- Email: [email protected]
Analysis of interpolated arrays with spatial smoothing
SIGNAL
PROCESSING Signal Processing 54 (1996) 261-272
Analysis of interpolated
arrays with spatial smoothing
K. Maheswara Reddy”, V.U. Reddyb,* “CASSA. Defence Research & Development, New Thippasandra, Bangalore 560 075. India bDepartment of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India
Received 15 January 1996; revised 24 June 1996
Abstract An interpolated array is a means of extending the rooting techniques and preprocessing schemes like spatial smoothing, which are available for uniform linear arrays, to arrays with arbitrary geometry. This paper analyses the effect of spatial smoothing on the performance of the interpolated arrays in the presence of interpolation errors and finite data perturbations. An attempt is made to bring out the capability of smoothing to reduce the effect of finite data perturbations and interpolation errors on the performance of the Root-MUSIC with interpolated arrays. Simplified expressions are obtained for certain special cases. Computer simulations are provided to demonstrate the usefulness of the analysis.
Ein interpoliertes Array ist ein Mittel zur Erweiterung von Rooting- und Vorverarbeitungstechniken wie rgumlicher Glgttung, die man fiir gleichfiirmige lineare Arrays zur Verfiigung hat, auf Arrays beliebiger Geometrie. Dieser Beitrag analysiert die Auswirkungen rtiumlicher Glgttung auf die Leistungsfihigkeit interpolierter Arrays beim Auftreten von Interpolationsfehlern und begrenzter Datenverfilschungen. Es wird versucht, die FHhigkeit einer Gllttung zur Reduktion der Auswirkung endlicher Daten- und Interpolationshfehler in die Leistungsfihigkeit eines ROOT-MUSICVerfahrens mit interpoliertem Array einzubringen. Fur einige Spezialfille ergeben sich vereinfachte Ausdriicke. Rechnersimulationen werden dargestellt, welche den Nutzen der Analyse demonstrieren.
Un rbseau interpolit est un moyen d’itendre aux rCseaux g gkometrie arbitraire les techniques d’extrdction de racines et les mitthodes de prC-traitement telles que le lissage spatial, qui sont disponsibles pour les rkseaux linkaires uniformes. Cet article analyse l’effet du lissage spatial sur les performances des rCseaux interpolCs en prksence d’erreurs d’interpolation et de perturbations de donn&es finies. Une tentative est faite pour mettre en relief la capacitC du lissage i rCduire l’effet des perturbations de donnCes finies et des erreurs d’interpolation sur les performances du Root-MUSIC avec des rCseaux interpolCs. Des expressions simplifites sont obtenues pour certains cas particuliers. Des simulations informatiques montrent I’utilitC de cette analyse. Keywords:
Interpolated
*Corresponding
arrays; Spatial smoothing
author. Tel.: 080 309 2278; fax: 080 334 1683; e-mail: reddy@,ece.iisc.ernet.in.
262
K.M. Reddy, V. U. Reddy / Signal Processing 54 (1996) 261-272
1. Introduction Subspace-based direction-of-arrival (DOA) estimation algorithms such as MUSIC [13], ESPRIT [12] and minimum norm [4] yield high resolution and asymptotically exact DOAs. The performance of these algorithms, however, degrades when the impinging sources are correlated. In particular, when two or more impinging sources are fully coherent, the algorithms break down, and in the case of finite data, the DOA estimation performance degrades even when the sources are partially correlated. Correlation between the impinging sources arises because of either multipath propagation or due to deceptive jamming. Some preprocessing techniques such as spatial smoothing [14], weighted smoothing [l l] and redundancy averaging [S] have been proposed to alleviate the ill effects of correlation. Also, the smoothing reduces the effects of finite data pertubations thereby improving the estimation accuracy [6]. But all these techniques can only be applied to uniform linear arrays. Furthermore, the rooting techniques which are shown to yield better resolution and accuracy [S] are also applicable to uniform linear arrays only. However, in most of the real world applications, the array size and shape are often not determined based on the requirements of the direction finding system; but they are determined based on either the size and shape of the platform on which the array is to be mounted or other structures and mechanical limitations of the platform. For example, the array may have to be mounted in the nose or near the wings of the aircraft. The array geometry resulting from such physical constraints will not have a uniform linear array configuration. Also, when 360” azimuth coverage is required, we prefer a circular array to a uniform linear array. Thus, the uniform linear array configuration may not be encountered in many practical situations. It is, therefore, desirable to develop techniques which extend the preprocessing and rooting methods applicable to uniform linear arrays to the arrays of other geometry. In this paper, all the arrays other than uniform linear arrays are referred to as arbitrary arrays. In [2], Friedlander has proposed interpolated arrays for extending the use of rooting techniques
