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Spatial smoothing for localized correlated sources Its effect on different

localization methods in the nearfield

Wolfram Pannert

University of Applied Sciences Aalen, Department of Mechanical Engineering, Beethovenstr. 1, 73430 Aalen, Germany

a r t i c l e i n f o

Article history:Received 2 February 2011

Received in revised form 10 May 2011

Accepted 19 May 2011

Available online 15 June 2011

Keywords:

Spatial smoothing

Beamforming

Coherent signals

Covariance matrix

a b s t r a c t

The spatial smoothing (SS) technique has been proved to be effective in decorrelating coherent signals byrestoring the rank of the signal covariance matrix R. Averaging the covariance matrices of subarrays of

the original array, is a technique which increases the rank of the smoothed matrix RSS. Algorithms like

MUSIC or Capon, which rely on the use of the signal covariance matrix Rand fail in the case of correlated

sources, can be applied to scenarios with correlated sources after spatial smoothing.

However, SS is most practically applied to uniformly spaced arrays or to arrays which have a transla-

tional symmetry.

In addition the formulation is strictly applicable only to such farfield conditions, where the incoming

waves are plane waves and the steering vectors to the sources of the different subarrays are identical.

These conditions are not fulfilled in the nearfield.

Spatial smoothing is now applied with an acoustic camera in the nearfield and it is shown that up to

some limits this technique is applicable. Effects/limitations are studied using simulation and measure-

ments with several Beamforming algorithms (MUSIC, Capon and Orthogonal Beamforming) are carried

out.

The results demonstrate the benefits of SS even in the nearfield up to some limits, which are given

through the distance of the different subarrays in comparison to the spatial resolution of the Beamform-

ing algorithm. Especially at lower frequencies SS in connection with MUSIC- or Capon-Beamforming givebetter resolution in comparison to D + S Beamforming.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

Array processing as a field is well developed for signal process-

ing in acoustics and radar applications. It combines the signals of

the sensors in the array into a single output. A summary of stan-

dard and advanced Beamforming methods in acoustics was written

by Van Trees [1]. Much research has been focused on high-resolu-

tion Beamformers. A high-resolution Beamformer was developed

in 1969 by Capon, known as Minimum Variance Distortionless

Response (MVDR) Beamformer [2]. One of the largest problemswith high resolution methods is performance degradations with

correlated sources. Historically, the problem of correlated sources

has been studied exclusively in the farfield. Only recently the prob-

lem of correlated sources has been analyzed in the nearfield [3,4].

The original use of spatial smoothing was introduced by Evans

et al. [5]. The modification to the Beamforming problem was done

by Shan and Kailath [6]. The performance of the method for direc-

tion of arrival estimation (DOA) was also studied by Shan et al. [7].

Spatial smoothing remained the most widely used method for

separating correlated sources. Reddy et al. in 1987 studied the

performance of MVDR and how spatial smoothing improved the

method. A representation of the output power of MVDR and the

specific effects of correlation were mathematically proven [8].

Spatial smoothing was also shown to be a toeplitzization of the

covariance matrix by Takao et al. [9]. The covariance matrix

becomes more diagonal as signals are decorrelated. The work

was extended to a more general form by Tsai in 1995 [10]. Bresler

et al. [8] provided a proof of the effectiveness of spatial smoothing

through signal subspace and parameter estimation. The proof

allowed iterative methods to be used for estimating signal param-eters [11].

Another high-resolution Beamformer is the MUSIC Beamformer

[12]. The MUSIC algorithm is an example of a subspace-algorithm,

which divides the vector space of the covariance matrix in a signal-

subspace and a noise-subspace. For the MUSIC-algorithm corre-

lated sources have the effect that part of the signal subspace is

indistinguishable from the noise subspace and results in a dshift

of signal eigenvectors into the noise subspace. As a result, the

observed noise subspace is no longer orthogonal to the steering

vectors and the MUSIC algorithm fails. Similar effects make

problems with the Capon algorithm, where an inversion of the

covariance matrix is necessary. In the case of correlated sources

0003-682X/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.apacoust.2011.05.011

E-mail address: [email protected]

