Generalized Eigen Value Decomposition_00328059

8/2/2019 Generalized Eigen Value Decomposition_00328059

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IEEE TENCON'93 / Br i j n g

GENERALIZED EIGENVECTOR DECOMPOSITION METHOD FOR POLEEXTRACTIONAND GPOFMETHOD

Weidong WANG, Haizhou LI and Youan KEDept. of E. E.,Beijing Inst. of Tech.100081, Beijing, China

ABSTRACTA method of generalized eigenvectordecomposition (GEVD) is proposed to used forextracting the poles of radar target by using theautocorrelation of target late-time impulse responsesin this paper. Unlike to the g e ne d h d pencil-of-function (GPOF) method proposed by Hua andSarkar [2], the additive noise in signal is consideredin this method. And it can be used to evaluate thepoles when only the amplitude data of multiplefrequency returns is available. The simulation resultsin this paper illustrate that the new method hasadvantages over GPOF method in the noisesensitivity and the computation.

INTRODUCTIONIn the past decades, the pole extraction by usingthe late-time response of the targets has been thefocus of much research work. Linear prediction (LP)and GPOF methods were found to extract the polesfrom an exponentially damped sinusoids. LP methodtreats the pole extraction as a linear least squareproblem. GPOF method treats the pole extraction asa general eigen-analysis problem. But both of themdo not consider the additive noise in their solutionsto the pole extraction problem. Therefore, thesemethods can, usually, only be used in the case ofhigh signal-to -noise ratio.This paper will develop an efficient method forpole extraction. In its mathematical model, theadditive noise is considered. Theoretically, thismethod is insensitive to noise. The simulation resultspresented in this paper illustrate that the new methodhas better performance than GPOF method.This paper will be divided into three parts. In part

one, The brief descriptions of GEVD method and itsalgorithm is presented. In part two, theperformanceof GEVD and GPOF methods are compared. In partthree, an example of pole extraction from thetransient response data, which is impulsed by thenarrow Gaussian pulse and measured practically byour experiment system, s presented.G E N E R A L I Z E D E I G E N V E C T O RDECOMPOSITION METHOD FOR ESTIMATINGPOLES1. Theoretical Results

It is well known that a late-time impulse responseh(t) of a radar target can be describexi as thefollowing sum of exponential signal

Y

Where ai and ai re the damping coefficient andangular frequency of the i-th natural mode, a, and 4,are the amplitude and phase of the i-th natural mode,and pi, p;=ui+aj and ri, I;=&& are called as thei-th pair of radar target pole and residue respectively.It is noted that the target poles are independentwithaspect angles but the target resides am related toaspect angles. M s the total number of distinctivenatural modes. In practice, M can be estimatedbasedon the frequency spectrum of the interrogatingradar pulse. Our goal is to obtain a noise-insensitiveand accurate method, which is efficiently determinedby target poles under the presence of noise. Assumethat the experimentaldataof the radar target impulseresponse h(t) under the presence of noise isexpressed as

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where w(n) is a Guassian white noise with zero meanand d variance. By the simple computation, theautocorrelation r(m) of target impulse response canbe obtained in the following formr(m) = ~ < m > ~ i t=1,2, ..,"-I (3)

zya, = 5,=1#2,..m (6 )

b l P,+P*

and the correlation matrix asai)= (di ) (i.1) ... di+M2-1) ) M22 W(8 )

In the case hat M,>M, M2 2Mand N>M,+M2-1,

where * equals M1 or M2, To look into theunderlying structure ofthe these matrices, and basedon the above equation, we can obtain

To solve the problem of radar target poleestimation, the concept of generalized eigen-analysisplays an important role in the process. The generaleigcn-analysis of singular matrix pair {B(i),B(j)} isformally defined by

R (&?e = A, &I%, i=l, 2, ...,W (14)The knowledge of the eigenvalue can be seen toconvey important information concerning the targetpoles. Hence, we can obtain the following theorem.[Theorem] The M nonzero eigenvalues of theabove eigen-analysis system a, espectively,expressed as

The proof of the above theorem can easily beobtained from the equation (13). The theorem makesit clear how to solve the pole estimation problem ofradar targets. The remainder is how to get theautocorrelation function r(m) of target impulseresponse. With exception of the direct computation ofthe auto-correlation from radar target impulseresponse, Usually, two approaches can also beadopted:The first is to estimate the autocorrelationr(m) by using the amplitude I (oJ I' in multiplefrequency radar. The sccond is to calculate theautocorrelation r(m) by using the &ved signal innoise radar. In the following, a simple algorithm forestimating poles is presented.2. Eigenvector Decomposition Method forComputing Generalized Eigenvalues

To develop and illustrate the use of algorithm forcomputing generalized eigenvalues of the matrix pairproblem, we can write the correlation matrix BO)into the following form of matrix eigenvectordecompositionB O - U A V B (16)

where the superscript H denotes the complexconjugate operator. A s the dwwhich consist of thenonzero eigenvalues of Bo), and is thecorresponding unitary matrix.Calculate the followingmatrix as

From the above equation, it can be seen that the

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matrix 3 as the same eigenvalues as the generalizedeigen analysis equation (14). Note that for noisy dateh(n) the matrix A should choose the 2M largesteigenvalues of BQ). This can reduce the effect ofnoise in the signal.According to the equations (7) (8) (16) and (17),the target poles can be estimated from theautocorrelation function of its impulse. response.