and preprocessing schemes to arbitrary arrays. This method involves developing a transformation which matches manifold of the arbitrary array to that of a chosen uniform linear array (ULA), which we refer to as the virtual ULA, within the field of view of the array. If it is not possible to match the manifolds over the entire field of view, it can be divided into smaller sectors and different transformations can be chosen for different sectors. The output of the arbitrary array, when passed through these transformations, resembles the output from the virtual ULA. It is now possible to apply rooting and smoothing techniques to the transformed data. As it is not possible to exactly match the manifolds of the arbitrary array and the virtual ULA over the entire chosen sector, there will be some interpolation error. In [ 151, Weiss and Friedlander analysed the performance of spatial smoothing with interpolated arrays. They provided theoretical expression for the mean square error (MSE) in the DOA estimate. In their analysis, however, they assumed the interpolation error to be very small and consequently its effect on the performance of the algorithm was ignored. They compared the performance of the algorithm with the Cramer-Rao bound (CRB) and also with the performance of spectral MUSIC as applied to the arbitrary array output. In this paper, an attempt is made to analyse the effect of forward spatial smoothing on the DOA estimation performance of the Root-MUSIC [l] in the presence of finite data perturbations and interpolation errors for the case of both correlated and uncorrelated sources. It is shown that in addition to reducing the correlation between the sources and the finite data perturbations [lo], the spatial smoothing also reduces the effect of interpolation errors on the performance of Root-MUSIC applied to the virtual ULA output. Simplified expressions are obtained for certain special cases. The paper is organised as follows. In Section 2, we briefly introduce the interpolated arrays and the associated spatial smoothing algorithm. In Section 3, we analyse the performance of the spatial smoothing and obtain explicit expressions for some special cases. We present the results of computer simulations in Section 4. Section 5 concludes the paper.
263
KM. Reddy, V. U. Red+ / Signal Processing 54 (1996) 261-272
2. Background and problem statement Consider an arbitrary array composed of L, sensors. We assume that the elements of the array are in the xy-plane and all the elements are omnidirectional. The steering vector of such arbitrary array can be written [3] as a, (0) = [ej2Mx,sin0+ ~1~0s 01, . . , e j2a(x,.dsm
0 + !,.,cosO)
1> T
(1)
where (x,, y,,J are the coordinates of the mth sensor in wavelength units, 8 is the angle in the xy-plane with respect to y-axis and the superscript T denotes the transpose of a vector. In the interpolated array approach [2], the field of view of the arbitrary array is divided into different sectors. For each sector, we choose a virtual ULA and match the manifold of the arbitrary array with that of the chosen ULA at a predetermined set of angles using a transformation. Let L (L d L,) be the number of elements of the virtual ULA for a given sector and d be the spacing between the elements. The steering vector of the virtual ULA is given by a(@ = [l, ePj*l, . . . ,e-j”(L-l))]T,
(2)
where IX = (2nd sin 0)/n with ,J denoting the wavelength. Let Jt’ be the number of angles at which the manifolds of the arbitrary array and the virtual ULA are to be matched in a chosen sector and assume that &i, flb2, . . . , eb, are the angles. Let Aba be the matrix obtained by stacking the steering vectors of the arbitrary array along the chosen angles A,, = [a,(e,,), . . . db,
)I.
(3)
Similarly, let Ab be the matrix obtained by stacking the steering vectors of the virtual ULA along the chosen angles d46= [a(e,,), . . . ,a(eb, )I.
The approximation in (5) is because of the inconsistency of the equations which may arise in trying to match the two different manifolds at as many angles as possible. Solving (5) using least-squares technique, we get B = AbAg = AbAya (A,,Ara)- ‘,
where (.)” represents the pseudo inverse and (.)” represents the complex conjugate transpose. This transformation can now be applied to the output of the arbitrary array to obtain the data which would resemble the output from the virtual ULA. We can then apply the techniques available for the ULA to the transformed data. Assume that M narrowband sources impinge on the arbitrary array. Let us assume that the sources are in the same plane as the elements of the array and that the DOAs of these sources are 8i, e2, . . , eM. The array output vector can then be written as x,(t) = &s(t) - n(r),
(7)
where s(t) is the signal vector, n(t) is the complex circularly Gaussian-distributed noise vector which is assumed to be spatially white and uncorrelated with s(t), and A, is the matrix of direction vectors of the arbitrary array A, = [aa(
a,(e,), . . , dedi.
