High Resolution DOA Estimation Using Matrix Pencil
Jinhwan Koh
*'
Department of Electronic Engineering, Engineering Research Institute, GyungSang National University, Jinju, Korea,
[email protected] Tapan K. Sarkar Department of Electrical Engineering, Syracuse University, Syracuse, NY, USA
1. Introduction
Many algorithms for the problem of DOA (direction of amval) estimation have been studied to increase their resolution capability as well as to reduce their computational cost. Capon's
minimum variance technique attempts to overcome the poor resolution problems associated with the delay-and sum method [I, 21. More advanced approaches are so called superresolution techniques. Techniques based on the eigen-structure of the input covariance matrix including MUSIC (Multiple Signal Classification), Root-MUSIC and ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) generate high resolution DOA estimation. Music algorithm proposed by Schmidt [3] retums the pseudo-spectrum at all frequency samples. Root-music [4] r e m s the estimated discrete frequency spectrum, along with the corresponding signal power estimates. Root-music is one of the most useful approaches for frequency estimation of signals made up of a sum of exponentials embedded in additive white Gaussian noise. In this research, the Matrix Pencil Method (MPM) has been compared to the well known rootMUSIC algorithm. The term Pencil originated with Gantmacher [5] in 1960 and Sarkar [6, 71 developed the mathematical model to obtain the exponent of a sum of complex exponentials. Actually, MPM is a variation of the eigen-structure approach and it uses the input pencil matrix instead of the input covariance matrix. By mapping the noise components to the null space, the signal components can be distinguished exactly. Simulation results show that the MPM has a superior ability of discriminating closely spaced target f?om a single target to the root-MUSIC algorithm.
2. Matrix pencil method
The sampled signal x(kT,) is to be modeled by a sum of complex exponentials, i.e.,
I
This work was supported by Korea Research Foundation Grant (KRF-2003-003-DOO316).
0-7803-8302-8/04/$20.00 02004 IEEE 423
where R I = Residues or complex amplitudes,
s, = -a,
+&,
a,
=
Damping factors,
U ,=
Angular frequencies, e',', = Z, , for i = l,2, ...,M The objective is to find the best estimates of M,R,and
Z,
from x(kT,)
Consider a matrix Y (Assume we have N sampled data) and two sub-matrices Y,. Yt,
40) M
41)
A
M
x(L-1)
O
M
41)
42)
42) M
43)
A
dL+O
A
4N-I)
>
M x(N-L+I)
where L is called the pencil parameter, L is chosen in between N/3 to N/2 for efficient noise filtering 171. One can write
Yo =Z,,RZ,,(3) Y, = ZnR,Z,Z,, (4)
I
A
1
where
2, =diag[z,,z,,A , z M ] , R, =diag[R,,R,,A ,R,].
Consider the matrix pencil Y, - A Y, = Z,R, [Z, - A 112,.(5)
Therefore,/2 = z, , for i=l,2,,..,M would be the eigenvalue of the generalized eigenvalue problem, Y,-AY,.(~)
It can be shown that this is the same as solving the ordinary eigenvalue problem [Y"+Y&A
I , (7)
where Y,'
is the Moore-Penrose pseudo-inverse of Yo which is defined by
Y,' =[Y.f~YJ'Y.'l. (8)
Once /z = 2, are known, the residues R, are solved for from the following least square problem
The frequency component is computed from
424
(10)
0, = 1 ~ l n ( z , ) ] .
and the magnitude A, for a single frequency CO, is evaluated from ~ , e = R , ~ Jovp
A, =
R,[- Re(.,)]. (11)
Since we do not know how many frequency components exist in the signal, the number of estimated frequencies M should be determined using some criteria.
Typically the singular
values beyond Mare set equal to zero. The way M i s chosen is as follows [7]. Consider the singular value uc such that u & , ~ l O p ,wherep is the number of significant decimal digits in the data.
