Complex natural resonance extraction from cross-polarized measured scattering data

WAVE MOTION 8 (1986) 259-265 NORTH-HOLLAND

COMPLEX MEASURED

NATURAL RESONANCE SCATTERING DATA

259

EXTRACTION

FROM

CROSS-POLARIZED

D. L. M O F F A T I " a n d C.-Y. L A I Department of Electrical Engineering, Ohio State University, Columbus, 0H43212, U.S.A.

Received 12 August 1985

Complex natural resonances (CNR) of realistic aircraft models are extracted from both co-polarized and-cross-polarized measured scattering data. Two principal points are demonstrated in this paper. First is the fact that natural resonances can be meaningfully utilized using 6.0 to 12.0 GHz data. All previous resonance extraction results used 1.0 to 6.0 GHz data. The above frequencies are model frequencies. The full scale frequencies are in the HF band. Second, it is illustrated that if the co-polarized returns are dominated by specular scattering, then there is considerable merit to using cross-polarized scattering data for resonance extraction. It is shown that resonances extracted from cross-polarized data are much more stable (with respect to target orientation) than resonances obtained using co-polarized data.

1. Introduction

2. Theoretical background

T h e c o m p l e x n a t u r a l r e s o n a n c e s ( C N R ' s ) o f an o b j e c t are useful for i d e n t i f i c a t i o n p u r p o s e s . The success o f this p r o c e d u r e , however, d e p e n d s on careful e x t r a c t i o n o f the C N R ' s as well as the i d e n t i f i c a t i o n process. In this study the e m p h a s i s is on the r e s o n a n c e e x t r a c t i o n process itself a n d we illustrate the usefulness o f c r o s s - p o l a r i z e d scattering d a t a in this context. T h e a v a i l a b i l i t y o f a c c u r a t e , b r o a d b a n d scattering d a t a has m a d e the p r e s e n t s t u d y p o s s i b l e [1]. At most o r i e n t a t i o n angles the c r o s s - p o l a r i z e d returns are at least ten dB b e l o w the c o - p o l a r i z e d returns b u t the d a t a are still useful for n a t u r a l r e s o n a n c e extraction. In fact, a very d o m i n a n t s p e c u l a r return in the c o - p o l a r i z e d d a t a u s u a l l y m a s k s the r e s o n a n c e effect. This is f o u n d to be the case for aspects n e a r b r o a d s i d e for the aircraft. F o r b o t h the c o - p o l a r i z e d a n d c r o s s - p o l a r i z e d d a t a the m e t h o d o f C N R e x t r a c t i o n is the r a t i o n a l f u n c t i o n fit m e t h o d [2]. T h e targets are e l e c t r o p l a t e d m o d e l s o f m o d e r n aircraft.

F o r s i m p l e o b j e c t s such as the sphere a n d infinite circular c y l i n d e r , r i g o r o u s analytical s o l u t i o n s exist for the l o c a t i o n (in the c o m p l e x s p l a n e ) o f the C N R ' s . F o r g e o m e t r i c a l l y c o m p l i c a t e d targets, however, the C N R ' s m u s t be e x t r a c t e d from m e a s u r e d scattering data. Even m o d e s t electrical sizes c a n n o t be t r e a t e d using a m o m e n t m e t h o d f o r m u l a t i o n a n d n u m e r i c a l search p r o c e d u r e . A s c a t t e r e r has an infinite n u m b e r o f C N R ' s a n d is t h e r e f o r e a system o f infinite order. T h e i m p u l s e r e s p o n s e o f a g e o m e t r i c a l l y s i m p l e target or, for that matter, the r e s p o n s e o f the target to any a p e r i o d i c i l l u m i n a t i o n , consists o f first a f o r c e d r e s p o n s e as the w a v e f r o n t moves across the target a n d then a free or n a t u r a l r e s p o n s e as the w a v e f r o n t moves b e y o n d the target. As d i s c u s s e d later, the s e p a r a t i o n is n o t distinct for c o m p l i c a t e d targets. In the s ( L a p l a c e ) d o m a i n then the transfer function o f the target can be m o d e l l e d as a r e s i d u e series p l u s an entire f u n c t i o n :

0165-2125/86/$3.50 t~) 1986, Elsevier Science Publishers B.V. (North-Holland)

D. L. Moffatt, C.- Y. Lai / Resonance extraction flora scattering data

260

F(s) = ~ C,(n.......~).~_G(s, n). .=1

(1)

S-- S.

