direction of arrival estimation in the presence of distributed noise sources: cumulant based...

Direction of arrival estimation in the presence of distributed noise sources: Cumulant based approach

Prabhakar S. Naidu and Raghavan Subramaniyan Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560012, India

(Received 25 February 1994; revised 12 April 1994; accepted 18 December 1994)

In the past few years there have been attempts to develop subspace methods for DoA (direction of arrival) estimation using a fourth-order cumulant which is known to de-emphasize Gaussian background noise. To gauge the relative performance of the cumulant MUSIC (Multiple S_jgnal Classification) (c-MUSIC) and the standard MUSIC, based on the covariance function, an extensive numerical study has been carried out, where a narrow-band signal source has been considered and Gaussian noise sources, which produce a spatially correlated background noise, have been distributed. These simulations indicate that, even though the cumulant approach is capable of de-emphasizing the Gaussian noise, both bias and variance of the DoA estimates are higher than those for MUSIC. To achieve comparable results the cumulant approach requires much larger data, three to ten times that for MUSIC, depending upon the number of sources and how close they are. This is attributed to the fact that in the estimation of the cumulant, an average of a product of four random variables is needed to make an evaluation. Therefore, compared to those in the evaluation of the covariance function, there are more cross terms which do not go to zero unless the data length is very large. It is felt that these cross terms contribute to the large bias and variance observed in c-MUSIC. However, the ability to de-emphasize Gaussian noise, white or colored, is of great significance since the standard MUSIC fails when there is colored background noise. Through simulation it is shown that c-MUSIC does yield good results, but only at the cost of more data.

PACS numbers: 43.60.Gk

INTRODUCTION

The estimation of direction of arrival (DoA) forms an important problem in digital array processing literature. Schmidt I and Bienvenu 2 were pioneers in the application of the subspace method previously used by Piserenko 3 for estimation of sinusolds. Schmidt and Bienvenu's subspace method popularly known as MUSIC requires the background noise to be spatially uncorrelated or of known correlation so that it can be decorrelated before the application of the MU- SIC algorithm. This strategy is not practical since a sample of pure noise waveform just before DoA estimation may not be available; hence its correlation cannot be accurately estimated. In the context of underwater application, the (acoustic) sea noise is known to be correlated and highly variable. Higher-order statistics in particular bispectrum and trispec- trum may be used to discriminate Gaussian noise (white or colored) in favor of a non-Gaussian signal, for example, a sum of random sinusolds.4 A cumulant (fourth-order) matrix has been used for estimation of frequencies of sinusolds 5'6 and DoA of plane wavefronts on a linear array. 7 It has been clearly demonstrated that when the background noise is colored, the MUSIC algorithm based on higher-order statistics (cumulant matrix) succeeds, but MUSIC, based on a second- order statistic (covariance matrix), fails. 8 What is not widely discussed is the need for a large amount of data, particularly at low SNR commonly encountered in underwater signal processing problems, so that the discrimination of Gaussian colored noise by cumulant based MUSIC is effective.

The present work is directed to achieve this objective. By means of computer simulations and intuitive reasoning

we demonstrate that the cumulant MUSIC or c-MUSIC re-

quires, to achieve comparable results, two to twenty times more data than that needed by the covariance-based MUSIC. The need for large data arises because, in the estimation of the cumulant matrix many cross terms do not vanish unless the data length is large. We assume a physical model for the background noise, namely, a collection of point noise sources distributed over an angular sector which generates a spatially correlated noise field. 9 The signal source which radiates a sum of random sinusolds is embedded in a cloud of Gaussian

noise sources (see Fig. 3).

I. CUMULANT MUSIC (c-MUSIC)

Consider a linear equispaced array of P-omnidirectional sensors and M-uncorrelated narrow-band sources with center

frequency co o and bandwidth Aco. The sources are in the far- field region and hence the wavefront at the array is assumed to be linear. The waveform received at the pth sensor will be given by

M-1

Xp(t) = • Sm(t)e j•øøprrn q- rip(t), p = O, 1,...,P- 1, rn----0

(1)

where r m = d sin(0m)/C, d is sensor interval, Om is the direction of arrival of the mth source, c is the speed of propaga- tion, rip(t) is the additive background Gaussian noise which may be white or colored. The outputs of P sensors may be written in a compact vector form:

X(t) = AS(t) + ,•(t), (2)

2997 J. Acoust. Soc. Am. 97 (5), Pt. 1, May 1995 0001-4966/95/97(5)/2997/5/$6.00 ¸ 1995 Acoustical Society of America 2997

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 150.135.239.97 On: Fri, 19 Dec 2014 08:06:19

where

A= {a0

a m = col{ 1 ,e jøøø rm,... ,e jøøø(f- 1)rm} ' and

S(t) = col{S0(t),S•(t),...,SM_ •(t)}.

