweenr the two reflectors, the bandwidth of the transmitter, and the repetitionperiod of the LFN waveform.

$*9CURIlY CLA5SIPICATIom OW THIS PAS~iwk..DNt .~

CONTENTS

Section Page

IIntroduction ...................... .................................. 3

2. Circular P1olarization ................................................. 4

3. ie Polariiation Matrix .............................................. 6

4. Simple Ref'lectors..................................................I

5. Varget Model................ ....................................... 13

6. [)epolari/ing [argu. s................................................. 23

7. Linearly IPolari/ed Radar ............................................. 27

X. ('nCILuSiOnS..................................................................... 28

References ............................................... ....... .. 29

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DPC rA!BUamnawced

Availability 0oaeSAvail and/or

Dist special

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ILLUSTRATIONS

Figure Page

1. Transmitter - Receiver System........................................ 5

2. Right Circularly Polarized Wave (RCP) ................................ 7

3. RCP Incident on Flat Plate ......................................... 12

4. RCP Incident at Angle with Respect to Flat Plate........................ 13

5. Linear Frequency Modulation ....................................... 20

6. Frequency Spectrum for Target, Clutter Model .......................... 22

2

I. INTRODUCTION

Recent interest has been shown in using a technique referred to as polarimetric phase

processing (or simply, p, arimetric processing) for clutter rejection and target discrimination

in acquisition and fire control radars 111, 121, 131. 141. The technique has also been called

pseudo-coherent detection (discrimination). A circularly polarized wave is transmitted, and

separate horizontal and vertical channels receive the return simultaneously. These two signals

are then mixed, using one channel as the local oscillator (hence, "pseudo-coherent"). The

resulting signal depends on the relative phase between the horizontal and vertical channels,

and this phase difference is called the polarimetric phase.

The underlying assumption of the discrimination technique is that, for any given radar

range cell, the resulting polarimetric phase is a function of the transmitted frequency. For

example. if a frequency agile radar is linearly frequency modulated (LFM I. then the measured

signal voltage at the mixer output is a function of time. If this time varying voltage is Fourier

analyzed, then the resulting spectrum contains frequencies which are characteristic of the

target clutter.

In the present work. target models are used to generate a mathematical analysis of the

output signal expected from polarimetric phase processing in an effort to determine the utility

of the technique for target acquisition. rhe general thrust of this effort is toward tactical land

combat operations, where discrimination between threat targets (such as tanks) and non-

threat targets (jeeps for example) is desired.

In Section 2, a simplistic transmitter receiver system model is described, and the sign

convention for a circularly polarived wave is established. In Section 3. the polarization matrix

is introduced, and the matrix elements are related to the radar cross section.

Simple targets, consisting of flat plate type reflectors, are described in Section 4, and it is

V. shown that odd bounce reflectors. suozh as a flat plate or a trihedral corner reflector, reverse the

rotational sense of the circularly polari/ed wave.

A target model, in which the target is assumed to consist of a collection of even bounce and

odd bounce type reflectors, is developed in Section 5. An equation is derived which contains

the various contributions to the dc signal expected at the mixer output. Some of the terms

sho%% a sinusoidal dependence on frequency. while others are frequency independent. It is

shown that the interference effect between each pair of individual reflectors contributes a

frequency to the Fourier transformed signal. For LFM, this frequency depends only on:

* The differe,,ce in radar range between the two reflectors.

e The bandwidth of the frequency agile transmitter.

e l he repetition period ofthe frequency span (the time during which the transmitter spans

the band~idthl. Thus. targets can be characterized on the basis of difference in distances

bet\kcen major rellectors.

In the model developed in S.,ction 5. the targets were assumed to cause no depolarization.

That is. if circular polarization were transmitted, then any given reflector would return

cicular polarization. In Section 6. dipole-like targets are considered as reflectors which return

elliptical polarization (depolarized returns). It is shown that such reflectors do not contribute

frequency independent dc terms to the mixer output. Thus. to the extent that clutter can be

characterized as dipoles. the model predicts clutter rejection.

For comparison, an analysis of a system transmitting and receiving linear polarization is

developed in Section 7. It is shown that the frequencies obtained in the Fourier transformed

signal are the same as those obtained with polarimetric processing.

In Section 8. conclusions drawn from the analyses are given, and the usefulness of

polarimetric processing for target acquisition is evaluated.

