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An Adaptive Filtering Approach to Chirp Estimation and ISARImaging of Maneuvering Targets

Genyuan Wang*

AbstractIn inverse synthetic aperture rada r (ISAR) maging, due

to the noncooperative motion of maneuvering targats theDoppler shifts of the scatterers are usually time-varying,and th e radar return signals are usually chirps. T he chirpestimation plays an important role in the performance ofthe ISAR imaging. Li and Stoica recently presented a nadaptive FIR filtering approach to estim ate the amplitudesand phases of sinusoidal signals and applied this methodto the synthetic aperture radar (SAR) imaging with thesinusoidal signal model. In this paper, we extend the Liand Stoica's algorithm to estimate the chirps and thenapply it to the ISAR imaging of maneuvering targets. Theextended algorithm is verified by simulated data.

1 IntroductionChirp signals form an important class of signals that

have applications in radar and sonar For example, in SARimaging, when targets have motions, the radar return sig-nals are chirps, and in ISAR imaging, when targets havemaneuvering motions (rotations), the radar return signalsare also chirps. T he chirps in the radar r etur n signals in-clude the importan t information about the moving targets,such a s th e velocities and the location param eters of themoving targe ts in SAR imaging. Furthe rmore , the properdechirping is necessary in the ISAR imaging, where thechirp estimation is also needed. There have been manyalgorithms for the chirp estimation, for example [K-161. Inthis paper, we propose a new chirp estimation algorithmand in particular apply it to th e ISAR imaging. T he newchirp estimation algorithm is an extension of the adaptiveFIR filtering approach for the amplitude and phase esti-mation of sinusoidal (APES) signals recently proposed byLi and Stoica [l ]and applied in SAR and ISAR imagings[3-71 based on sinusoidal signal models. Although the pro-posed chirp estimation algorithm in this paper applies togeneral chirp signals, in the second half of this paper we

'Department of Electrical and Computer Engineering, Uni-versity of Delaware, Newark, DE 19716. Email: (gwang,xxia)Qee.udel.edu. Phone: (302)831-8038. Fax: (302)831-4316.This work wa s partially supported by the Air Force Office ofScientific Research (AFOSR) under Grant No.F49620-00-1-0086and th e 1998 Officeof Naval Research (ONR) Young Investiga-tor Program (YIP) under Grant N00014-98-1-0644.

Xiang-Gen Xia*

focus on the ISAR imaging. ISAR imaging is a two di-mensional high resolution imaging of a flying target, suchas an airplane, where one dimension corresponds to therange (the fast time) direction and the other dimensioncorresponds to the cross range (the slow time) direction,and the Fourier transform is used in both dimensions. Theresolution in t he range direction is obtained by transmit-ting wide-band radar signals while the resolution in thecross range direction is obtained by th e spectrum analysis,far example the high resolution spectrum estimation meth-ods have been applied to E A R and SAR imagings in [3, 5 ,71. It is well-known that the Fourier transform works wellfor stationa ry sinusoidal signals but not for nonstationarychirp signals. Based on th is observation, improved ISARimaging algorithms have been obtained in [17-25],such asthe joint time-frequency analysis approach in [17-221. Highresolution spectral estimation based on the radar returndata in a short time duration was also studied , for example13, 5 , 71. In this paper, we propose to use the chirp esti-mation algorithm generalized from the algorithm ob tainedby Li and Stoica [l] or sinusoidal signal parameter esti-mation. Th e generalized algorithm is then applied to thesimulated radar data, which outperforms both the conven-tional ISAR imaging method and the superresoltion E A Rimaging method using the Li and Stoica's APES algorithm

This paper is organized as follows. In Section 2 , weextend the APES algorithm to the chirp estimation. InSection 3, th e extended chirp estimation algorithm is usedto the E A R maging of maneuvering targets. In Section 4,the extended algorithm proposed in this paper is verifiedby the simulated radar data.

PI .