(8)
The array covariance matrix for the arbitrary array can be expressed as R, = E[x,(t)x;(t)]
= A,SA,H + 0’1,
(9)
where S denotes the signal covariance matrix, c2 is the variance of the additive noise, I is an identity matrix and E(.) denotes the expectation operator. Applying the transformation B to the covariance matrix R,, we obtain the transformed covariance matrix as R, = BR,B”
(4)
(6)
= BA,SA;B”
-I- a2BBH.
(10)
Then, the desired transformation B which matches the manifold of the arbitrary array to that of the virtual ULA is given by
Noting that the matrix B is chosen for matching of the manifold of the arbitrary array to that of the virtual ULA, we can write the transformed matrix Rb as
Ab ‘v BA,,,.
R,, 2: ASAH + 02BBH,
(5)
(11)
264
KM
Reddy, V. U. Reddy / Signal Processing
where A is the matrix of direction vectors for the virtual ULA, ‘4 = [a(&), ... ,4&f)l.
(12)
Note that the approximation in (11) appears because of interpolation errors. We can now apply the prepocessing and rooting techniques to the matrix Rtl. In spatial smoothing, we divide the ULA into K overlapping subarrays. In the present case of interpolated arrays, the ULA is a virtual ULA and hence the subarrays are also virtual. The spatially smoothed covariance matrix, denoted by Rbs, is the average of the virtual subarray covariances and can be obtained from the covariance matrix Rb by computing (for forward smoothing) (13) where Z, = [er, el+l, . . . , ef +LO_r] with el denoting the Ith column of the identity matrix of size L and L,, (L, > M) is the size of the virtual subarray. Note from (11) and (13) that the noise part of the covariance matrix Rbs is not in the form of aZZ.We, therefore, apply prewhitening to Rbs. The prewhitening matrix is given by
-l/2 1
(14)
The smoothed covariance matrix obtained after prewhitening can be expressed as R,, = Z&.&Z& = Rnw;
i Z;BA,SA;BHZ,Rfw z-1
+ 02Z.
(15)
In MUSIC [ 131; we perform the eigen decomposition of the covariance matrix R,,. The minimum eigenvalue of R,, is a2 and its multiplicity is L, - M. The eigenvectors corresponding to the minimum eigenvalue are called noise eigenvectors and they are orthogonal to the matrix R,,A, where A, is the direction matrix of the virtual subarray. The MUSIC exploits this property and estimates the DOAs of the sources by searching for the nulls (minima in the present case) of the spectrum D,(O) = u:!(e)R,H,E,E,HR,,v,(e),
(16)
54 (1996) 261-272
where E, is the noise subspace of R,, and u,(0) is the normalized steering vector for the subarray of the virtual ULA. Because of the structure of uf(0), D,(8) given in (16) can be written as a polynomial. We can thus apply Root-MUSIC to R,,. The paper analyses the performance of the RootMUSIC with smoothing applied to the transformed data in the presence of correlated sources, finite data perturbations and interpolation errors.
3. Performance
of the Root-MUSIC for interpolated arrays
smoothing
with
The transformation B matches, in the leastsquares sense, the manifold of the arbitrary array to that of the virtual ULA in a given sector. Consequently, there exists some deviation between the transformed arbitrary array manifold and the virtual ULA manifold. This deviation, referred to as the interpolation error, is given by Aa (e)
= a(e) - Baa(e).
(17)
Because of this interpolation error, the transformed arbitrary array direction vector matrix deviates from that of the virtual ULA. This deviation can be expressed as AA=A-BA,.
(18)
As a result, the noise subspace of the covariance matrix R,, is not perfectly orthogonal to the direction vectors of the virtual ULA and this introduces errors in the DOA estimates obtained by the RootMUSIC algorithm. Substituting for BA, from (18) in (15), we get R,, = Rnwf i Z; [ASAH + AASAAH l-1 - ASAAH]Z,Rfw
- AASAH
+ a2Z.
(19)
Eq. (19) can be broken into R,, = R, + AR,,,
(20)
where R, = i
i
R,,ZFASAHZIRfw
+ a2Z
I-1 = R,,AfSfA~R~w
+ a2Z
(21)
KM.
AR,,f = - 4 i R,,ZT I-1
Reddy, V.U. Red& J Signal Processing
[AASAH
+ ASAAH] Z,R;‘,.