If the data is noisy, it is desirable to develop a statistical estimate for the z,that comes close as possible to the Cramer-Rao bound [7].This can be achieved by performing a TLS (total least square). Perform a SVD of Y to get Y =UZV".(12)
The modified matrix pencil becomes
Y,-~Y,=u~~v~-IY,H],(~~) where Vf and V, are obtained by eliminating the last and first rows of V. The noisy data matrix
Y is pre-filtered using SVD and (8) is transformed to a generalized eigenvalue problem, V, V r -AV, V," = O . (14) 3. Numerical Simulations As a first example, consider a signal of unit amplitudes arriving from PI=n/3and +pd,
P,=nN
i.e., /=exp(,~x,)+exp(,p,x,);x, = 0, 1, 2, ... , 99. The frequency error in the
estimation defined as
((p- p,, (II
/l(o*- p,Ill , where pel is the estimated frequency of p, and
and P =[PfPJT.In Fig1 shows the Monte Carlo simulation of the angle, p d , versus the error in the estimation. The pencil parameter, L, was equal to IO and 30 and the size of the corresponding covariance matrices were IO by IO and 30 by 30. The superior resolution performance of the MPM is demonstrated in comparison with conventional root-MUSIC algorithm in Figl. Fig2 illustrates the angle,
Pd, versus the error in the
estimation with the
different number of samples. Again, we observed that the MPM shows better performance than the root-MUSIC algorithm. L was equal to 113 of the number of samples. As a second example, consider a signal of unit amplitude arriving from pl=lu2 and a jammer
arriving p>=m'2 +
p,,with amplitude Jnq,
i.e., f = e x p ( j P , x , ) + ~ , , e x p ( j P ~ x , ); x,= 0,1,2,
.._,9. We are now interested in finding the jammer strength that is going to produce an output error of I% in the estimation of the signal strength. This is corresponding to an equivalent output signal to noise ratio of 40dB. In Fig3, we plot the
pd versus
the jammer to signal
strength to produce a 40dB signal to noise ratio at the output. MPM approach stays an advantage over root-MUSIC algorithm. 4. Conclusion
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In this paper, the performance of the Matrix Pencil Method and well known root-MUSIC algorithm for high resolution DOA estimation has been compared. Simulation results show that the MPM has a superior ability of discriminating closely spaced target from a single target to the root-MUSIC algorithm.
Reference [I] Liberti, J.C, and Rappaport, T. S.: ‘Smart antennas for wireless communications’, (Prentica Hall, Upper Saddle River, NJ, 1999) [2] Capon, J.: ‘Maximum Likelihood Spectral Estimation’, Nonlinear Methods of Spectral Analysis. 1979, pp. 155-179 [3] Schmidt, R. 0.:‘Multiple Emitter Location and Signal Parameter Estimation’, IEEE Trans. On Antennas and Propagation, 1986,34, (3), pp. 276-280 [4] Barabell, A. J.: ‘Improving the Resolution Performance of Eigenstructure-based Direction Finding Algorithm’, Proc. Of the IEEE Int’l Conf. on ASSP-83,1983, pp. 336-339 [5] Gantmacher, E R.: ‘Theory of Matrices Volume 1 ’ (Chelsea, New York, 1960) [6] Hua, Y. and Sarkar, T. K.: ‘Generalized pencil-of function method for extracting poles of an EM system from its transient response’, IEEE trans. on Antennas and Propagation, 1989, 37, (2), pp. 229-234 171 Wemer. D.H. and Mittra. R.: ‘Frontiers in Electromagnetics’ (IEEE press, Piscataway, NJ,
,
.., .. ‘.
..‘MPM, L;a
,I>;;
0 0032 Om4 O U 6 O m 3 001 0012 00140016 0018 $1.
Om
0 O W Z O m 4 0 a S O m 3 001 0012001400160010 18l,
Irrdlraq
Fig. 1 Resolution and errors in the estimations, different L and the size ofcovariance matrix 70,
,
,
,
.
,
,
,
Om
II.dlBLC1
Fig2 Resolution and errors in the estimations, different number of samples ,
Fig. 3 40dB output SNR criterion
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