The C,'s are the residues which are aspect (fl) dependent, the s,'s are the natural resonances and G(s, 1-1)is the entire function. Note that the entire function is also dependent on aspect. Any practical measurement system is a bandlimited system so only a limited number of CNR's are significant for calculations in a particular frequency range. The response to an impulse input is actually of the form N Y(s)= E n=l

C, S--Sn

+G(s,I'I)+N(s),

(2)

and N

y ( t ) = Y.

C, eSo'+~,(t,O)+n(t),

(3)

With this approach the forced response portion of the return is avoided. This is possible for simple target shapes although the return may be many dB down. For complicated targets, however, the response in the time domain is really a very complicated combination of free response and forced response. Some substructures of the target are resonating long before the interrogating wavefront moves beyond the target. The authors agree with Felsen [4] that subtracting the physical optics approximation would be a good first step to negate the effects of the entire function. Unfortunately, for geometrically complicated targets obtaining the physical optics approximation could be a major effort. We offer an alternative approach as outlined below. The rational function is written as

n=l

F(s)-

where G(s,/'2) ={0G(s''O)

within passband, (4) outside passband.

In a practical system one obtains the response as a function of real frequency, corresponding to information over a certain region of the ira axis. Neglecting the noise for the moment, if the unknowns of eq. (2), i.e. C,, s, and G(s, O) can be found somehow, the equation suggests that one has knowledge of the portion of the complex s plane delineated by the real frequency limits. For geometrically complicated targets there is no known method for finding the entire function G(s, ~). Therefore, some sort of approximating function must be employed in searching for the CNR's. Since the known data points are all on the ira axis, one is really performing an extrapolation even though complete knowledge of the entire function is not the goal. For this reason no method of CNR extraction can claim absolute accuracy. The entire function part of the response, being unknown and aspect dependent, causes difficulties. In Prony's method, where the time-domain response is sampled, it has been suggested that the late-time portion of the response be sampled [3].

bo+ bts + b2s 2 + " " "+ bN+l SN+I l+ats+a2s2+. ..+aNsN ,

(5)

rewriting eq. (5) F ( s ) = Y. n =.1

S -- Sn

+a+bs,

(6)

where b = bN+t/aN etc. The expression a+bs in eq. (6) is a simple attempt to model the entire function. It is motivated by the fact that the impulse response of a number of simple objects have impulse or doublet singularities. Therefore, as long as the frequency range over which one extracts CNR's is not too large (the optical range is avoided) the approximation remains good. The unknowns a and b are both aspect and frequency range dependent. A distinction should be made between the complex natural resonances of the target and the zeros of the denominator of eq. (5). The poles of eq. (5) are not necessarily true CNR's as there are usually curve fitting poles which must be eliminated. For a geometrically complicated target the true CNR's will not always appear in each test because some of the resonances may be unexcited or only weakly excited for particular orientations of the target. Equation (6) does not account for the different time delays for various components of the specular

D. L. Moffatt, C.- Y. Lai / Resonance extraction from scattering data

returns. Furthermore, these time delays are also aspect dependent. A possible formulation is as follows: F(s)= E n=l

S--Sn

+ E ( a i e - g ' S + b i s . e - h ' s ) • (7) i=l

The parameters gi and hi are real numbers that are aspect dependent. Equation (7) introduces more unknowns and is actually nonlinear. It may, however, lead to an even better fit of the measured spectrum. The individual titne delays only change the phase of the corresponding specular terms. However, the time delays of eq. (7) would affect the magnitude of F ( s ) also. Therefore, the use of an algorithm that requires only magnitude information will not offer any improvement. One way to remove the non-linearity is to estimate a priori gi and hi. This would require data which is sufficiently broadband that impulse and doublets with delay could be discerned. It would also require aspect and geometrical information of the object. The series expansion of the exponential terms is not useful since it may not converge. Details of the application of the rational function method are given in [2]. Note that all of the coefficients in eqs. (5) and (6) are real. Therefore the set of poles consists primarily of complex conjugate pairs with corresponding complex conjugate residues. The form ofeq. (6) can lead to a different method of target identification as given in [5]. With the method of identification given in [5], a and b play an important role and the identification technique is aspect dependent. For a given set of CNR's and a given frequency range, one first finds the ranges of variation of a and b for different signal to noise ratios. These parameters as well as the CNR's are stored. Given frequency scattering data from an unknown target, the set of CNR's are used to obtain the a and b ofeq. (6) and are then compared to their corresponding ranges of variation to make a decision on the identity of the unknown target. This method actually takes advantage of the existence of the entire function and works