Applied Acoustics 72 (2011) 873883

Contents lists available at ScienceDirect

Applied Acoustics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p a c o u s t
http://dx.doi.org/10.1016/j.apacoust.2011.05.011mailto:[email protected]://dx.doi.org/10.1016/j.apacoust.2011.05.011http://www.sciencedirect.com/science/journal/0003682Xhttp://www.elsevier.com/locate/apacousthttp://www.elsevier.com/locate/apacousthttp://www.sciencedirect.com/science/journal/0003682Xhttp://dx.doi.org/10.1016/j.apacoust.2011.05.011mailto:[email protected]://dx.doi.org/10.1016/j.apacoust.2011.05.011


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the covariance matrix is bad conditioned. (For details to the Delay

and Sum Beamforming, MUSIC-Beamforming and Capon-Beam-

forming see Appendix A.)

As a third example Orthogonal Beamforming uses directly the

eigenvalues of the covariance matrix and in the sense of a principalcomponent analysis, the P largest eigenvalues can approximately

be assigned to the P sources (see Sarradj [13]). The small eigen-

values correspond to noise. If now two sources are correlated one

of the associated eigenvalues is approximately zero and the algo-

rithm fails in separating the two sources. The Beamformers consid-

ered in this article do not reconstruct the complete soundfield like

holographic methods, but they estimate the direction and the

power of point sources.

1.1. Nearfield model

The term nearfield is used in this article not in the sense of

acoustical holography, where evanescent waves play an important

role in reconstructing the soundfield. The evanescent waves decay

exponentially within several wavelengths which is the present

consideration smaller than about 0.34 m (for fP 1000 Hz). The

simulation and measurements are carried out at a distance of

1 m or 2 m away from the sources, where the evanescent waves

have decayed, but the soundfield is assumed to be well decribed

by a spherical wave.

A nearfield representation of a narrow-band signal represent-ing a point source is a spherical wave. The free-space wave s(rq)

for a point source with the power r2 located at rq can be repre-

sented by Eq. (1.1). For a uniform linear array (ULA) the distance

from the source to the sensors is modeled by rn(h, rq) as shown

in Eq. (1.2). The range from the source to one of the n sensors, given

by rn, is determined by the distance from the source to the phase

center of the array rq, and the distance of the sensor from the phase

center of the array xn = n d where d is the distance between two

sensors. A single range bin, which is a circle from the phase center

of the array, can be selected by holding rq constant and varying h

(see Fig. 1.1).

snrq rejkrn

rn1:1

rn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

rq cosh2 rq sinh n d

2q

1:2

For farfield conditions (rq?1), the spherical wave becomes a

plane wave and the angle h for a single source is the same for every

sensor, whereas it varies in the nearfield.

The signal for a source located at (rq, h) which impinges on the

array with M microphones, can be represented by the vector s.

s s1rq; s2rq; . . . ; sMrqT

1:3

where si(rq) is the complex amplitude, at a given frequency, of signal

of the ith microphone.

1.2. Conventional Beamforming

The simplest form of Beamforming is the conventional Delay-and-Sum Beamforming (D + S) which uses the time delay between

the sensor signals for an assumed source position and sums the

channels up after correcting for the time delay. The time delays

are contained in a so call steering vector v. To achieve unity gain,

the steering vectors v for a steering location rs are scaled by the

number of sensors M.

v s1rs; s2rs; . . . ; sMrsT

1:4

The array-pattern is now given through the scalar-product between

the array-vector s and the steering-vector v.

Bh 1

Mv

H s 1:5

Fig. 1.1. Nearfield geometry: ~rq = vector from origin to source, ~rn = vector from

sensor to source, h = angle of source, DF = distance to focus-plane xn = n d position

of the microfones, d = distance of the sensors.

Fig. 1.2. Forming the smoothed covariance matrix RSS.

Fig. 2.1. Subarrays in the case of an ULA.

874 W. Pannert / Applied Acoustics 72 (2011) 873883
http://-/?-http://-/?-


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where the superscript H means the hermitian-conjugate of a vector

or matrix.

The array pattern is the output of a Beamformer with fixed

steering position rs, while varying the source location rq. This de-

scribes how the output is affected by signals from different loca-

tions. A null in a direction means that the power is low for

signals coming from that direction. A peak in a direction means

the signals from that direction are passed with up to unity gain.

Delay-and-Sum Beamformers have 13 dB sidelobes, if equal

weight is given to the signals from all microphones.