2 Chirp Estimation Using AdaptiveFiltering

In this section, we extend the APES algorithm obtainedby Li and Stoica [ l] o the chirp estimation.2.1 Chirp Estimation with Filter

We first consider single chirp signal model with additivenoise. Let {Z, , ] fz; denote a discrete time data secquenceof the form:

12Z, =A(u,w)exp(j(-an* +nu)) + e , , (2 .1)

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where A(a,w)denotes the complex amplitude of a chirpsignal with chirp rate a and initial frequency w, and endenotes the additive noise. Similar to the APES algorithm,we first consider how to estimate the complex amplitudeA ( a , w ) when the parameters a and w are known. Let

Z, =(2, ~ , + 1 ... 2,+M-11T, m =0 , 1 , ...,N -M ,be an M-dimensional vector sequence with length N - M +I , where stands for the transpose. The mo +1-th termZ+ in the vector Z, can be expressed as:

1Zm+mo = A(a,w)exp(j-a(m +mo)' + m+mo )w )2

(2 .2 )

+em+mo 1= A(a,w )exp(j( ami+mow))12xp(j( -am2 +mw))exp(jammo) +e+,

(2 .3 )Define

+Em+mowhere Gm+,, is the modulated noise

em+,, exp (-jammo) of e,+,, . LetX , =[x, , + 1 ... X , + M - I I ~ , m =0 , 1 ,...,N -M

be th e vector sequence obtained by the da ta XmfmO ivenin (2.4).

Let h ( a , w ) denote the impulse response of an M-tapfilter, where

(2 .4 )

h ( a , W ) = hi(( ^ h z ( c ~ , w )..h l l . ~ ( a , ~ ) ] ~ . 2.5)The impule response h(a , ) varies with the frequencyw and the frequency rate a. Note that the output ob-tained by passing the vector sequence X, through thefilter h ( a , w ) is

hH a , )X m=A(a,w)[hT(a,w)a(a,w)]exp(j(5am2+mu))

+wm, m =O , l , ...,N -M (2 .6 )where denotes the complex conjugate transpose, wm =hH ( a , w ) [ Em n + l ... Em+M-1IT denotes the pertorbationof the filter output due to the additive noise, and

which does not depend on the vector sequence index m ofZ,. If the filter h H ( a , w )s chosen such that

hH(a,w)a(a,w)=1 , (2.8)( 2 . 6 ) becomes

H 1h ( a , w ) X , =A(a,w)exp(j(-am2 + m w ) ) + w , . (2.9)

Therefore, the least-squares est imate of A(a,w ) from thedata h H ( a , ) X , is

1 H&,U) = N - M +1 [h (a )w)1- M( X m exp(-j(-am2 +~ > > ) 0 2 . 1 0 )

Let X ( a , w ) denote the following chirplet transform of(X,}",f divided by N -M +1 , i.e.,

m=O

N - M 12m xp(-j(-am2 +mu)> .m = ON - M + 1(a ,w )=

(2.11)A(a ,w)=h H ( a , w ) X ( a , w ) . (2.12)

Then, (2.10) can be represented as

In the previous derivation, we have assumed that the pa-rameters a and w of the chirp are known. In practice, theamplitude a(a , ) is estimated in terms of the parametersor variables a and w. The true parameters a and w arethe ones that maximize the magnitude of A(a,w),whichis similar to the sinusoidal signal estimation in [l].

The problem now is how to o btain th ~eilter h (a , ) usedin (2.12) , which shall be discussed in the next subsection.2.2 The Adaptive Filter Design

Let R denote the forward sample covariance matrixN - MCxmxL (2.13)m=ON - M + 1

=and

Q(a ,w )=R - X ( a , w ) X H ( a , w ) (2.14)denote the estimation of th e noise covariance matr ix. Sim-ilar to the derivation of th e adaptive FIR filter h A p E s ( W )obtained in [l], he adaptive filter h ( a , w ) in the last sub-section can be given as:

Note that the above filter h ( a , w ) satisfies property ( 2 .8 ) .Therefore, by combining (2.12) and (2 .15) , the amplitudeof the chirp signal can be obtained as

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Similar t o [l], he estimate A(a,w ) n (2.16) is the max-imum likelihood ( M L) estimate of A ( a , w ) when the noisewere independently and identically distributed zero meanGaussian random vectors.