261

extremely well for high (>15 dB) signal to noise ratios. For signal to noise ratios lower than 15 dB, the variation of a and "b become too great for the method to be useful in its preseiat form. We now have a basic understanding of the theory of CNR extraction and are ready to examine the results of C N R extraction using co-polarized and cross-polarized scattering data.

3. Computation and analysis The specular portion of the backscatter response from geometrically complicated targets is predominantly in the co-polarized component. For all but the most anomalous target one can safely conclude that only a weak entire function component exists in the cross-polarized return. As will be demonstrated, this means better numerical stability for the CNR's extracted from the cross-polarized returns. m compact radar cross section measurement range [ 1] was used to obtain calibrated amplitude and phase scattering data for the targets. A digital filter is then used to numerically time gate and remove unwanted clutter from the time regions where the target return is known to be zero. The time response of the target is not altered. Figure 1 shows a plot of the cross-polarized frequency response of an aircraft target at broadside incidence (90°). Figure 2 is the corresponding bandlimited impulse response of the aircraft. In Figs. 1 and 2 vertical polarization was transmitted and horizontal polarization was received. Figures 3 and 4 show corresponding frequency and time domain plots after the data have been passed through a digital filter. The purpose of the filter is to somewhat smooth the data without removing any salient features. The data shown in Fig. 3 are then used for pole extraction. Equation (5) is used to fit portions of the spectrum, typically applied over a 1.0 GHz bandwidth. This is done because of the extrapolation nature of the calculations. A wider bandwidth is not desirable from a theoretical viewpoint because windowing must be used to help

D. L Moffatt, C,- Y. Lai / Resonance extraction from scattering data

262 0 0o

O GO

~g

~ LIJ0 LU ~ n~

fie (.9 ILl r't

LU E3 Z o

\

0

Z

ILl

t~J ¢D

o= 2.0

'1.O

3.0 FREQUENCY

4.0 IN GHz

I

5.0

6.0

1.0

3.0 4.0 5.0 FREQUENCY IN GHz

2.O

6.0

g m o aod Z

mo QN Z

LU

~o

g_ I--

Z (.9 ~0

0

o i

I

1.0

2.0

3.0 4.0 F R E Q U E N C Y IN GHz

5.0

I.O

6.0

Fig. 1. Cross-polarized amplitude and phase response of aircraft at broadside (90°).

w? 03

Af

v

2".0

3.0 4.0 FREQUENCY IN GHz

5.0

6.0

Fig. 3. Cross-polarized amplitude and phase response of aircraft at broadside (90 °) after digital filtering.

0

Z 0 o_ (/)

Z 0 (2. UJ ~0

A,-~ A ~. V tV vv .

^.

0

A•r rV v v

-'

. . .

_J

_J O-

-o 7 0

-3.0

-2.0

-I.0 TiME

0 IN

1.0

2.0

3.0

NANOSECS

Fig. 2. Cross-polarized impulse response of aircraft at broadside (90°).

-3.0

- 2.0

-

1.0 TIME

0 IN

1.0

2.0

3.0

NANOSECS

Fig. 4. Cross-polarized impulse response of aircraft at broadside (90 ° ) after digital filtering.