(22)
Note that RF represents the smoothed covariance matrix of the virtual ULA (computed directly from the ULA configuration) with prewhitening. In expressing ARbf by (22) we implicitly assumed that the interpolation errors are small and neglected the terms containing more than one AA. In (21), S, and A, represent the smoothed signal covariance matrix and the direction matrix for the subarray of the virtual ULA, respectively. In practice, we estimate the covariance matrix R,, from finite number of snapshots. This also introduces certain errors in the covariance matrix. The finite data covariance matrix can be expressed as ffbs=~hf+hRF=~ft~F+~bS,
265
54 (1996) 261-272
the covariance matrix R,, ufl (0) is the derivative of uf (0) w.r.t. 0 and Re(.) represents the real part of (.). The above equations hold when the SNR or the number of snapshots is relatively high, and when the interpolation errors are small. The first and the second terms in (24) are due to finite data perturbations and the third term is due to interpolation errors. Thus, the overall MSE in the DOA estimate is a simple addition of two parts, one due to finite data perturbations and the other due to interpolation errors. As the number of snapshots tends to infinity, we get the MSE due to interpolation. The MSE in the DOA estimate due to interpolation errors only is deterministic and is given by
CRe(~“A% B)l"
E(AeZ) =“’ = [UT, ((ji)Rfwpn~n,ufl
((j,)]” ’
(29)
3.1. MSE due to jnite data perturbations
(23)
where ARr represents the perturbation due to finite data. Note that the perturbation due to finite data is random while the deviation in the covariance matrix due to interpolation errors, AR,,, is deterministic. Following the steps given in [9], we can obtain the mean square error (MSE) in ith DOA estimate due to finite data perturbations and interpolation errors as E [A($]
If the interpolation (24) can be simplified
errors as
are very small,
then
Let us consider the case with no smoothing, i.e., K = 1. The expression (30) can then be simplified as E(A0;) CT’ [(S-‘)ii NL
+ aZ(S-‘(AHR~~R~,A)-‘S-‘)ii] 2$(~JR!P~R,wU,
(oil
-’ (31)
(27) H
H
Rs, = R .wA,S,A,R nw,
(28)
whereR,, = Ryp = Eb,(t)y,H(t)],y,(t) is the output vector of the pth virtual subarray obtained after prewhitening. N is the number of snapshots, P, is the projection matrix on to the noise subspace of
where u1 (0) is the first derivative of u(0) w.r.t. f1,u(e) is the normalized steering vector of the virtual ULA and (.)ii represents the ith diagonal element of (.). Note from (31) that the denominator contains L and the terms (U’:(ei)R~~P,R”,U,(Bi)) and (AHR:wR,,,A) are also dependent on L, the size of the virtual ULA. Thus, as L decreases, the MSE performance of the Root-MUSIC deteriorates. Further, the value of L cannot be more than L,, the number of elements in the arbitrary array, as this choice makes the covariance matrix R,, to be singular. Hence, care must be exercised while selecting the number of elements in the virtual ULA.
266
KM
Reddy, V.U. Reddy 1 Signal Processing 54 (1996) 261-272
Let us consider the case with smoothing, i.e., K > 1. From (30), it can be shown that (see [lo] for the steps involved) E[A@] d
where Y = diag[e-j”l, . . . ,e-joM] and (.)* denotes complex conjugate of (.). Combining (35) and (34) and noting that S, !I’ are diagonal matrices, we get aHAR& = - f CX~R,,~: ZTAA !l’*(‘- ” I=1
G2 [ST ‘)ii + G’(Sy 1(A~RF;‘wR”wAf)- ‘Sy l)ii] NLo
x
2vfH1(ei)R~~wPrPnwUf1(ei)
(32) We note that the equality in (32) holds if the number of subarrays is one, i.e., when there is no smoothing. We also note that by replacing the source covariance matrix S by the smoothed source covariance matrix S, (which accounts for the reduction in the source correlations due to smoothing), L by Lo, A by A, and v1 (QJ by ufl (0,) (which account for the reduction in the array size because of smoothing) in (31), we obtain the RHS of (32). Thus, if the spatial smoothing were to reduce only the source correlations, then the MSE with smoothing should have been given by the RHS of (32). But, as seen from (32), the MSE can be smaller. This suggests that the smoothing also reduces the noise perturbations due to finite data, in addition to reducing the correlation among the impinging signals.
SAY RfWR$uf(Oi).
(36)
From (28), R$ can be written as R$ = &&(A~Rj;lyR,J/)-l xS;l
(A~RfWRwR.,AJIA~RH IlW.
