THE GENERALIZED POLARIZATION MATRIXThe polarization matrix in the mixed coordinate -system, say p,...
Transcript of THE GENERALIZED POLARIZATION MATRIXThe polarization matrix in the mixed coordinate -system, say p,...
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THE GENERALIZED POLARIZATION SCATTERING MATRIX
December 1968
This Study Was Performed UnderContract F 30602-68-C-0260
by
Norbert N. Bojarski16 Circle Drive
Moorestown, New Jersey 08057
Consultant to
Syracuse University Research CorporationSpecial Projects Laboratory
Merrill Lane, University HeightsSyracuse, New York 1321.0
Prepared for 0 0 0Rome Air Development Center -- 7- i'
Griffiss Air Force Base LRome, New York 13440 D SEP 22
011A
-Ld
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TABLE OF CONTENTS
LI
Section Page
1. INTRODUCTION
2. THE MONOSTATIC MONOCHROMATIC POLARIZATIONSCATTERING MATRIX 2
3. THE BISTATIC GENERALIZATION 5
4. THE SHORT PULSE GENER.ALIZATION 8.
15. THE BISTATIC SHORT PULSE GENERALIZATION 9
6. THE MEASUREMFNT OF THE POLARIZATION MATRIX 11
7. THE FARADAY ROTATION ANGLE 12
8. THE ORIENTATION ANGLE OF A SYMMETRIC TARGET 14
[
L 0
L'h
I
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U
ILIST OF ILLUSTRATIONS
Figure Page
1. The Target Coordinate System 3
2. The Mixed Coordinate System 6
i
Ii
Ii'4e
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THE GENERALIZED POLARIZATION SCATTERING MATRIX
[
( 1. IN'RODUCTION
The conventional definition of the monostatic monoiromatic
polarizationmatrix is first extefided to the bis~atic case, then to
the short pulse case, and finally to the bistaticshort pulse case.
The tran formations and convolutions involved are discussed in
some detail.
'The method of determining the Least Square Best Estimate
of the Generalized Polarization matrix from a set-of measurem4:nts
is then developed.
It is shown that the Faraday r3tation angles introduced by aI magneto ionic medium intervening the radar and the target are
determinable from measured short pulse monostatic polarization
: [ matrix data.
It is then shown that the Least Square Best Estimate of the
orientation angle of a symmetric target is also determinable from
Faraday rotation contaminated short.pulse monostatic polarization
matrix data.
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2. THE MONOSTAtIC MONOCHROMATIC POLARIZATION SCATTERINGMATRIX
11The monostatic monochromatic polarization scattering rn.a.t-ix(1)'p1 is defined by the equation
where Iand S are the range normalized incident and scattered electric
(or magnetic; depending on the convention adopted) far-fields re-
spectively, the phases of which are referred to a coordinate system
11 in which the scatterer is described (See Figure 1, p. 3); i. e.,
.ik ._Ei(x) =I e L (2)
I fk •E (x) S e (3j
where Ei(x) and Es(x) are the incident and scattered electric fields
at the point x respectively, and ki and k are the propagation vectors
of the incident and scattered fields respectively.
The rationale of the range',normalization of the scattered
field (see equation 3) is to insur e consistency within the power
cross-section definition " )
11 4Trx (4)
r2
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,ii
Scatterer N
FSLI [ =--j- )
X3
[I:
Figure 1
The Target Coordinate System
V 3
I [°i'
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- - Q o
13 and the relationship:
=P PI
-where p is the Eigen Value of the polarization matrix 2'or the specialcaseof certain geometries (to be discussed later) for which this
;polarization matrix is. fully degenerate; i. e.:
P = (6)
For this monostatic case, by the very definition of the mono-
static case as k. (See Figure 1, p3):
k~ =-k (7)8~ C
k k (8)
tA A
where k is the unit vector specifying the viewing direction, the bore-
sight vector of the radar, and the aspect angles (of the radar relativeto the target) for this monostatic case (See, Figure 1, p. 3).
I /
I It shouli be noted that the matrix pp. Y is not invariant to thechoice of coordinate systems.
It should further be noted that for this monostatic case, by
virtue of the transversality of the far-fields, the matrix p., can beI t
fully described as a second rank matrix if x is chosen as colinearwith k (see Figure 1, p. 3); i.e., since I 3 =S3 =0 for this co-
ordinate system, it follows that p p = 0.