We now introduce a fourth-order cumulant defined as fol-

lows:

Cum(XoXlX2X3 ) = E {XoXlX2X3} - E {XoXl }E {X2X3 }

-E{XoX2}E{X•X3}

- E {XoX3} E {X•X2}. (3)

Note that Cum(XoX•X2X3)=0, when X 0, X•, X 2 and X 3 are Gaussian random variables. Next, we introduce a cumulant matrix C whose (p,q)th element is given by

Cv,q= Cum{XpX•X• } (4) Using (1) in (4) and the definition of cumulant, we obtain a cumulant matrix having the following form;

C=AFA", (5)

where

1'= diag{ To, T• ,-.., TM- •},

and Tm= Cum{Sm(t)Sm*(t)Sm(t)Sm*(t)}. Note that since the background noise is Gaussian, its cumulant (fourth-order) is zero regardless of whether the noise is white or colored. Equation (5) is the same (except for the missing noise term) as one encounters in MUSIC where a covariance matrix is

used. • Hence, we expect the cumulant matrix [Eq. (5)] to possess the same eigen properties as those of the covariance matrix. We have, therefore, a cumulant MUSIC or c-MUSIC.

II. WHITE GAUSSIAN NOISE

0 200 400 600 800 1000

Snapshots

(a) Mean of DoA

0.8

0.6

0.4

0.2

0.0

0 500 1000 1500 2000

Snapshots

(b) Variance of DoA

FIG. 1. The mean and variance of the DoA estimate of a source at +3 ø in

the presence of another equal-power uncorrelated source at -3 ø and white Gaussian noise, SNR= 10 dB. Average of 200 experiments: (a) mean of the estimated DoA (in degrees) and (b) variance (units: deg 2) of the estimate as a function of number of snapshots. El, MUSIC and., c-MUSIC.

When the background noise is white Gaussian, c-MUSIC differs from MUSIC with respect to the SNR, which is now infinite for c-MUSIC, as the noise term does not appear at all in Eq. (5). It is therefore tempting to con- jecture that, since in the asymptotic case the SNR is infinite, the finite data performance of c-MUSIC ought to be better than that of MUSIC. But, as shown in Figs. 1 and 2, this is far from true. c-MUSIC requires much a larger quantity, 2 to 25 times, of data to achieve a performance comparable to that of MUSIC. Previous workers 7 have made a similar ob- servation, but they did not elaborate. We provide the following intuitive argument to show why c-MUSIC requires large data. In the definition of cumulant [see Eq. (3)] there are four terms, all of which are to be estimated from finite data. Let Xi-Si + Vi, i =0,1,2,3. The expected operation in (3) will be replaced by a summation over all snapshots. The terms of the following type,

(1) Z SriSr;'SrkT]r•--4 terms, r

10 0

10-1

10-2

10-3

10-4 11

10-5 -10 0 10 20 30

SNR

FIG. 2. Variance of the DOA estimate as a function of SNR. Thick curve, c-MUSIC and thin curve, MUSIC. Number of snapshots=500 and number of experiments =200.

2998 J. Acoust. Soc. Am., Vol. 97, No. 5, Pt. 1, May 1995 P.S. Naidu and R. Subramaniyan: DOA estimation 2998


(2) • SriSr%. T]r k rlrl--6 terms, r

(3) • S ri 7'/rj 7'Irk •7rl •4 terms,

(4) • SriSr• • Srk 7'/r%--8 terms, o Array

*ZSrk * (5) • S r 7-/r j 7'/rl•8 terms,

(6) • SriSr• • 7'/r k •7r•--4 terms, ,,

(7) • Sri 7'In j • 7'In k 7']rl--8 terms,

known as cross terms, are perhaps the' most troublesome. They vanish only when the data length is infinite. Note that in the estimation of the covariance function from finite data, there are only two cross terms. As a first-order approxima- tion, let us assume that each cross term gives rise to an uncorrelated, zero mean, fixed variance random error. It is straightforward to show that the variance of the net error in the estimation of the cumulant would be roughly 20 times that in the estimation of the covariance. This ratio will also

depend upon the actual SNR. At high SNR the cross terms of types (1) and (5), that is, 12 terms are more dominant. At low SNR all terms excluding (1) and (5) will be dominant. In view of the presence of such a large number of cross terms, any analytical study of their effect is bound to be highly complex. Instead of attempting this difficult problem we have resorted to a numerical study in order to gain some insight into the nature of the estimation errors in c-MUSIC.