(37)
Substituting this in (36) and noting that S = S, for the uncorrelated sources, we can simplify (36) to get 1 ff%Rbf/ZI= - K&O
ctHRnw,il #-Aa
ej”i’z- ‘), (38)
where Lo is the size of the virtual subarray and is given by Lo = L - K + 1. Let Aa, be the effective interpolation error vector (with smoothing) in the direction ei, Aar(Bi) = f i ZTAa(Bi)ej@ic’- ‘).
(39)
I-1
Eq. (38) can now be written as 3.2. Asymptotic performance uncorrelated
for the case of
CruAR&? = - &
sources
In this section, we try to bring out the asymptotic performance of the Root-MUSIC with smoothing for the interpolated arrays when the sources are uncorrelated (S and S, are diagonal). Substituting for fi from (27) in the numerator of (29), we obtain mHARbf/I= u~AR~,-R,#~R,,,v~(~~).
(33)
Substituting for AR,, from (22) and noting that P, is orthogonal to R,,A,, we get “HA%fP 1 = - K CX~R,,~5 Z~AASAHZIRfWR,#fR,,vf(Oi). I=1
(34)
From the definition of A and Z,, we can express AHZl = y*“- “&!,
(35)
uHR,,,,,Aaf(Bi).
(40)
Substituting (40) in (29), we get the expression for the asymptotic performance of the Root-MUSIC with spatial smoothing in the presence of uncorelated sources as A@
=’
CRe(~HR,,A~,#J)12
LO u~l(ei)R~~,P,Rnwvr1(ei)12'
(41)
Note from (41) that the MSE in the ith DOA estimate (which is square of the error in the present case) is dependent only on the effective interpolation error in the direction of the ith source. In Appendix A we have shown that the norm of the effective interpolation error vector reduces with smoothing. Hence, we can expect the smoothing to improve the asymptotic performance of the RootMUSIC by reducing the effect of interpolation errors.
KM
3.3. Asymptotic performance correlated sources
Reddy. V. U Reddv 1 Signal Processing 54 (1996) 261-272
4. Discussion of simulation results
for the case of
We now attempt to arrive at the asymptotic performance of the Root-MUSIC with smoothing when the sources are correlated. Following the steps of the previous section, we can show that c?‘AR,/j = - &
AfHIR,,P,,R,,, 0!
5 .ZTAASY*“-“Sj’
,
I=1
(42) where A,,
is given by (43)
If we consider the case of no smoothing, K = 1, then (42) can be simplified as
i.e.,
(x~AR~SB = - 1 JL
(44)
CIHR,wAa(
267
Substituting (44) in (29), we obtain the asymptotic MSE in the presence of correlated sources which is same as (41) with K = 1. When K > 1, simplification of (42) leads to unwieldy expressions even for a two-source case and hence is not presented here. However, using simulations, we show that the asymptotic performance improves with smoothing for correlated sources also. We conclude this section with the following remark. While designing the transformation matrix B, it is possible to incorporate the location inaccuracies of the array elements and the differences among these elements. Whenever the calibration data is available, the data can be used while constructing the matrix Aha given in (3). The resulting transformation B and the smoothing then reduce the effects of location inaccuracies and the use of non-identical elements on the performance of the algorithm. Noting that the Root-MUSIC and spatial smoothing (in their original forms) can only be applied to linear arrays with identical elements, the interpolated arrays can be viewed as a way to extend the use of rooting and preprocessing techniques to non-ideal arrays also. This appears to be one of the major advantages of using the interpolated arrays.
We conducted simulations to test the usefulness of the analysis of the previous section. In our simulations, the arbitrary array was taken as a semicircular array. The semicircular array was assumed to lie in the xy-plane with the first element at the origin and the last element on the x-axis at a distance equal to the diameter of the array. The rest of the elements were placed at equal distance along the semi-circle. The semicircle is towards the positive jj-axis. The virtual ULA was chosen to be along the x-axis such that the positions of the end elements of the ULA match with those of the semicircular array. The angle % was measured w.r.t. the y-axis. In the simulations and numerical evaluation, a semicircular array consisting of 25 isotropic elements with half wavelength inter-element spacing (straight line distance between the elements) was assumed. The number of snapshots was fixed at 100. We considered two-source equi-power scenarios in our simulations. We generated the snapshot data with the noise vector n(t) consisting of zero mean, unit variance independent complex circularly Gaussian random variables and the signal vector consisting of zero mean complex circularly Gaussian random variables whose variances were fixed to give the desired signal powers. In the case of uncorrelated signals, the random variables comprising the signal vector were generated independently. In the case of correlated signals. the second signal was generated as SZ(L)= Psi(t) + JY&(t),
(45)
where sl(t) is the signal from the first source with variance 0,’ and sj(t) is a zero mean, complex circularly Gaussian distributed random variable with variance a:. The signal s3(t) is uncorrelated with the signal s1 (t). The snapshots were made independently by changing the seeds randomly from snapshot to snapshot. The simulation results were obtained by averaging over 100 independent Monte Carlo runs. We may also point out that in any Monte Carlo run if the DOA estimation error is more than l”, then we treated that run as the one in which the algorithm failed to estimate the DOA.