When there are multiple chirps in a received signal, theone with the maximal energy is estimated first and thensubstracted from the signal. Repea t this procedure forthe remaining signal until t he energy of the remaining sig-nal is insignificant. The multiple chirps can be estimatedby the procedure. The accuracy of the above chirp esti-mation algorithm is higher than the traditional RandonWigner transform transform. Fig. 2 shows the chirp es-timation result using the Randon Wigner transform whileFig. 3 shows the chirp estimation result using the abovealgorithm. One can clearly see the improvement.

3 ISAR Imaging of ManeuveringTargets

In ISAR imaging, the radar is stationary and trans-mits wideband electromagnetic waves to a moving target.Suppose the radar transmits a linear frequency modulated(LFM) signal

s , ( t ) =Aexp{j27r(fct +Et')}, (3 .1)where A is the amplitude, fc is the carrier frequency, K isthe chirp rate.

imaging p r o j ec t i o n planei/x J,

Figure 1: Target image projection plane.Let 0 be a scatterer treate d as the center of the target,v o ( t ) be the velocity of th e scatterer 0. Thus, the motion

of the target can be determined by vo(t) and the rotationwith the angular velocity R(t) of the target with respect toscatterer 0. Let X - Y -2 be the coordinate system shownin Fig. 1 and P with location (21, 1, z1 ) be a scatterer o,fthe moving target, the radar is sitting on the X axis, Ris the rotation ve_tor of the targe t with respect t o thecenter 0, where Re s th_e effecqive rotating vector and /3is the angular between !2 and_&, ?is the vector betweenscatterer P and scatterer 0,R is the vector of the center

0 to the rada r, i.e., the radar line-of-sight. Then, th ereturned signal of scatterer P is

ns ( t , T ) =Ap exp{j2r[fc(t - ;1)+T ( t- d ) ~ ] } , 3 .2 )where t is the fast time for the range dimension, T is theslow time for the cross-range dimension, Td =~ R ~ ( T ) / csthe time delay, R P ( T )s the range from the radar to thepoint scatterer P at time 7 , nd c is the wave propagationvelocity. Th e baseband signal in this case is

Ks b ( t , ~ ) = Ap exp{-j27r[fcrd - (t - ~d)']}2= A P exp(-j@(t)), (3 .3)

where the phase @ ( t )s2RP(.) 2

( 3 . 4 )@(t )=27r[fCTd-qt-Td)7 =~-7rRp(T)7rK(t-- ) .2 xAfter the range compression by multiplying exp(j7r&)and taking the inverse Fourier transform of the basebandsignal in (3 .3) it becomes

) I ,(3 .5)

~ R P ( T )s ( t , T )=Ap exp(-j- 47rRp( )sinc[mTCp t--where, without confusion in understand ing, we still use thenotation s(t,.r) for simplicity, and T,, is the compressedpulse width. Let &(T) be the range between the center 0and the radar at time T. Then

R P ( T ) =& ( T ) +(d(T) .'((.>)E(.),and the signal s ( t , T ) in (3.5) can be written as

4 4

11,RP ( 7 )sinc[mT,,(t --where 2 represent the unite vector of the line of radarsight, and 47~&(7)/X is the phase variation of scatterer 0.When the pulse compression is good enough, for simplic-ity we assume the sinc function in (3.6) o be the deltafunction,

1.R P ( T )a(t --f scatterer 0 is used as he hot point t o obtain the motion

compensation, the returned signal s ( t ,7) f scatterer Pafter t he motion compensation becomes

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where 7 d 0 =F.fter the returned signal of scattererP s aligned to t he same range, the signal can be writtenas