2. CIRCULAR POLARIZATION

A simplified representation of the transmitter-receiver system used in polarimetric phase

processing is shown in Figure 1. The output from a frequency agile oscillator is divided and

sent to horizontal ( H) and vertical (V) channels. The phase of the horizontal channel is delayed

by a 90 degree phase shift. It is assumed that the target is in the far-field of the antennas and

that the electric fields are plane waves. Using the right-hand coordinate system shown in

Figure 1. the vertical component of the electric field incident on the target can be written

tE = E 0 cos (kz w t) . !

x o

i

L4

z4i

U..

Eo is the electric field amplitude, to is the carrier angular frequency, and k is the wave

number, which depends on the carrier wavelength, A, according to

k = 2Tr/ . (2)

The horizonal component is

E E os kz w -)(3)y 02

E sin (kz - t) • (4)

In vector form, the transmitted electric field is

Et =E cos (kz - A) x + E sin (kz- wt) (5)0 0

%khere x and y are unit vectors along the x and y axes, respectively. Figure 2a shows the

individual components of the electric field, and Figure 2h traces the tip of the total rotating

electric field vector about the z axis, This polarization, with the wave propagating in the

positive z direction. is defined here as right circular polarization (RCP).

For left circular polarization (LCP), the y component of the electric field leads the x

component. viz.

Ext = E0 cos (kz -wt) (6)

St =E cos (kz- t+1) (7)y 02

= -E sin (kz - t) (8)

3. THE POLARIZATION MATRIX

In order to relate the electric field incident on the target to the field returned to the antennas,

the polarization matrix is introduced. The"received field". E' is the plane wave impinging on

the antennas, assuming that the antennas are in the far-field of the target. If only the vertical

component E,' is transmitted, the received electric field can be written

r t=axx EXt x + axy E t y (9)

L 6

Ili -. 1 I

IE

EV,

a. Electric field components.

b. Tip of resultant electric field vector.

Figure 2. Right circularly polarized wave (RCP).

I

where the a's are the electric field coefficients. The a, coefficient is a measure of depolarization

since it represents the horizonal reflection obtained from a vertical transmission. A simple

example of a depolarizing target is a long thin wire in the x - y plane, but tilted at some angle

with respect to the vertical.

Similarly. if only the horizontal component is transmitted, the return can be written

- ra E t -E xYx Ey ,a Y (10)

If both components are transmitted, the return is

Sr = r rE Ex + E y (IIx y

t t + t t)(a E + a E X + (a + a E y .(12)ax Ex yx y xy yy y

In matrix representation. this is

x axx yx Ext (13)

Ey axy a yy Eyt

where the two-by-two matrix is called the scattering cross section matrix. It can be shown that,

for the case where the receiving antennas and transmitting antennas are at the same location

(monostatic radar), the matrix is symmetrical: a- = a, 15 1. Setting a, = a,. a%, = a. and a,, =

a,, = a,, (where d implies depolarization), the polarization matrix is

( a d(14)

In general, the matrix elements not only contain amplitude information, but also phase

information. For example. consider the case of two long thin wires, one hori/onal and the

other vertical, but separated in range by a distance D. One received component will be shifted

in phase with respect to the other by w 2D

It is clear that the electric field coefficients of the polarization matrix (a's) are related to the

target radar cross section (RCS). The RCS is defined by

a lim 4rR 2 1E__

R-0 E1-1 2 (15)AS~

where:

* R = radar range,0 E = electric field strength at the receiving antenna, and*E' = electric field strength incident on the target 161.

The infinite limit on the range R assures that the electric fields impinging on the target and on

the receiving antenna are plane waves. If the target is in the far-field of the radar and vice versa.,

then, to a good approximation, Equation (15) becomes

u=41TR2 •rl2 (16)IE tj 2

or

I t (17)i. v4i.

Note that there is no phase information in Equation (17).