The variance of the DoA estimate obtained from

c-MUSIC as shown in Fig. 2 is always greater than that from MUSIC based on the covariance function, namely, 2 times, at high SNR, to 20 times, at low SNR. Hence, to achieve a comparable performance, c-MUSIC will naturally require a data length 2 (high SNR) to 20 times (low SNR) greater than that for MUSIC. The cross terms pf type (4)-(7) are contrib- uted by the covariance terms in Eq. (3). It may be noted that in the fourth-moment method, suggested in Ref. 7, since the covariance function is not subtracted, the cross terms of the types (4) to (7) are absent. As a result, the performance of c-MUSIC is worse than that of the fourth-moment method, by a factor of 2, at high SNR and by a factor of 8, at low SNR.

III. DISTRIBUTED NOISE SOURCE

The real strength of the cumulant approach probably lies in de-emphasizing the colored Gaussian noise, an environment in which the standard MUSIC is known to fail. We

have considered the following physical model for the colored Gaussian noise: a large collection of point sources distributed uniformly over an angular sector (see Fig. 3). Each point

FIG. 3. Distributed noise sources and a signal source at 3.75 ø. The noise received by the array is assumed to be Gaussian but the signal is non- Gaussian (a sum of random sinusoids).

source radiates a white Gaussian noise waveform. We shall

call such a source a distributed source. The eigen properties of the covariance matrix of the field due to such a distributed

source and received by a linear array were studied by Subbarayudu. 9 Considering the distributed sources alone with some background white noise, it is shown in Ref. 9 that the signal subspace is spanned by multiple eigenvectors depending upon the angular width of the distributed source and the array size. There is a smooth transition from signal subspace to noise subspace. For an angular width of 15 ø and an 8-element array, most of the power is contained in the first four eigenvalues. If we now apply the MUSIC algorithm, the resulting spectrum will show several peaks, depending upon the chosen dimension of the null space. For example, when the dimension of the noise space is 7, there is just one peak at the center of the distributed source, when the dimension is 6, there are two peaks and so on. There is another interesting feature, namely, that the MUSIC spectral peak asymptotically gets saturated instead of going to infinity, as would be the case had there been a point source.

The eigenstructure of the cumulant matrix, in particular, the largest eigenvalue, is of interest. Ideally, since the distributed noise sources are Gaussian, its cumulant (fourth order) is zero; hence the largest eigenvalue of the cumulant matrix ought to be zero. But, for finite data the largest eigenvalue will turn out to be finite, and it would represent the corre-

.Ol

o 200 400 600 800 lOOO

Snapshots

FIG. 4. The largest eigenvalue of the c matrix of distributed Gaussian noise source (averaged over 80 experiments). As shown, the largest eigenvalue asymptotically tends to zero.

2999 d. Acoust. Soc. Am., Vol. 97, No. 5, Pt. 1, May 1995 P.S. Naidu and R. Subramaniyan: DOA estimation 2999


2O

o

-3o -2o -lO o lO 20 3o

Angle (in degrees)

4O

ml 30

o 20

10

0 500 1000 1500 2000

Snapshots

FIG. 5. Distributed noise source (see Fig. 3) and a signal source at 3.75 ø. The true source position is shown by an arrow. SNR=0 dB; snapshots=25. Thick curve, c-MUSIC and thin curve, MUSIC.

lated Gaussian power that has apparently leaked into the cumulant matrix. A plot of the largest eigenvalue (averaged over 80 trials) as a function of number of snapshots is shown in Fig. 4.