Fig. 2.2. Two sources in the farfield with/without spatial smoothing: f= 2000 Hz

(frequency of the sources), theta1 = +10 (angle of the 1st source), theta2 = 10(angle of the 2nd source).

Fig. 2.3. One source in the nearfield with/without spatial smoothing: f= 4000 Hz

(frequency of the source), DF = 1.0 m (distance array-source), d = 0.05 (sensorspac-ing), L = M d (size of the ULA).

W. Pannert/ Applied Acoustics 72 (2011) 873883 875


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1.3. Covariance matrix

High resolution Beamformers use the covariance matrix R

(R= s sH) of the sensor signals, since it contains all the data

needed for each source: power, direction and correlation. In order

to understand the effects of correlated signals, a covariance matrix

model for a two-source-situation is used. The two-signal model is

specified by Eqs. (1.6)(1.8), where v1 and v2 refer to the steeringvector to the 1st and 2nd source. P is the correlation matrix of

the two sources, r2i (i = 1, 2) is the power of the signals and r2n

the power of uncorrelated noise. The statistical correlation be-

tween two signals is p and increases the off-diagonal elements. p

has a range from 0 (no correlation) to 1 (perfect correlation).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-25

-20

-15

-10

-5

0

x[m]

dB

Delay+Sum - and Capon- Beamforming

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-25

-20

-15

-10

-5

0Delay+Sum - and Music- Beamforming

x[m]

dB

Fig. 2.4. Two sources in the nearfield with/without spatial smoothing: f= 2500 Hz

(frequency of the sources), X1 = 0.3 m (position 1st source), X2 = 0.3 m (position2nd source), DF = 1.0 m (distance array-source), L = 0.8 m (length of the array).

Fig. 2.5. Two sources in the nearfield with/without spatial smoothing: f= 1000 Hz

(frequency of the sources), X1 = 0.3 m (position 1st source), X2 = 0.3 m (position

2nd source), DF = 1.0 m (distance array-source), L = 0.8 m (length of the array).

Fig. 3.1. Two-dim. subarrays in a 7 5 regular array; red (full line) = position of the

central array. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)



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The eigenvectors of covariance matrix R can now be

investigated.

The eigenvectors which belong to the large eigenvalues of R

(signal subspace) span the same subspace as the steering vectors

pointing to the targets and the magnitude of the eigenvalues corre-

spond approximately to the power of the signals.

A v1 v2 1:6

P r21 r1r2p

r1r2p r22

" #1:7

R APAH r2nI 1:8

The correlation p from Eq. (1.7) causes many problems for Beam-

formers based on covariance matrix inversion. In the uncorrelated

case, the number of sources is the same as the number of dominant

eigenvalues of the covariance matrix. Johnson and Dudgeon [14, p.

386] show that two perfectly correlated signals with |p| = 1 and the

same power r2, result in a Rof the form shown in Eq. (1.9). The re-

sult are M 1 eigenvectors with eigenvalues ofr2n due to noise and

a single eigenvector with an eigenvalue ofMr

2 + r2n

due to signal

plus noise.

R r2nI r2v1 v2v1 v2

H 1:9

The two-dimensional signal subspace of two uncorrelated signals

has collapsed to an one-dimensional space, where the two corre-

lated signals are mixed into an indistinguishable signal.

Fig. 3.2. Capon-Beamforming: f= 25892690 Hz (frequency range), DF = 2 m (distance array-source), distance between speakers = 0.4 m.



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The solution for this problem in the far-field case is spatial

smoothing unless a better method is found. For an ULA it averages

along the diagonal of the covariance matrix as shown in Fig. 1.2.

For a two-dimensional regular array it is a little more complicated.

The averaging process assumes a plane-wave because only now a

source signal will have the same direction of arrival for any section

of the array. However this fact is not true in the nearfield.

Looking to the mechanism how spatial smoothing works, theaveraging process reduces the correlation by combining spatially

separated parts of the array (subarrays). The correlation of the

sources has different effects on different subarrays. This can be

seen in the terms proportional to p in the expression for the nth

and mth sensor in Rn,m in Eq. (1.10) which describes the two-signal

model used in Eq. (1.9). Note that rn(hj) represents the distance

from the nth sensor to the source located at an angle hj, rm(hj)

the same for the mth sensor. Under farfield conditions the first

two terms depend only on the difference (n m) in an ULA and

so the corresponding elements in the covariance matrices Ri of

the subarrays are the same. The terms proportional to p vary and

are reduced by averaging.