The above definition of RCS assumes that the receiving antenna can receive whatever

polari/ation that the scattered wave possesses. If one breaks the plane waves into components

as before. then an 'RCS matrix" similar to the scattering matrix can be defined,

Ex - l---- , (8)E y - '--R N 'r--'Z I/0 '- Ey

where no phase information is retained. Thus, the magnitudes of the scattering matrix

elements are given by

aij -(19)

For the monostatic case, the above notation can be simplified as follows:

Ia x l =

____ (20)

la I = , (21).Y V' R

Id (22)

IL

Ihe electric fields can be expressed in complex notation to facilitate the manipulation ofphase shifts. Let transmitted RCP be

E Re t1 (23)

iRe{ej (kz - wt) + ej (kz -wt - /2)41 (24)Re= X + e°

where Eo = I for simplicity. Also for simplicity, the e- "' time dependence will be implied but

not expressed. so

jkz -j r/2= { +e "2 ~1(25)t e {x + e y ) .(5

Then

tex r)a X e j6 X je d )j k( r~ lax e ad e- 1oY_\ 5 / (ad eay e y e 2 (26

d Y e T(6

%%here the a's are now amplitudes only, 0,. 0, O,, are the corresponding phase shifts, and R is theradar range to some reference point on the target. The phases have been related to that of the

transmitted signal at the antennas, so that z = 2R.

Equation (26) holds for all targets. no matter how complex. The polarimetric processing

in olhes frequency agility. ho\ ever. and it is apparent that the a's and 0's of the polarization

matrix depend on frequency, and hence on time. in a complicated way. The analysis is

simplified if the complex target can be considered as a collection of simple, independent

reflectors. Then, the resultant phase shift can be found from superposing the returned electric

fields from all reflectors. For the (' reflector in a given range cell,

=.r (Z: 2:' 10(7Sr. a d j2kR. (

Tr ad

Yil

Vwhere Ri is the range to the /h reflector.

For a given range cell with n reflectors.

nx r = : rIXl x (28)

r n r (29)i=l Yi

4. SIMPLE REFI.ECTORS

Consider no% the polari/atioii matrix for a few simple reflectors, beginning with a flat metal

plate perpendicular to the incident RCP \&a\c (see Figure 3aL. The vertical component is

reflected totally vertical (i.e., there is no depolarization) but with a phase shift of 180

degrees at the plate surface. This phase shift is the result of the requirement that the electric

field inside a perfect conductor is always /ero. The hori/ontal component also experiences a

I KO dcgree phase shift, and one would be led to believe that there is no net phase change

hct'%ccn vertical and hori/ontal components. Uhis is not the case, however, because the

propagational direction has changed. In order to understand the phases seen by theantennas,

consider the primed coordinate system in Figure.1h. With respect to this "antenna"coordinate

s\stcm. it is easy to see that the reflected wave is [LCP. [he net eflect of the reflection~from the

\ic%\point of the antennas, is that one component has been shifted in phase by 180 degrees

rclati\ c to the other component. RClP will be changed to I.CP. and vice versa. So. the phases

of the polari/ation matrix can be %ritten

(y 0) (30)

\o\\. in prelude to anal\ /ing returns from multiple bounce reflectors, consider reflection

from an infinite plate at an angle w\ith respect to the plane ol' the incident wave (see Figure 4).

Fhe \crtical component of the transmitted wa\e is reflected as before with a 180 degree phase

shift. I he hori/ontal component is broken do\% n into two further components:

e A component perpendicular to the plate. which experiences no phase shift

e A component parallel to the plate. which is shifted by 180 degrees

As can be seen from 4ture 4. the reflected wave is still the opposite polari/ation from that

transmitted.

ItII

*+ • ,+.. ... . ..... .77 7 l K.

cc

LU 0uii

a.

ILILo 4c0- 0- L a.

0 L 0

0

LCL

.0

Ci

00

U..

lu+

12

11

INCIDENT WAVEREFLECTED (RCP)

WAVE (

(LCP)

.---------------- , y

Figure 4. RCP incident at angle with respect to flat plate.

The above result shows that at every bounce from a surface, the sense of the polarization is

changed. For a dihedral corner reflector, the wave is returned in its original polarization. In

general. .in odd bounce target changes the polari/ation sense. an even bounce target does not.

Ili polari/ation matrices (phases only) are:

(-1 0 )for odd bounce targets. (31)

( 1 0) for CeCn bounce targets. (32)

5. TARGET MODEL

For a target model consisting of flat plates only (dihedral, trihedral corners, etc.), fromEquations (19), (26), (28), and (29), the reflected wave is given by

13

. . .

n fl - j 2kR.r= ( -1) e 1x i=1 4IT R.

odd Ihounfce.refIlectors

(33)

m j 2kR2 ,

bouncereflector,,

r = ____/a j 2kR._ E - e ei=1 AF Toddbouncereflectors

(34)

+Ee ek=1-,4TR

bouncereflectors

T he -I in the first term of equation 133) Is thle onl% ef fect of phase sfiif ts due to reflection. I n