KM.
268
Reddy, V.U. Reddy 1 Signal Processing
54 (1996) 261-272
sector. This sector was selected from - 60” to 60” and JV was chosen as 50. Fig. 1 shows the norm of the effective interpolation error vector (defined in (39)) as a function of the angle in the chosen sector for K = 1 and K = 4. It is seen from this figure that the effective interpolation error vector norm reduces as the number of subarrays increases. This reduction in the interpolation errors improves the performance of the Root-MUSIC especially when the errors are not very small. Table 1 shows the asymptotic performance of the Root-MUSIC with smoothing applied to transformed data as a function of the number of subarrays in the presence of uncorrelated sources. The DOAs of the sources (55’ and 60”) were chosen such that the interpolation errors are maximum (see Fig. 1). From the manifold of the semicircular array and the transformation B, R,, was computed and the Root-MUSIC with smoothing was applied to this matrix to determine the asymptotic performance. We refer to this result as the one obtained from the algorithm in our Tables 1 and 2. The theoretical values of the MSE (which is square of the error in the present case) were obtained by numerically evaluating (41). Note from the results
Thus, the simulation results presented here also contain the number of times the algorithm failed to estimate the DOAs. The search for the minima in the case of spectral MUSIC was conducted in steps of 0.005”. For all simulations in this section, SNR referes to the signal to noise ratio without the array gain. The particulars of the signal scenario, number of subarrays and the number of snapshots used are given in the captions. For obtaining the transformation B, the angles at which the manifold of the arbiteb,, eb2, . . . , t!9b.y., rary array is to be matched with that of the virtual ULA, can be placed uniformly over the sector or can be chosen using the Chebyshev distribution. When the angles are uniformly placed, the norm of the interpolation error vector keeps increasing as we move towards the ends of the sector. If they are chosen using the Chebyshev distribution [7], the norm remains within a given limit throughout the sector. The number of angles chosen for matching (J+‘) in the sector must always be greater than or equal to the number of elements in the arbitrary array to obtain a transformation B which gives least interpolation error. In the simulations, the angles were placed uniformly over the sector and the sources were assumed to belong to that single
-15
-45’ -60
I -40
-20
0
20
40
60
angle in degrees Fig. 1. Norm
of the effective interpolation
error vector
with smoothing
(La = 25, L = 25, sector = - 60” to 60”, A” = 50).
KM
Table 1 Asymptotic performance of the Root-MUSIC with smoothing applied to the transformed data for uncorrelated two-source scenario (L, = 25, L = 25, DOAs = 55” and 60”, SNR = 6 dB, p = 0.0, sector = - 60” to 60”, ,V” = 50)
Number of subarrays
From
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.5474 0.4323 0.1374 0.3708 0.6230 0.7539 0.8177 0.7559 0.5686 0.3285 0.1054 0.2367 0.3707 0.1432
algorithm
Evaluation
Spectral MUSIC applied to arbitrary array output
Root-MUSIC with smoothing applied to transformed data
x x x x x x x x x x x x x
0.5132 0.9318 0.3389 0.5825 0.7297 0.8135 0.8482 0.7664 0.5709 0.3298 0.1075 0.2803 0.3584 0.1420
No. of subarrays
MSE in DOA estimate
(K)
(deg’)
No. of failures
MSE in DOA estimates
lo-’ 10-Z IO-’ 10-l 10-Z lo-’ 1om2 lo-* lo-* 1o-2 10m4 1om3 10-l
x x x x x x x x x x x x x
1 2 3 4 5 6 7 8 9
0.0924
2
of (41)
lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-* lo-’ 1o-2 1O-4 10m3 1O-2
Table 2 Asymptotic performance of the Root-MUSIC with smoothing applied to the transformed data for correlated two-source scenario (L, = 25, L = 25, DOAs = 55” and 60”, SNR = 6 dB, p = 0.9, sector = - 60” to 60”, ,Y’ = 50)
Number of subarrays
Table 3 Finite data performance of the Root-MUSIC with smoothing applied to the transformed data for closely spaced sources (L, = 25, L=25, N = 1000, DOAs =O” and 3”, p -0.0, SNR = 0 dB, ,V = 50, sector = - 60” to 60’)
Asymptotic performance of the Root-MUSIC with smoothing (A@ in deg’)
(K)
Asymptotic performance of the Root-MUSIC with smoothing (A@ in deg2)
(K)
From algorithm
Evaluation
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.5379 0.4054 x 0.1505 x 0.3828 x 0.6086 x 0.7165 x 0.7786 x 0.7892 x 0.7204 x 0.5825 x 0.3659 x 0.1268 x 0.1435 x 0.1154x
0.4296 0.1222 x 0.4948 x 0.6601 x 0.7003 x 0.7429 x 0.7736 x 0.7664 x 0.6901 x 0.5527 x 0.3424 x 0.1132x 0.3744 x 0.1254x
lo-* lo-* lo-’ 10-2 lo-’ lo-’ IO-’ lo-’ 1O-2 lo-’ lo-* 1o-4 10-Z
269
Reddy, V. U. Red& / Signal Processing 54 (1996) 261-272
of (29)
10-i lo-’ 1o-2 1o-2 lo-’ lo-’ lo-’ lo-* lo-’ 1O-2 lo-’ 1o-5 1O-2