Rn;m r21e

jkrnh1rmh1 r22ejkrnh2rmh2 p

r1r2ejkrnh2rmh1 pr1r2e

jkrnh1rmh2 1:10

In the nearfield the first two terms will also vary from subarray to

subarray. This results in a widening of sources the amount of

spread depends on the spatial shift of subarrays against each other.

The sources are still decorrelated since the p terms still vary more in

the nearfield.

A further effect of spatial smoothing is a loss of spatial resolu-

tion through reduction of aperture size due to the forming of smal-

ler subarrays out of the original array.

2. One-dimensional spatial smoothing

In this part of the work the effect of spatial smoothing for an

ULA under farfield conditions and nearfield conditions is studiedwith simulations. The set-up was an array with M equidistant

sensors and d = 0.05 m spacing. Three subarrays were built and

averaged (Fig. 2.1). The 2nd array is the central array which is used

for focusing and the 1st and the 3rd array is shifted for 0.05 m

compared to the central array. The simulated data had a SNR of

40 dB.

Fig. 2.2 presents the effect of spatial smoothing on MUSIC-,

Capon-, and D + S Beamforming with an ULA (L = 0.8 m) and two

correlated sources with same amplitude and position at 10

inthe farfield. The spatial smoothing has decorrelated the two

sources which has also an effect on the D + S Beamforming. The

number of dominant eigenvalues of the correlation matrix R has

changed from one to two. Without spatial smoothing there is only

one virtual source in the middle between the real sources.

The next series in Fig. 2.3 show the result for Capon and D + S-

Beamforming for one source under nearfield conditions at differ-

ent values of the array length L. As the spatial resolution DX

becomes better (smaller DX) with increasing size of the array

one can notice the effect, that the three subarrays see the

source under different angles. So the averaged covariance matrix

RSS contain three sources. From the result it is clear, that there is

only a nearfield-effect if the spatial-resolution DX is smaller than

the shift of the subarrays in comparison to the central array,

which is used for focusing. Similar results are known for

Synthetic Apertur Radar (SAR), where range migration effects

need not to be compensated, when they are less than the size

of the range-resolution.

In Fig. 2.4a similar scenario is studied now with two sources in

the nearfield. The results in the nearfield show that spatial smooth-

ing has not the potential as in the farfield situation nevertheless,

the dynamic is improved for both algorithms MUSIC and Capon-

Beamforming as compared to the situation without SS. D + S Beam-

forming is nearly not influenced by SS.

Further simulations are carried out for different frequencies,

source separations and correlation coefficients. The results show

that especially in situation, where the resolution of D + S

Beamforming, Capon- and MUSIC-Beamforming without SS is

insufficient to separate the sources, the application of SS improvesthe effectiveness of Capon- and MUSIC-Beamforming.

Fig. 3.3. Capon-Beamforming with two uncorrelated sources; same parameters as in Fig. 3.2.



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OneexampleisgiveninFig.2.5. Capon- andMUSIC-Beamforming

with SS are able to separate two sources where D + S Beamforming

and Capon- and MUSIC-Beamforming without SS clearly fails.

2.1. Summary

Especially at low frequencies where the spatial-resolution is

poor, Capon and MUSIC Beamforming show their potential of sep-arating two sources after application of SS. The resolution is even

better as with the D + S Beamforming.

So one can draw the conclusion that mainly at low frequencies,

one can apply spatial smoothing without having the effect that a

source is split up into several parts.

3. Two-dimensional formulation and experimental setup

In the two-dimensional case the array must have a translational

symmetry like for example the regular 7 5 array shown in

Fig. 3.1. The sensors are clustered into overlapping 6 4 subarrays.

One possibility with four subarrays is shown in Fig. 3.1. Processing

a scenario with the smoothed covariance matrix, RSS is calculated

with a central 6 4 array (red in Fig. 3.1), which only existsvirtually.

For the experiments a commercial system was used (CAE

Noise Inspector) with a 8 6 regular array with 0.1 m distance

between the sensors. The system has the benefit, that it offers

an interface for user-defined algorithms in Matlab-Code. From

Fig. 3.4. Music-Beamforming: f= 25892690 Hz (frequency range), DF = 2 m (distance array-source), distance between speakers = 0.4 m.