Equations (33) and (34) above, the subscrip "" is used to denote odd bounce reflectors onkl.%s hie the subscript "."indicatcs c~en bounce reflectors. I he RCS matrix elements. o. nowk

have ts% o subscripts, the first indicating x or . direction. and the second inrd icating the

particular reflector.

rhe voltaii.esat the mixerimust no"s he related to the electric fields at the antennas. Referringto Fh~'ure /. the terminal %.oltages at the antennas are related to the electric field vector at the

antennas by-

V E (35)

14

I he \ector h is delined as the effective height of a given antenna. i.e., the effective distancealong the antenna over which the field acts. The complex voltages for identical ideal

horizontal and vertical horns are

V =h r r (36)

rVH = h f (37)

I he horiiontal component is delayed by 90 degrees. so at the mixer.

r -.V h er e (38)

I lie output of the mixer is proportional to the product of the two voltages (neglecting

atten uation losses):

V V VH , (39)

\\here the constant of proportionality has been set to i. Using Equations (33). (34). and (38).

Equation (39) becomes

h 2 Fn a xi j2kR iV = E(- e

odd

m /X7 j2kRt

even

15

n___ 1j j jkk

i~ R1 e e

odd

m .r j2kR 1Se -3 e (40)+ =i R£

even

h n 2xi j2kR m j(2kR4)4 E for e

odd even

n(jkR- t) (43)

i~ i Z=1 k£

lodd even

wvhere in Equation (411 e-w has been set equal, to -1.

Consider nthe tle to simpl cases ofa single odd bounce target and a sinle even bouncetarget. For an odd bounce target.

Vm h Ros (2kR- wt) (42)

and for an e*en bounce target

h2 V /C x a YCos 2(2 kR - wt) (43)Vm 4n Tr 2

where the voltages are now real. and the time dependence is given explicitly. Since

Cos 2 (2 k R w t) = + cos [2 (2kR -wt)] (44)

16

" " :'" 2 '......................" " '..

the dc %oitagc colmpoent the mixer output is

S8T 2 for single odd bounce targetV dcR

hc = t R2 for single even bounce target.

Tr R

I- ora complex target consi, log of a collection ol rn ccn and n odd bounce reflectors (usinp

'i" and p' tor odd bouncc rllector subscripts and and -q- to denote even bouncereflectors) lquation 41 1 hccomes

h2 [ n n ~ .V ER=i Z -- ''YP cos 2kR.m = 4 = 1 p .1

dd odd

n m aO.iwt) cos (2kR - wt) + xi - cos (2kR.p i=l i Ri Rj 1

odd even

n m xY o (- wt) cos( 2kRk - wt) - R.R cos (2kR

i l =l 1 £

odd even

m m yOx

- wt) cos(2kR - wt) - kx cos(2kR£9=i q=l k qeven even

wt) cos(2kR - t . (47q

17

Using the formula

cos A cos B = 1/2 cos (A - H) + '/2 (A + B),keeping only the dc terms, and combining the two middle terms gives

odd

R. z EP Cos 2k (R. + Ei=1 P=1 R i R 1 Rp)J i=1 k.=1odd odd odd eveni "p

R RCos [2k(R. - d 2

even

X =I. q=1 R.Z jRq ]even even

Z~g (48)

Upon examining Equation (48) term by term, one finds two terms with no cosine factors:

h2 n ai joyl h 2 m CrX aY8w and -

odd even

These terms represent the -self-inter ~erence" effects for each individual reflector in the

complex target. These terms are the same as found in Equations (45) and (46) for single

reflectors. The term

h2 n n8-ff --- E E RR Cos [2k(R1 - Re)]

-w =1 P=1 RiRPodd odd

i#p

18

j represents the odd bounce reflectors interfering with each other. For the case of only two odd

bounce reflectors. this term becomes

h 2 OX 2 +V' X~j Cos [2k (Ri R)81 R1 R 2 1 2

Similarly, the term

h2 m -,Yc xg8 R) x2q81 E R R cos [k (R£ - R q)]

Z=i q=1 R q

even even

represents the interference effects bet\%een even bounce reflectors.

The remaining term.

h2 n in -1 J rycx8-7 Z E R cos [2k(Ri Re)

i=1 =i i .odd even

represents the interference betw\een each even bounce reflector and each odd bounce reflector.