\
It
L~ \- 4
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3. THE BISTATIC GENERALIZATION
For the monostatic case equation 1, by virtue of equations 7 and 8,
can clearly be written as:
A A A
S'(k) = p (k) I (k) (9)
For the bistatic case, for which k. * "k5 , the natural gen-
eralization of the above defining equation 9 is clearly (see Figure 2,
A A A As (k8 ) = ft ILYkski) I (k.) (10)
Except that it is now no longer possible to find a single coordinate
system such that the rank of the matrix p, (k 5 I 1i) reduces to two.A A L 8
for all k. i -_k. It is, however, possible to find two coordinate, systems-
such that the matrix p (ks, ki), expressed in these mixed coordinate
systems, reduces to a matrix of rank two; i.e., any two coordinate
systems x ( i ) and Z(s) chosen such that x(') and x are colinear with
ki and k s respectively. The details of the transformation of equation
10 into such a mixed coordinate system will be presented rriext.
Let T (i) aind , (s) be the coordinate transf0oition matrices11Y ILY
from the target coorzdinate system (see Figure 1, p. 3) to the
coordinate systems x and (s) respectively (see Figure 2, p. 6);
i.e.
(I) M T7 (i) ly(1
S (S -T S) S(12).
55
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I\
X3..
U1
112'3 x, -i
Xll
£111
1X2
1I
Figure 2
II The Mixed Coordirate System
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where (see Figure 2-, p. 6):
r-sine tnCs
IL Y n n ~ l~ nCOnS? 1 (3
Co C. n Co I1-n sin tnCi j sinqn
[ It thus- follows fromr equation 10 that (since- T for a real
unitary transformation):
The polarization matrix in the mixed coordinate -system,
say p, is thus:
P, r~D)P LM (16)
where, '.Y virtue of the transversality of the incident and scattered
L far-fields, -the matrix p' is a second rank matrix; i. e.:
LY Lct yP 3 P,
a2 LP 1,2,
LRelative to the x 3-axis, the components 1 and 2 of the primedsecond rank matrix are clearly the TRANSVERSE and LONGITUDINAL
Icomponcnts respectively (see Figure 2, p. 6); e.9. , if x 3 ' isthe axis of sym-metry of an axially symmetric scatterer.i
1~ 7
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4. THE SHORT PULSE GENERALIZATION
For the monochromatic case,equation I, by virtue of a~ ins
7 and 8, can clearly be written as:
For the time domain, equation 18 thus yields:
S 1t') =. M * 1 119)
where:
p M) = - je t (ca) d (20)
IL Y Zf-O ILY
S (t) = 1 7 e'iats 1c1 da (21)
.1M I(t) - f e iWtI (w) dw (22)
-oo 7 G
In the time domain, the polarization matrix equation (equa-
tion 19) is thus a matrix- convolution equation; i.e.,iGoII IL M ~ p (t') I *Y(t- t)dtt (23)
For finite transmitted power, i. e.
J I Mt I dt < oo(2.)Ii0
8
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r
rit follows that:
00
K~Y y
It thus follows that the integration and implied summati6n in
equation 23 can be intez -hanged; i. e., written out:
1 [ 1 2
ILS(t) =Y f p 0t') I (t-t') dt' (26)
1 5. THE BISTATIC SHORT PULSE GENERALIZATION
For the monostatic monochromatic case equation 1, by virtue
of equations 7 and 8, can clearly be writteii as:
A A A(),k) p (w. k) I (wk) (27)
Thus, for the bistatic monochromatic case, by virtue of
equation 10:
A A A
' IL (u) (,k) p= WY (w k a ki) I (to.ki)1 (Z8)
tion 26:Thus, in the time domain, by virtue of the results of equa-
I i A 26:
" ~S (t, ks k ,, t s ki) I(t- t', ki) dr
. 4i l (29)
F: I.t9
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I A"
In the mixed coordinate system discussed in Section 3,.
equation 29 yeilds: ."
where by equations 11, 12, and 17; and equations 20, 21, nd 22:
[ 1 340 t
3 * t A
s S)(tI-y 8"- (w .k
3 3
li 3 (33)•;L M, ,T (k( ' P 0, k ki) d
where the logical notation:
7 (k) M 7 (n) n = i,s (34)11Y T ILY
has been adopted.
Iio 10
U
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[6. THE MEASUREMENT OF THE POLARIZATION MATRIX
A single Measurement with a monochromatic: monostatic radar
possessing polarization diversity capabilities (i. e., a radar capable
of transmitting an arbitrary incident polarization and measuring the
jscattered polarization associated with this transmission) clearly
yields knowledge of the pair of vectors I and S (see equation 1, p. 2).
For a set of such measurements, a set of pairs of vectors, say, -I (n)
and S(n) is clearly yieldet1 The Least Square Best'Estimate of the
polarizatin matrix (say p ) is thus that matrix p which best satisfies
equation 1, p. 2; i. e.. the Least Square Best Estimate of p by the
I; [equation: n
5(n) I (n): S - (35)
Defining the matrices:
S s(n) (36)
I - I (n) (37)
|" this yields for equation 35:
.Sn IL. yn (38)
or, in pure matrix notation:
IfS PI (39)
If the matrices, I,S, and p were real, when the Least Square
Best Estimate of p would have been given by (3) :
P Si -1)-(40)p[S(II
L1
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InIt can rea'dily be shown (4 ) that for complex matrices I, S, and
p, the Least Square Best Estim'a'te of p is:
(] sit (I I+ - (41)
Iwhere the error associated with the Least Square Best Estimate is
calculable by the conve'tional method 5 ).