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the simulation results in the last section one can deduce, that the

shift of the different subarrays compared to the central array

(which is virtual and only used for focusing) should not be great-

er than the spatial resolution of the high-resolution-algorithm.

This is a limitation for the number of subarrays and therefore

like in Fig. 3.1 four subarrays are used. So the shift of a source

seen from a subarray compared to the central array is 0.05 m

and one should expect only a splitting of sources, if the spatial-resolution is better than 0.05 m.

3.1. Experimental results

The following series in Figs. 3.23.6 show the results for two

loudspeaker signals and different Beamforming algorithms. In Figs.

3.2 and 3.4 the signals are perfectly correlated (white noise mono-

signal), whereas in Figs. 3.3 and 3.5 uncorrelated signals are used

for comparison. The potential of spatial smoothing is tested with

three different Beamforming algorithms high resolution Capon-

Beamforming, high resolution Music-Beamforming and Orthogonal

Beamforming which can display different uncorrelated sources

separately.

The results are displayed with 6 dB dynamic. The bandwidth ofapproximately 100 Hz contains the summed results of a narrow-

band FFT-processing.

In Fig. 3.2 the result for Capon Beamforming is presented and

the effect of spatial smoothing is compared.

In Fig. 3.3 for comparison there is the result for Capon-Beam-

forming applied to a similar scenario with two completely uncor-

related sources. From the result it is clear, that spatial smoothing

in the nearfield does not work perfectly, but the potential of resolv-

ing correlated sources is given.

Next there is the MUSIC-algorithm applied to the same

scenario.

In Fig. 3.5 for comparison there is the result for MUSIC-Beam-

forming applied to a similar scenario with two completely uncor-

related sources as in Fig. 3.3.Fig. 3.6 shows the result for Orthogonal Beamforming. Orthog-

onal Beamforming is able to separate uncorrelated sources approx-

imately. Without spatial smoothing the scenario is resolved in two

peaks but they belong to one eigenvalue of the cross spectral ma-

trix. The following pictures show the result after application of spa-

tial smoothing. The signal of the two speakers has been

decorrelated and Orthogonal Beamforming can separate the sce-

nario in source 1 and source 2. The number of dominant eigen-

values in the cross spectral matrix has changed from one to two.

In Fig. 3.7 for comparison there is the result for OrthogonalBeamforming applied to a similar scenario with two completely

uncorrelated sources as in Fig. 3.6.

4. Conclusion

The spatial smoothing technique allows the use of Beamforming

algorithm which usually are applicable only for scenarios with dec-

orrelated sources. In the present investigation three Beamforming

algorithms are compared Capon, MUSIC, and Orthogonal Beam-

forming. Whereas spatial smoothing in the farfield with plane

waves is known to work properly, problems come up in the near-

field, because sources appear under different look-angles which

correspond to the different subarrays. With the simulation andmeasurements its shown that under certain conditions this tech-

nique is also applicable in the nearfield. The conditions are the

distance of the centre of the subarrays from the centre of the cen-

tral array (which is used for focusing) should be less than the

spatial resolution of the Beamforming algorithm. Otherwise single

point sources are split up. The experimental results show that the

potential of the different algorithms is worse compared to the ideal

case of not correlated sources, but there is great improvement in

resolution compared to the processing without the spatial smooth-

ing technique. Especially at low frequencies, Capon and Music

Beamforming provide better resolution than standard D + S Beam-

forming. Whereas SS has less influence on D + S Beamforming, the

is a great benefit of SS for the high resolution algorithms Capon-

and MUSIC-Beamforming. In the case of Orthogonal BeamformingSS also shows its potential to decorrelate two previously correlated

sources.

Fig. 3.5. Music-Beamforming with two uncorrelated sources; same parameters as in Fig. 3.4.



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Fig. 3.6. Orthogonal Beamforming: Df= 1/12 octave at 3140 Hz (frequency range), DF = 2 m (distance array-source), distance between speakers = 0.4 m.



10/11

Appendix A. Appendix

Expressions for the Beampattern with different Beamforming

algorithms.

~Vhi is the steering vector to a location hi.

R is the M M cross spectral matrix of the M microphone

signals.