of the table that the MSE in the DOA estimate is maximum when there is no smoothing and is much less with smoothing. Note also that the predicted results using (41) match well with the simulation
(dcg)’
No. of failures
0.0192 0.0216 0.0303 0.0412 0.0536 0.0632 0.0737 0.0876 0.1011
0 0 0 0 0 0 0 0 0
results. Small fluctuations in the values with the number of subarrays is because of the change in the virtual subarray size as the number of subarrays increases. Table 2 gives the asymptotic performance in the presence of correlated sources. Note from the results that smoothing improves the asymptotic performance in this case too. However, the improvement is slightly less than that with the uncorrelated sources. Table 3 shows the simulation results obtained for the Root-MUSIC with smoothing for two closely spaced uncorrelated sources when the SNR is 0 dB. The DOAs of the sources were chosen such that the interpolation errors in the corresponding directions are negligibly small. It can be observed from the table that the spectral MUSIC applied to arbitrary array output fails in some runs because the algorithm could not resolve the sources. However, the Root-MUSIC is able to resolve the sources very well and gives the best performance when the number of subarrays is one. As the number of subarrays increases, the MSE performance of the RootMUSIC degrades. This is because the sources are closely spaced and the aperture plays a dominant role in determining the MSE performance. Note, however, that the MSE performance of the
270
KM. Reddy, V.U. Reddy / Signal Processing
Table 4 Finite data performance of the Root-MUSIC with smoothing applied to the transformed data for the correlated sources (L, = 25, L = 25, N = 100, DOAs = 0” and 15”, p = 0.9, SNR = - 6 dB, sector = - 60” to 60” JV = 50)
No. of subarrays (K) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Spectral MUSIC applied to arbitrary array output
Root-MUSIC with smoothing applied to transformed data
MSE in DOA estimate (deg’)
No. of failures
MSE in DOA estimates (deg)’
No. of failures
0.2496
0
0.2431 0.1404 0.0845 0.0430 0.0442 0.0412 0.0419 0.0444 0.0441 0.0489 0.0503 0.0506 0.0553 0.0572 0.0622 0.0722 0.1021 0.1698
31 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Root-MUSIC with smoothing is better than that with spectral MUSIC as long as the subarray size is not too small compared to the source spacing. Table 4 shows the simulation results for the case of two widely spaced correlated sources with SNR of - 6 dB. The correlation between the sources is kept at 0.9. The table shows the MSE in the DOA estimate for the source at 15”. Observe from the results that the Root-MUSIC fails to estimate the DOAs in some of the runs, particularly when the number of subarrays is 1. This is because of a spurious root (within the unit circle) which has an amplitude greater than that around 15”. The spurious root occurs especially at low SNRs. Because of low SNR and N being the same as in the earlier case, the finite data errors are more in the present case. Until these finite data perturbations and interpolation errors are brought down to a lower level by
54 (1996) 261-272
smoothing, Root-MUSIC can encounter spurious roots leading to some failures. It is seen from the table that the MSE performance of Root-MUSIC improves as the number of subarrays increase. The performance is optimum when K = 6. We note that the MSE performance of the Root-MUSIC with smoothing applied to transformed data is approximately six times better than that with spectral MUSIC applied to arbitrary array output. From the results of Table 4, it may also be noted that as K becomes too large, the error in the DOA estimate increases. This is because the aperture of the subarray becomes small compared to the source spacing when K is made very large. For example, when K = 15, the beam width for the subarray is approximately 20” while the source spacing is 15” (The spacing between the elements in the virtual ULA is 0.321 approximately.) In order to get the best performance with smoothing, the value of K should be chosen so that the beam width of the subarray is less than the source spacing. Fig. 2 shows the MSE in the DOA estimate obtained by applying the Root-MUSIC with smoothing for a correlated two-source scenario for a relatively high SNR. The figure also gives the plot of numerical values obtained from the bound in (32). The difference between the bound and the simulation results can be attributed, to a large extent, to the capability of smoothing to reduce the finite data perturbations. Note that the more the number of subarrays, the higher is the reduction in the finite data perturbations. The slight deterioration in the performance for larger values of K (K > 16) is because the aperture of the subarray becomes small compared to the source spacing.