D. L. Moffatt, C.-Y. Lai / Resonance extraction from scattering data

eliminate pattern-fitting poles. Choosing endpoints of the band at local minima helps to increase the accuracy of the solution. Slightly shifting this band, e.g., using 4.0 to 5.0 G H z and then using 4.05 to 5.05 G H z helps in weeding out curve or patternfitting poles. C N R ' s have been extracted for four incidence angles 0 ° (nose-on), 45 °, 90 ° and 180 °, using three polarizations (vertical-vertical, horizontalhorizontal and vertical-horizontal). The C N R ' s extracted are listed in Table 1. Only the oscillatory part of the C N R ' s are listed. It is clear from Table 1 that no incidence angle yields all of the extracted poles, but the crosspolarized results have a lower rate of absenteeism than the co-polarized results. Reasoning with regard to the specular returns, it seems clear that even near nose-on and tail-on where the crosspolarized return is 10 dB down from the co-polarized returns it is still advantageous not to have to deal with specular components. In the neighborhood near broadside (90°), considerable difficulty was experienced in extracting C N R ' s from the co-polarized returns. This was anticipated as it is clear that the entire function is quite large in this region. Figures 5 and 6, which show the co-polar-

263

ized returns, are both relatively fiat compared to the cross-polarized return in Fig. 3. Table 2 lists the average C N R ' s as found over the four angles. Maximum deviation from the average over the angles are listed inside parentheses. Both of the co-polarized results are essentially equally different from the cross-polarized returns. Table 1 also shows that the co-polarized returns are more aspect dependent in that more poles are missing at many aspects compared to the cross-polarized returns. As would be expected for a complicated object, all of the poles shown in the cross-polarized returns are also present in one of the two co-polarized results. The cross-polarized returns cannot contain any C N R ' s that'are not in any of the co-polarized returns.

4. Use of higher frequency data The rational function method has been applied to the horizontal polarization response of another aircraft model at 6.0-12.0 GHz. The C N R ' s (Table 3) appear to have the same type of behavior as those found earlier using 1.0-6.0 G H z data. Therefore, this suggests that this frequency range is also

Table 1 Oscillatory parts of CNR's

extracted from backscatter

0* W

responses of different polarizations

45 °

and angles

90 °

VH

HH

VV

VH

HH

W

VH

180 ° HH

VV

VH

HH 1.100

1.143

1.017

1.215

1.134

1.106

1.259

1.050

1.090

1.208

1.198

1.408

1.470

1.474

1.494

1.574

1.626

|.427

1.490

-

1.473

1.488

1.642

1.682

1.770

1.765

1.797

-

i.714

1.821

1.742

-

1.956

1.958

1.919

2.097

-

2.015

2.330

2.195

2.280

2.210

2.247

2.418

2.420

2.720

2.512

2.592

2.657

2.656

2.717

-

2.905

3.073

2.965

2.871

-

2.947

3.310

3.287

-

3.298

3.271

3.276

3.349

3.586

3.747

3.654

-

3.707

3.675

-

3.639

4.026

4.059

4.159

3.893

4.067

4.103

4.192

4.239

-

-

4.718

4.543

4.627

-

5.056

-

5.505

5.533

-

5.442

-

3.971

4.208

-

4.727

4.684

4.717

5.024

5.136

-

-

5.51

1.439 -

2.128

2.097

2.038

1.926

-

2.298

2.341

2.233

2.647

2.643

2.657

3.065

-

3.047

2.870

-

-

3.321

3.781

3.645

3.572

4.113

4.100

4.021

-

4.226

4.241

4.671 5.547

2.597 3.219 4.145

-

4.596

-

-

4.924

4.796

4.510

5.511

-

5.677

5.622

264

D. L Moffatt, C.-Y. Lai / Resonance extraction from scattering data o co J

(~

0O

f

wo w~

r~ t.9

w Q

t:3

zo

o z

hl ¢/I

W or)

~o I

o

o El

"T

I

2.0

1.0

3.0 4.0 FREQUENCY IN GHz

5.0

1.0

2.0

o "T 1.0

2 .O

6.0

3.0 4.0 FREQUENCY IN GHz

5.0

6.0

5.0

6.O

0

El

oo

/-

Z hi t:3

/

z W a

~_o

~o t,-

i--

z

z (.9

o

I.O

2,O

5.0 4.0 FREQUENCY IN GHz

5.0

6.0

3.O

4.0

FREQUENCY IN GHz

Fig. 5. Vertically co-polarized amplitude and phase response of aircraft at broadside (90*) after digital filtering.

Fig. 6. Horizontally co-polarized amplitude and phase response of aircraft at broadside (90 ° ) after digital filtering.