Note that if each reflector is symmetric with respect to RCS, i.e., if o, o for all reflectors,

then this term is zero.

Recall that the wave number k is related to the angular frequency of (he radar w by ( = kc.

w.here c is the propagation velocity. From the above terms it is clear that, if the frequency is

varied or stepped as a function of time, the 'dc" signal also v-aries as a function of time. This

time variation, howeser. is not only due to the fact that the cosine arguments area function of

time. but also is due to the dependence of RCS on wavelength. For the flat plate type targets

(including corner reflectors) assumed in this analysis. the RCS for normal incidence is giv'en by

= = 4Tr A2 (49)

x y 2

19

............ ......... -. -7-- 7 -i

where A is the area of the aperture [7]. Using Equation (48) for two odd bounce scatterers; with

aperture areas A I and A2 gives

2 j2 + A2

Vdc 2X 1[ 2~ R2

+ 2 RA 1RA 2 cs[kR -R)](50)+ 1 2-

Assume now that the frequency increases linearly with time, (see Figure 5), so that

fe+ t (51)

Ir

T 2T 3T

Figures5. Linear frequency modulation.

where:e L = starting frequency,o fi, = bandwidth of frequency agile transmitter, and

o T =repetition period of' frequency span.

20

Note that a linearly stepped frequency is equivalent, as long as the signal is sampled once for

each step. The 1, X2 gives a nonlinear decrease of the overall signal level during T. For L = 35

(ilI and fiH = 500 MHz. this variation amounts to 2.8 percent. The information of interest, i.e.,the frequency spectra characteristic of the target, is contained in

V f =cos [2k(Rj - R 2 )] (2

Using k- - 27T f/ gie

V f C 4 0 T

%%hich can he w~ritten

Vf cos (2nr f t + (54)f 12

where 6 is at constant phase angle.

4Tr (R1 - R ) (5C 0

and I', is the target characteristic frequency given by

2 12 c I (56)

Note that this frequency is independent of the starting frequency fo. lf,forexampleIR,-R)

I m, fi, = 500 MHz, and T = 3 ins, then f,2 =I KHz.

From Equation (56). it is clear that the frequency spectrum of the received signal for a

complex target contains frequencies given by

f. . 2 R3. R f 1 (57)

so that at given target can be characterized by the difference in range

d -jR R. (58)

21

between individual reflectors. For example, a man-made target (vehicle) might consist of two

major reflectors separated in range by - 1 m. The clutter might consist of a large number of

randomly distributed reflectors whose d,. and hence f, could be described by a gaussian

probability density, f 2

(f _ 1 e 1 o (59)

-V7 a

where 1 is the mean frequency and a is the standard deviation of the distribution. Suppose the

clutter can be characterized as reflectors separated by a mean distance of 2m with a standard

deviation of 0.25m. Using the same numbers as before for the target (d: =7 I m) and the radar

(fi, = 500 M Hz, T = 3ms), then the frequency spectrum will resemble Figure 6. The target can

be distinguished from the clutter if the difference between the target characteristic frequency

and the mean clutter frequency is large compared to the clutter frequency spread.

W

CLUTTER

Zi TARGET0.

SII -

1KHz 2KHz 3KHz

Figure 6. Frequency spectrum for target/clutter model.

Similarly. if two different targets have sufficiently different reflector range characteristics,

then target classification, based on measured data at various target aspect angles, may be

possible.

22

6. DEPOLARIZING TARGETS

lp to now, no targets which depoiaiize have been considered; metal edges and thin wires

(dipoles) are targets giving depolarized returns. If the radius of curvature of the wire (or edge)

is small compared to the transmitted wavelength, then the RCS for normal incidence and

electric field vector parallel to the wire is

Cy L -(60)"T

where L is the length of the wire 151. If the wire is at an angle 0 relative to the vertical, the

polarization matrix is given by

r 26xi L. cos20. - cosO. sin~i E,

r - 2TR. cos0 sin0. sin .2 Ey/ (61)

ta yi 1 t)

for an odd bounce reflector; the minus signs go out for an even bounce reflector. It is expected

that. since the RCS of a wire is much less than that of a plate, the returns from multiple

bounces between wires, and between wires and plates, can be neglected.