4xx(7 ) =A p exp(j-fL(7)ycosp) =A p exp(jw(7)). (3.7)Notice that t n (3.7) is the slow time. In a small viewinginterval, the target rotations can be regard as constantacceleration, and without loss of generality, ,f3 = 0, i.e.,the target rotates in the X -Y plane, then the angularvelocity is 0:n,(T)=R(T) =ROT +-2,2and

W ( 7 ) =-(Ooy.rx +-7-Y 2).x 2Therefore, the returned signal of scatterer P after therange compression and the motion compensation can befinally written as

where T, s the length of the aperture, WO =4xRoy/x andWI =47my/X. This means t he re turned signal of scatte rerP is a chirp signal within a small viewing interval, which isverified by the Wigner-Ville distribution of the raw radarda ta shown in Fig. 4 and Fig. 5. In general, the radarreturn signal after the range compression and t he motioncompensation in a range bin has the form:

Ns ( t ) =C A , xp(j(a,t2 +brit+C n ) ) (3.10)

n = l

which is the summation of several chirp signals with dif-ferent chirp rates, N is the number of scatte rers in therange bin. In the classical ISAR imaging, the FFT of theabove s ( t ) in a range bin is taken, which is based on thesinusoidal signal model, i.e., the signal s ( t ) in (3.10) hasonly sinusoidal signals. Th e APES algorithm was appliedto t he ISAR imaging in [7]. Because of the variation of theDoppler in te rms of th e time, i.e., the chirps, the instanta-neous image should be made for the maneuvering targets,see for example [18-251. In the following, the chirp esti-mation method proposed in Section 2 will be used to formthe ISAR image in each range bin.

4 SimulationsIn this section, we want to apply the chirp estimaion

algorithm obtained in Section 2 to the ISAR imaging ofsimulated dat a of b727. The processed da ta has 128 slowtime sample points. The airplane rota tes maneuverly dur-ing the observation.

The received signal of each scat terer is approximately achirp. Fig.G(a) is the imaging result by using the first half(first 64 slow time sample points) received data . Since theconventional imaging met hod is based on the Fourier trans-form and one can see tha t the image especialy the head of

airplane is smeared. The super resolution imaging algo-rithm using the TVSE proposed in [3] is used to the firsthalf raw data. The image is shown in Fig. 6(b). The imag-ing quality is proved compare with that of Fig.G(a). How-ever, the imaging quality of Fig.G(b) is not high enough,the head of airplan is also smeared , because APES is theparameteres estimate algorithm to sinusiod signals and thereceived signals are chirp ones. Fig. 6(c) shows th e ISARimaging result to the data used in Fig.G(b), by using thechirplet decomposion imaging algorithm (CDIA) proposedin [15]. Th e image result in Fig.G(b) has higher imagequality especialy the head of airplane. Fig. 6(d) showsthe imaging result t o the da ta used in Fig. 6(a) by thenovel imaging algorithm. Fig 6(d) have higher resolutionthan t ha t of Fig.G(a), Fig.G(b) and Fig.G(c) (especialy inthe head of airplane. Fig. 6(e), Fig. 6(f ), Fig.6 (g) andFig. 6(h) show the image results of th e conventional ISARimaging algorithm, the TVSE algorithm, the CDIA algo-rithm and t he novel imaging algorithm respectively to thewhole 128 slow time sample points data. It is shown thatFig. G(h)obtained by the novel algorithm has higher res-olution than tha t of Fig. 6(e) and Fig. 6(f). Alough theFig. 6(h) has almost th e same resloution t o the Fig. 6(g)at the head of the plane, it has a higher resolution thantha t of Fig. 6(g) a t the body of the plane.

The novel algorithm is also verified by raw data. Theimaging results are not shown here, because the data arenot open now.