For the Bistatic Short Pulse Case (see equation 28, p. 9),
1the defining equations 36 and,37 need thus merely be extended to:
SSk) = s(n. A
^ I (,k ) (42)
A ) AII yfl(w.k 5 )=11 (ck) (43)
The additional constraining condition that the polarizationII (6)matrix is true-symmetric has not been imposed on the Least
Square Best Estimate of the polarization matrix for reasons that
will become evident in-the next Section 7.
7. THE FARADAY ROTATION ANGLE
11If a mangeto ionic medium (such as the ionosphere in the
magnetic field of the earth) is intervening between the radar and
Ii the target, then the measured monostatic monochromatic polarizationmatrix is not true-symmetric because of the Faraday rotation effects
introduced by such a medium(7)
HGiven such a Faraday rotation contaminated monostatic mono-chromatic polarization matrix, the Faraday rotation angle 0 can be
(8)4.]] determined8; i. e.:
L{U 12.
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[
tanl 26 P1 1 (44)PlZ P22
and the uncontaminated monostatic monochromatic polarization matrix,-say- p , (namely that matrix that should have been measured had there
been no intervening m~agneto ionic medium) can also be determine ;
i. e.
[* p. = e (45)
where:
, \-sin 0 Cos (46)
I -The rationale for not seeking a true-symmetric Least Square
Best Estimate of the polarization matrix (see Section 6)Iis"thus
•evident; namely, the asymmetric components of the Least Square-i LBe t Estimate of the polarization matrix are utilized in equation 44.
-Since the preceding equations 44, 45, and 46 are in the fre-
. quoncy domain, it follows by an argument similar to the argumd.. of
Section 4. that for the short pulse-monostatic polarization matrix:
00'
0(t) e'tlw)dw
[ where O(w) is given by equation 44, and
p (t) = @ (t)*p(t)* G (t) (48)
I
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where:.
= os 0(t) sin O(t)(
\ sin 0(t) cos e(t)(
8. THE ORIENTATION ANGLE OF A SYMMETRIC TARGET
U It has previously been shown "(1 0 ) that the orientation angle i
of a- symmetric target is related to the Faraday rotation contaminated
monostatic monochromatic polarization matrix p by:
tan 2a - Pl " (50)11l P ZZ
"II Since the above equation 50 -is in the frequency domain, it
follows by an argument similar to the argument of Section 4 that
the above equation 50 must be written as:
iz(, ) + PZ(
tan Z Ca 0 ) = " (51)
.°P 1()-P Z2(W )
The orientation angle a is clearly, however, not a function
of the frequency, since it is a purely geometrical quantity.
It thus immediately follows that the Least Square Best Estimate
of the orientation angle c, say , is given for the short pulse mono-
static polarization matrix case by:
WZ.
Ct W (a a(w dw (52)
U f14
Li
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where:
ta- P [1(a) + 02(40J (53)
where p is given by equation 41 of Section 6.VtI(U) It should be noted in)' conclusion that the determination of the.
Least Square Best Eiatof the orientation angle is. accomplished
directly, without first determining the Faraday rotation angle and
then removing its effects.
'iis
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REFERENCES
1. R.S. Berkowitz, "Modern Radar," Sect VI. 5-5, pp. 560-565.
2. D. E. Kerr and H. Goldstein, I"Propagation of Short Radio Waves,"M.I.T. Radiation Laboratory Series, Vol., 13, Sect. 6.2, p. 455Boston Technical Lithographers, Inc., 1963.
3*. C. Lanczos, "Applied Analysis, " Sect. I. 25, p. 156 et seq., PrenticeHall, Inc. 1964.
4. N. N. Bojarski, "Electromagnetic Short Pulse Inverse Scattering forDiscontinuities in an Area Distribution, " Sect. VU, pp. 41-44,Syracuse University Research Corporation, Special Projects Lab-oratory Report, June 1967, AD 845 125.
5. C. Lanczos, op. cit.
6. N. N. Bojarski, "A Solution for the Orientation Angle of CylindricallySymmetric Targets, and Faraday Rotation Angle, in Terrs of FaradayRotation Cbntaminated Scattering Matrix Radar Data," RCAr-MoorestownReport, January 1964.
1 7. Ibid.
8. Ibid.
9. Ibid.
10. Ibid.
UU