For a collection of n wires,

n L. 2 j2ka.r = - . 2 c os . e i lR

ex 2TrR. C 1i=l 1

(62)

n Li -j j 2kRi1 cos e. sin 0. e e2TR. 1

n L j 2kR.s 2aR. csin i ei=l 1

(63)

n L . -j j2kR+ £ sin 6. e e

i=1 21Ri

23

I The voltage appearing at the output of the mixer is

V~~ ;2[n Li 2kR. /CS20

+cos e.i sin 0.i e- 37/ (64)

L. 2kR 2i 1 o 1i 1, ejW/2)]

n L2hn

* Osf 0. sn6COS (2kR. wtci)= 1 142 Li R.

n n L.iL.2R R Cos' 0. sin 0. cos(2kR

i=1 j=1 Ri~

n L.wtci) cos(2kR. - wta)- E -i-Cos 3 0.sn.

2 6 sin

n n L.L.cos(2kR. wtc) sin(2kR. wt) - E

1 1i=1 j=1 R.R.

i7 'j

Cos 2 0. Cos 0 i sin eOi cos(2kRi - wta) sin(2kR. wtci)

+ E co-s 0.i sin3 0. sin (2kR. wtci) cos (2kR. wtci)i=1 R. 1

24

n n LL.2+ ~co s e. sin e. s 2 6. sin(2kR. wt)

i=1 j=i RiR 1

i/i

no(2R L 1t 2 __ 2 2 2cos(2kRi= wtR-. 2 Cos 0. sin a0. sin (2kR. - wt)

n n LL- E cos e. sin 6. cos e. sin.e. sin(2kR. wt)

1= l R~R 1 JF 1

iysi

sin(2kR. wt) (65)

I. ~ ii ri~!ofllll~ nc dent ti nd kcepi11 kiI x ii the dIc trms LeI\ eS

2 n n L.L.Vd h ZICs 2 0 sin 2 0. cos 2k(R. R)

dc 87 i=l j=1 R~R [O R 1i/ij

+ Cos 2 e. cos e sin esin 2k(R i - R)

+ cos e. sin e. sin2 0. sin 2k(R. - R)

Co e o sin 0. cos e. sin 0. cos 2k(R. - R.1 (66)

or

h2 n n L.L. A. co 2kR -R)+B.Vdc 8 i=1 j=1 RR 1) 1R1

iqi i

- Ri)](67)

25

................- ---- --- -

where

A cos 2 0 sin2 . - cos 0i sin 6. cos 0. sin ej ;(68)

B COS2 i Cos 0 sin 6j + sin 0 c2 C. (69)

Notice that all of the "self interference" terms have gone out of the mixer dc output; a single

wire target gives no signal in polarimetric processing. If clutter can be characterized as being

more like wires (trees, long grass, etc.) and targets characterized more by flat plates and corner

reflectors, then the polarimetric processing technique should provide clutter rejection for the

general dc signal at any given frequency.

For the simple case of two wires, one horizontal and one vertical, the signal is

Vd =h 2 LIL2 (0Vd 28 IR 2 cos 2k(R -R) (70)

dc 8 R12 R 2

as one might intuitively expect. For the case of a collection of wire like reflectors which are all

oriented at the same angle. 0.

2 n n L.L.V h sin 20 Z Z 1 3 sin 2k(R. - R.) . (71)

dc i=1 j=l RiRj 1

iij

Thus, for 0 = 0 degrees or 90 degrees, there is no return. One can expect that for trees, weeds,

and tall grass which are approximately vertical, the frequency dependent dc terms are zero,

and this results in clutter rejection in frequency space. To the extent that clutter can be

characterized by randomly-oriented, dipole-like reflectors which depolarize, one can expect

some clutter rejection using polarimetric processing.

For an extension to the more general case of a targe: consisting of both wire-like and plate-

like reflectors, only the additional cross terms between Equations (62), (63) and (33), (34)

remain to be considered. It is obvious that the interference between each wire and each plate

target contributes a frequency to the spectrum.

* . -* - 26 - .

7. LINEARLY POLARIZED RADAR

For comparison, an analysis of a system transmitting and receiving linear polarization isdeveloped. For vertical transmit and receive, the electric field is given by

n a -X j JTj e 2kRi n n2 k

i=l 4-TR. = JIR,0odd bounce' even

reflectors bouncereflectors

L 2kRM+ E 7R sin2 e e m(72)

wire- typereflectors

I he output signal is lound from squaring the voltage signal at the antenna.