Table 3 Table 2 Average of oscillatory parts in Table 1 over the four angles (maximum deviation from the average value is shown in parenthesis)

Oscillatory parts of C N R ' s extracted from horizontal polarization response of another aircraft at 6.0-12.0 G H z between 0 ° and 90* 0°

W 1.227 1.451 1.707 2.008 2.315 2.678 2.968 3.319 3.616 3,998 4.209 4,647

VH (0.032) (0.043) (0.065) (0.089) (0.105) (0.042) (0.061) (0.030) (0.030) (0,105) (0,017) (0.070)

-

5,476

(0.034)

1.131 1.506 1.767 2.003 2.261 2.605 2.928 3.293 3.666 4,063 4,240 4,705 4.974 5,576

15"

30 °

45 °

6,245 7,385 7,906 8.516 9,141 9,776 10,117 10,469 11.078

6,520 7.282 8,000 8,556 9.156 9.550 10.053 11.094

60 °

75 °

90 °

HH (0.081) (0.068) (0.085) (0.047) (0.080) (0.093) (0.137) (0.028) (0.094) (0,042) (0,001) (0,034) (0,050) (0,101)

1.098 1.513 1.756 2.027 2.310 2.637 3.073 3.248 3.703 4.139 4,208 4,579 4.996 5.555

(0.008) (0.113) (0.014) (0.101) (0.108) (0.080) (0.029) (0.078) (0,026)

6.465 7,406 7.908 8.686 8.974 9,521 10.53 10.873

4,247 7,302 8,005 9,017 9,585 10,053 10.45

o n e w h e r e it is m e a n i n g f u l (0.105) (0,200) (0,067)

use them

6,479

6,559 7,413

8.077 8,696 9.924 9.562 10.516 10.95

9,153 9.582 10.059 10,475 -

to extract CNR

and to

for discrimination.

It is d i f f i c u l t t o r e l a t e t h e s e r e s o n a n c e s tural

6.410 7,342 8,040 8,730 8,918 9,621 9,951 10.59 11.072

features

of the

aircraft

to struc-

at this stage.

It is

D. L. Moffatt, C.-K Lai / Resonance extraction from scattering data

possible to postulate physical substructures that resonate. However, there is no analytical proof. Also, each physical substructure would have more than one complex natural resonance associated with it. One possible way to discern structural dependence would be to use absorbing material on the various substructures to study their effect on the CNR's.

265

fully understood. One could simply use crosspolarized data for C N R extraction. In this case the measured scattering data are controlled and the measurements are clutter-limited, not noiselimited. The extracted CNR's could then be used in prediction-correlation processing using copolarized a n d / o r cross-polarized scattering data. The idea of target identification from CNR's using cross-polarized real world scattering data needs to be very carefully explored.

5. Conclusions The practical implications and problems of C N R extraction have been studied using both co-polarized and cross-polarized backscattering data. The CNR extraction method and the required accuracy are directly related to target identification using prediction-correlation. The CNR locations must be quite accurate. The research reported here has demonstrated two main points. First, the frequency range which was used in previous studies can be doubled without encountering problems with the C N R concept. It is known that above some unknown high frequency limit the density of the CNR's will preclude target identification using prediction-correlation. Additional research is needed to establish this high-frequency limit for practical targets. Second, it has been demonstrated that there are advantages in terms of orientation sensitivity and stability to CNR extraction using cross-polarized backscatter data. The disadvantage is that the data may be as much as 10 to 15 dB down from the co-polarized return. This point must be care-

Acknowledgment The work reported in this paper was, supported jointly by the Office of Naval Research under Contract numbers N00014-84-K-0705 and N00014-82K-0037.

References [I] E.K. Walton and J.D. Young, "The Ohio State University Compact Radar Cross-section measurement range", IEEE Trans. Antennas Propagat. 32, (11), 1218-1223 (1984). [2] D.L. Mof/att and C.Y. Lai, "Natural resonance estimation", IEEE Trans. Instrumentation and Measurements, to appear. [3] J.R. Auton, T.L. Larry and M.L. Van Blaricum, "'Target identification via complex natural resonance extraction from radar signatures", N A T O / A G A R D "'Target Signature" Symposium, October 8-12, 1984, London, England. [4] L.B. Felsen, "Comments on early time SEM", IEEE Trans. Antennas Propagat. 33(1), 118-120 (1985). [5] D.L. Moffatt and C.Y. Lai, "Non-cooperative target recognition", Ohio State University ElectroScience Laboratory Report for Project 714190.