Now consider a point source of sufficiently large power embedded within the distributed source. We have carried out

a numerical study of the performance of c-MUSIC. 1ø For experimental purposes we have assumed 30 point sources, each emitting a white Gaussian noise waveform. The noise sources are distributed over an angular sector of _+ 7.5 ø and a signal source is at 3.75 ø, surrounded by the noise sources (see Fig. 3). The signal power was set equal to the total noise power (SNR--0 dB). A typical c-MUSIC output is compared with a MUSIC output in Fig. 5. As expected, the DoA estimates using MUSIC were found to be highly biased, but those from c-MUSIC were found to be asymptotically unbi- ased (see Fig. 6). However, when the source is at the center of the distributed noise sources, the MUSIC and the c-MUSIC outputs are found to be identical except for the peak height. For a large number of snapshots, the MUSIC spectrum peak becomes saturated but the c-MUSIC spectrum peak goes to infinity (see Fig. 7). Indeed, a signal source in

True ,

0 200 400 600 800 1000

Snapshots

FIG. 6. Average of DOA estimate using c-MUSIC. Distributed noise sources (see Fig. 3) and a signal source at 3.75 ø. Averaged over 100 experiments. SNR=0 dB.

FIG. 7. Signal source at the center of the distributed noise sources (SNR=0 dB). The c-MUSIC displays a peak at correct position and its magnitude continuously increases with increasing number of snapshots (thick curve). On the other hand, MUSIC spectrum displays a peak of constant magnitude at the center, unaffected by the presence of signal source (thin curve).

the midst of distributed noise sources would remain undetec-

ted unless a cumulant-based approach is used. Next, we study the variance of the DoA estimate in the

presence of distributed noise sources. The signal source was assumed to be at 0 ø and all other parameters were assumed to be same as stated previously. The results are shown in Fig. 8. Interestingly, at low SNR (<0 dB) the effect of colored noise increases the variance of the DoA estimate by a factor of 4 to 8 as compared to the white noise background. At high SNR (>5 dB), however, this difference disappears and the colored noise has the effect of white noise.

IV. SUMMARY

The cumulant MUSIC (c-MUSIC), while it is able to correctly estimate the DoA of incident plane waves in the presence of correlated noise, for example, by a collection of point radiators as in the underwater acoustic environment, requires a larger amount of data compared to that required by

lO 1

lO 0

10'1

10-2

10-3

10-4 -10 0 10 20 30

SNR

FIG. 8. Variance of DOA estimate using c-MUSIC in the presence of distributed Gaussian noise sources and a signal source at 0 ø (thick curve). For comparison the results obtained in the presence of white Gaussian noise are also shown (thin curve). Number of snapshots=500 and number of experiments=200.

3000 J. Acoust. Soc. Am., Vol. 97, No. 5, Pt. 1, May 1995 •.• P.S. Naidu and R. Subramaniyan' DOA estimation 3000


MUSIC based on the second-order statistics. Poor performance of c-MUSIC is traced to the presence of a large number of cross terms. We were able to provide only a qualitative explanation in terms of the relative importance of the various cross terms.

R. O. Schmidt, "Multiple emitter localization and signal parameter estimation," Proc. Rome Air Development Center Spectrum Estimation Workshop (October 1979).

2G. Beinvenu and L. Kopp, "Principe de la goniometrie passive adaptive," Proc. GRETSI (Nice, France) (May 1979).

3V. E Pisarenko, "The retrieval of harmonics from a covariance function," R. Astronom. Soc., Geophys. J. 33, 247-266 (1973).

4p. Forster and C. L. Nikias, "Bearing estimation in the bispectrum do- main," IEEE Trans. Signal Process. 39, 1994-2003 (1991).

5 A. Swami and J. M. Mendel, "Cumulant based approach to the harmonic retrieval problem," Proc. ICASSP-88 4, 2264-2267 (1988).

6Z. Shi and F. Fairman, "A cumulant based TK method for the harmonic retrieval problem," Proc. IEE, Part E, 139, 221-225 (1992).

7 G. Scarano and G. Jacovitti, "Sources identification in unknown Gaussian colored noise with composite HNL statistics," Proc. ICASSP-91 5, 3465- 3468 (1991).

s G. Bienvenu, "Influence of the spatial coherence of the background noise on high resolution passive methods," Proc. ICASSP-79, 306-309 (1979).

9M. Subbarayudu, "Performance of the eigenvector method in presence of colored noise," Ph.D. thesis, Indian Institute of Science, Bangalore (July 1985).

løS. Raghavan, "DoA estimation in presence of distributed noise sources: a cumulant based approach," Master of Engineering dissertation, ECE Dept., Indian Institute of Science, Bangalore (1993).

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