V = 2 (,r 2 7

Using the trigonometric identities. and keeping only dc terms as before. gives similar results asf'or polarimetric processing. rHere is a frequency in the spectrum resulting from theinterference hetueen each pair of' reflectors. but there are now dc terms independent offriequcnc% resulting f'romi the "self-interlerence- effects of' the wire-like reflectors. For alinearly fretquenc\ modulated signal. the f'requency spectra of the returned signal will contain

the

f =11) c T (57)

as before, but the amplitudes nou~ depend only on the o,. element of the RCS matrix. It shouldbe pointed out that the same -clutter reJection" obtained for vertical trees, grass. etc. withpolarimectric processing can be achieved here by transmitting a horizontal linearly polarizedwave .

27

8. CONCLUSIONS

For the flat plate/corner reflector target model assumed, Equation (48) shows that the

resulting signal can be a bipolar response as a function of frequency. Intuitively, one might

expect that a bipolar signal results from a complex target, because such a target is a collection

of even bounce and odd bounce scatterers which result in a combination of those terms given

in Equations (45) and (46). Equation (48) shows however, that these terms are frequency

independent. In fact, the minimum condition for achievinga bipolar response requires that the

target contain two even bounce scatterers and two odd bounce scatterers. Not only must these

four scatterers be present, but they also must be appropriately spaced in range, i.e., the two

odd bounce targets must be constructively interferring while the two even bounce targets are

destructively interferring, and vice versa. This seems a rather broad assumption on which one

presumes to be able to discriminate between a tank (complex target) and a decoy (single corner

reflector).

It was shown, however, that polarimetric processing does in fact provide a method of

obtaining a characteristic target spectrum (Fourier transform of the mixer signal) using

frequency agility. The characteristic frequencies depend on the range difference between

target reflectors, on the LFM bandwidth, and on the FM repetition rate (Equation (57)). If

targets and clutter have sufficiently unique range distributions for individual reflectors (for

example, a characteristic average inter-reflector distance), then clutter rejection and target

classification in frequency space may be possible.

It was also shown that, for clutter which can be characterized as depolarizing dipoles, a

reduction of the overall dc signal due to clutter results. There is evidence which does in fact

indicate that clutter depolarizes more than hard targets. It has been shown that, at microwave

frequencies. o,,/ o, is - 4 dB for distributed clutter (trees) and 8 - 10 dB for vehicles [3]. This

clutter rejection occurs, however, in the dc terms which are frequency independent. If the

desired target/clutter discriminants are the frequency spectra due to frequency agility, it is not

clear to what extent this "clutter rejection" enhances the signal to noise ratio in the frequency

domain.

Finally, it was shown that target/clutter discrimination based on the Fourier transform of

the received signal is not indigenous to polarimetric processing; the technique can be realized

using linearly polarized radar. In short, there are two entirely separate effects involved in the

system model:

28

- --- -

* Pohatmwtric processing which results in rejection of dipole-like clutter.

e Frequency agility which creates the possibility of characterizing targets in frequency

space due to interference eftects.

29

REFERENCES

I. Reedy, E. K., Eaves, J. L., Piper, S. 0., Parks, W. K., Brookshire, S. P., Wetherington,

R. D., and Trebits, R. N., (C) Stationary Target Detection and Classification Studies(U), 2nd Interim Report, US Army Electronics Research and Development Command,

No. ECOM-76-0961-2, March 1978.

2. Root, Jr. L. W. (US Army MIRADCOM)and Cullis, R. N., (C) "Multi-environment

Active RF Seeker (MARFS)(U)," US Army Missile Research and Development

Command, November I, 1976.

3. Lewinski. D. J.. Echard, J. D.. Rausch, E. 0., Piper S. 0., and Eaves, J. L., Stationar;

Target Detection and Classification Studies. Final Report, US Army ElectronicsResearch and Development Command, No. DELCS-TR-76-0961-F, April 1979.

4. Stovall, R. E.. A Gaussian Noise Analysis of the "Pseudo-Coherent Discriminant",

Lincoln Laboratory, Massachusetts Institute of Technology, Technical Note 1978-46,December 29, 1978.

5. Berkowitz, R.S., Modern Radar Anah-sis, Evaluation, and Srstem Design. New York,

Wiley, 1965.

6. Skolnik, M. I., Introduction to Radar SYstems, New York, McGraw-Hill, 1962.

7. Crispen, Jr., J.W. and Siegel. K.M., Methods of Radar Cross-Section Analysis, New

York, Academic Press, 1968.

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