5 ConclusionIn this paper, the APES algorithm for sinusoidal sig-

nals proposed by Li and Stoica was extended t o the chirps.The extended algorithm was applied to t he ISAR imaging.The ISAR imaging results indicated tha t t he extended al-gorithm outperforms the existing algorithms.

References111 J. Li and P. Stoica, An adaptive filtering approach tospectral estimation and SAR imaging, IEEE Tkans. onSignal Processing, vo1.44, June, 1996[2] M. R. Palsetia and J. Li, Using APES for interfero-metric SAR imaging, IEEE Trans. on Image Processing,(31R. Wu, Z. Liu, and J. Li, Time-varying complex spec-tral estimation with applications t o ISAR imaging, Thirtysecond Asilomer Conference on Signals, Systems and Com-puers, pp.14-18, Pacific Grove, California, Nov. 1998.[4] J. C. Wood and D. T. Barry, Linear signal synthe-sis using the Radon-Wigner transform, IEEE Trans. onSignal Processing, ~01.42 , p.2105-2111, 1994[5] M. Kram, K. Meraim and Y. Hua, Fast quadrticphase tranform form estimating the parameters of muti-component chirp signals, Digital Signal Processing,vol.7,pp.127-135, 1997.[6 ] W. Li, Wigner distr ibution method equivalent todechirp method for detecting a chirp signal, IEEE Trans,

~01.7: 9) 1340-1353, SEP T. 1998.

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Acoust. Speech Signal Processing, vo1.35, pp.1210-1211,1987.[7] T. J. Abatzoglou, Fast maximum likelihood joint esti-mation of frequency and frequency rate, IEEE Puns. onAerosp. Electron. Syst., v01.22, pp.708-715, Nov. 1986.[8] R. Kumaresan and S. Verma, On estimating the pa-rameters of chirp signals using rank reduction techniques,Proc. 2 ls t Asilomar Conf. Signals, Syst., Comput., PacificGrove, CA, pp.555-558, 1987.[9] P. M. Djuric and S. M. Kay, Parameter estimationof chirp signals, IEEE Pans. on Acoust., Speech, SignalProcess., ~01 .38 , p.2118-2126, Dec. 1990.[lo] S. Qian, D. Chen, and Q. Yin, Adaptive chirpletbased signal approximation, ICASSP98, Seattle, Wash-ington, U.S.A, May 1998.[ l l ]V. Chen, Reconstruction of inverse synthetic apertureradar image using adaptive time-frequency wavelet trans-form, (invited paper), SPIE Proc. Wavelet Applications,[12] V. Chen and S. Qian, Joint time-frequency trans-form for radar range-Doppler imaging, IEEE Bans. onAerospace and Electronic Systems, vo1.34, May 1998.[13] V. C. Chen and W. J. Miceli,Time-varying spectralanalysis for radar imaging of maneuver ing targets, IEEProceedings on Radar Sonar and Navigation, ~01.145, pe-cial Issue on Radar Signal Processing, pp.262-268, 1998[14] V. C. Chen and H. Ling, Joint time-frequency anal-ysis for radar signal and image processing, IEEE SignalProcessing Magazine, vo1.16, pp.81-93, March 1999.[15] Z. Bao, G . Wang, and L. Luo, Inverse syntheticradar imaging of maneuvering targets, Optical Engineer-ing, ~01. 37, p.1582-1588, 1998

~01.2491, p.373-386, 1995.

Figure 2: Chirp estimation result by the conventionalmethod.

Figure 3: Chirp estimation result by the extendedAPES algorithm.

Figure 4: Wigner-Ville distrib ution of th e received sig-nal from a range bin.

Figure 5 : Wigner-Ville distr ibution of th e received sig-nal from another range bin.

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I IO m 30 U) 80 60h R n p l

(a ) Conventional imaging resultto the fir st half data

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to the firs t half data(c ) C D I A maging resultto the firs t half data

(9)D CI A imaging resultto the whole data ( h ) Novel algorithm imaging resultto th e whole data

Figure 6: Imaging results by different algorithms

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