SIMULATION RESULTSIn order to facilitate comparison to previouspublished GPOF method, In this section, an exampleextracted from [2] was chosen for the simulation ofthe noise sensitivities of GEVD and GPOF methods.Specifically, Thirty data points were generatedaccording to the model

Yx(n) = CA,sin(o&++,)e-=C (18)

i-1

where n=O, 1 ..., N-1, N=30, M=2, A,=A2=1,and w1=O.2r, w2=O.35r, 41=42=0,a,=O.O2?r,a2=O.035?r. Note that where damping factors andresonant frequencies are, respectively, normalized bythe sampling frequency f,= 1/T.For illustrating the noise sensitivitiesof methods, itis assumed that the additive noise in the signal x(n)is a white Guassian random sequence with a varianceof $. Thus, we define the SNR as -lolog($) (dB).The mean square error (MSE) results of the poleestimates of Monte Carlo simulation (50 independenttrails) are shown in Fig 1.With corresponding to the 1-th pair of pole, theCrame-Rao bound for unbiased estimation, and theperformance curves for GPOF and GEVD methodsis given in Fig la. It is not difficult to find thatGEVD method has almost the same performance asGPOF. With corresponding to the 2-th pair of pole,however, the case will be different. In Fig lb, theC-R bound for unbiased estimation, and theperformance curves for the GPOF and GEVdmethods are shown. It is seen that GEVD method hasbetter performance than GPOF method, andpossesses the lower bound of SNR than GPOFmethod. And the failure of Hua's GPOF method toexceed the C-R bound at the low SNR stems fromthe bias in its estimate, which may probably beattributed to its signal model.The causal of GEVD's better performance thanGPOF's lies in the following ways: The first is that .Hermitican data matrices is approximated with thelower rank ones in GEVD method, but Toeplitz datamatrices is approximated by the lower rank ones inGPOF method. The second is that Two computationof EVD is needed in GEVD method, bu t threecomputation of EVD is needed in GPOF method.

AN APPLICATIONConsider that a EM pulse is generated by 0.84-mdipole antenna of radius0.003-m which impulsed bya narrow Gaussian pulse with time-width 2 ns and

amplitude 80 v. The EM pulse is received to get atransient response by using the same antenna. Thistransient response is measurkd through a digital storeoscillometer (SQ27) with the sampling interval 0.02ns. Its waveform is shown in Fig. 2. To get theinstrinsc poles of dipole itself, we only consider asegment of the transient response for 10.27 to 12.23ns, which consists of 100 sampled data (N=100).Applying GEVD method with MI=M2=N/2=50and M=6 as well as GPOF method with L=N/2=50and M=6 to the 100 sampled data,we calculate the17 natural modes of the dipole antenna by using themshown in Tab 1by different approaches for theidentical measured data. where L is the length of thedipole and c is the light velocity, and * denote thenon-appearance of the resonant frequency.With difference of the natural mode number M,GEVD method yielded the following poles:for M=2, 0.061, -8.6061*9.1182j,for M=3, -0.4472*12.6479j, -0.1424f 0.5815j,for M =4, -0.0377f 3.4O42j, -0.03 11f 1.697 j,for M =5, -0.0255f 3.3507j, -0.019 1f 1.6639j,

10.76+ 16.0854j-0.0284f 0.1801j-0.1394f 0.5815j, -0.0274f 0.1800j-0.3595f 8.886Oj, -0.1344f 0.5800j,-0.0259f 0.1806jWith the difference of the natural mode number M,the corresponding poles are estimated by usingGPOF method [2] for the same dat in the followingfor M=2, 0.60, -8.0600f9.1180j,

for M=3, -0.5414&12.9446j, -0.1526f 0,5916j,for M=4, -0.0979*13.7274j, -0.0713f11.974Oj,for M= 5 , -0.0232*13.6717j, -0.0539&11.9409j,

34.86+ 16.0856j-0.0319f 0.1823j-0.1514f 0.5915j, -0.0313f 0.1821j,-0.3767f 9.0998j, -0.1458f 0.5905j,-0.0296f .01829jFrom the above results, the first two pair poles isstable. two estimated resonant frequencies areobtained to be 0.18 and 0.57 GHz. and some otherfrequencies also appear with the increase of naturalmode number.

REFERENCES111 Weidong WANG and Zaigen FANG, "Newalgorithms for High-resolution array signal ~processing", Pro. ICASSP, Beijing, 1990 '[2] Yingbo HUA and Tapan K. Sarkar,"GeneralizedPencil-of-function Method for Extracting Poles of an

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EM system from Its Transient R esponse",IEE E Trans. Vol. AP-37, NO.2, 1989Table 1. The Estimated Resonant By Frequencies Various Algorithms

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, N \ ,0

----

" ' I L , . . , , . . . .

0c\1

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Generalized Eigen Value Decomposition_00328059