The superscript H means the hermitian conjugation (trans-

pose and complex conjugation).

A.1. Delay and sum Beamforming

BDShi ~VHhi R ~Vhi

A.2. Capon Beamforming

Capon Beamforming tries to minimize the influence of sources

of interference. Ideally it results in a very sharp Beampattern (seeRef. [2]).

A variational approach leads to:

BCaponhi 1

~VHhi R1 ~Vhi

The cross-spectral matrix R is bad conditioned in the case of

correlated sources and a regularization is necessary (diagonalloading).

A.3. MUSIC-Beamforming

The MUSIC (Multiple Signal Classification) algorithm belongs to

the class of subspace algorithms (see Ref. [12]).

The term subspace refers to the space of eigenvectors of the

cross-spectral matrix R.

In the presence of Psources R has P large eigenvalues and M-P

small eigenvalues which belong to the noise in the system.

The related eigenvectors ~USk and~UNk build the signal-subspace

and the noise-subspace.

The separation between the two spaces occurs through compar-ison with a threshold.

Fig. 3.7. Orthogonal Beamforming with two uncorrelated sources; same parameters as in Fig. 3.6.



11/11

Further on one can show that the steering vectors~Vhi pointing

to the P sources in the scenario, are orthogonal to the noise eigen-

vectors ~UNk with k = P+ 1. . .M.

~VHhi ~UNk 0 for i 1 . . .P and k P 1 . . .M

With this feature one can form a pseudo-Beampattern which has

very distinct maxima in the direction of the P sources (hm = hi)

BMUSIChm 1j~VHhm ~U

Nk

j2

The height of the maxima has no relation to the signal power of the

corresponding source.

References

[1] Van Trees HL. Optimum array processing. Wiley Interscience. ISBN 0-471-09390-4.

[2] Capon J. High-resolution frequency-wavenumber spectrum analysis. In: ProcIEEE, vol. 57; August 1969. p. 140818.

[3] Agrawal M, Abrahamsson R, Ahgren P. Optimum Beamforming for a nearfieldsource in signal-correlated interferences. Signal Process 2006;86(5):91523.

[4] Lee Ju-Hong, Chen Yih-Min, Yeh Chien-Chung. A covariance approximationmethod for near-field direction-finding using a uniform linear array. IEEETrans Signal Process 1995;43(5):12938.

[5] Evans JE, Johnson JR, Sun DF. Application of advanced signal processingtechniques to angle of arrival estimation in ATC navigation and surveillancesystems technical report. M.I.T. Lincoln Laboratory, Lexington, Massacusetts;

June 1982.[6] Shan Tie-Jun, Kailath T. Adaptive Beamforming for coherent signals and

interference. IEEE Trans Acoust Speech Signal Process 1985;33(3):52736.[7] Shan Tie-Jun, Wax M, Kailath T. On spatial smoothing for direction-of-arrival

estimation of coherent signals. IEEE Trans Acoust Speech Signal Process1985;33(4):80611.

[8] Reddy V, Paulraj A, Kailath T. Performance analysis of the optimum

Beamformer in the presence of correlated sources and its behavior underspatial smoothing. IEEE Trans Acoust Speech Signal Process 1987;35(7):92736.

[9] Takao K, Kikuma N, Yano T. Toeplitzization of correlation matrix in multipathenvironment, vol. 11; April 1986. p. 18736.

[10] Bresler Y, Reddy VU, Kailath T. Optimum Beamforming for coherent signal andinterferences. IEEE Trans Acoust Speech Signal Process 1988;36(6):83343.

[11] Tsai Churng-Jou, Yang Jar-Ferr, Shiu Tsung-Hau. Performance analyses ofBeamformers using effective SINR on array parameters. IEEE Trans SignalProcess 1995;43(1):3003.

[12] Schmidt RO. Multiple emitter location and signal parameter estimation. In:Proc RADC spectrum estimation workshop. Griffiths AFB, Rome, New York;1979. p. 24358.

[13] Sarradj E. A fast signal subspace approach for the determination of absolutelevels from phased microphone array measurements. J Sound Vibr2010;329:155369.

[14] Johnson Don H, Dudgeon Dan E. Array signal processing: concepts andtechniques. Englewood Cliffs (NJ): PTR Prentice Hall; 1993.


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