5. Concluding remarks In this paper, we have analysed the performance of the interpolated arrays with spatial smoothing in the presence of correlated sources, finite data perturbations and the interpolation errors. In particular, an attempt is made to bring out the capability of smoothing to reduce the effect of noise perturbations due to finite data and the effect of interpolation errors on the performance of the Root-MUSIC applied to the transformed data. Simplified
KM
Reddy, V.U. Reddy / Signal Processing
x
271
54 (1996) 261-272
simulation
al
$ 0.06 m 2 0.05 : 0.04 B 0.03
t 0.02
,,
x
x
x 0.01
x
I-\0'
0
x
x
*
.x
x
x
x
x
x
x
x
x -4
2
4
6
6
10 12 number of subarrays
I
14
16
18
20
Fig. 2. Performance of the Root-MUSIC with smoothing applied to the transformed data as a function of the number (La = 25. L = 25, DOAs = 0’ and 15”, SNR = 0 dB, p = 0.9, N = 100, sector = - 60’ to 60’) .)k” = 50).
expressions are derived for the asymptotic performance for certain special cases. Computer simulations are provided to support the theoretical assertions.
Combining (A.4) and (39) the expression for the effective interpolation error vector (with smoothing) can be written as Auf (0) = +fH
Appendix A
Aa,p (Q,
where the vector
In this appendix, we show that effective interpolation error vector smoothing, i.e., we show
the norm of reduces with
AafH(@Au,(O) d AaH (0)Aa (0). Let the interpolation
f is defined as
f = [I, e-jw, . .. ,e-jw(X-l)]T, From
(‘4.6)
(A.5) we get
Aa7 (@Au, (0) = $ Applying
Aa; (0) ffHAa,,
Schwartz
by Au(e) = [A a, (0) . . , Aa,(0)lT.
(A.5)
(H).
(AI)
error vector Au (0) be given
AuH (@Au, (0) d (A.2)
&f
inequality
(A.7)
to (A.7) we obtain
HfAu:p (fWa,,
(f4.
From (A.6) (A.4) and the definition can be simplified to obtain
From (A.2), we get
of subarrays
(.A.8) of Z,, 04.8)
AaH (@Au (0) A+ (@Au, (0) = ClAai (@I2 +
...
Let the vector Aas,
+ lAur.(0)12].
(4.3) d;
be given by
Au,~ (0) = [AaT (t?)Z, , . . , AaT (0)ZK]T.
(A.4)
[[Au1 (@I’ + 2lA~,(8)1~ +
~IA~L-IV)I~
+
...
+ IAaL(Q)121.
(A.9)
272
KM
Reddy, V. U. Reddy / Signal Processing 54 (1996) 261-272
From (A.9) and (A.3), we arrive at the inequality in (A.l). The equality holds when K = 1.
References
Cl1 A.J. Barabell, “Improving the resolution performance of eigen structure based direction finding algorithms”, Proc. Internat. Co& Acoust. Speech Signal Process. 1983,
pp. 336-339. PI B. Friedlander, “The Root-MUSIC algorithm for direction finding with interpolated arrays”, Signal Processing, Vol. 30, No. 1, January 1993, pp. 15-29. c31B. Friedlander and A.J. Weiss, “Direction finding for wideband signals using an interpolated array”, IEEE Trans. Signal Process., Vol. 41, April 1993, pp. 1684-1634. c41R. Kumaresan and D.W. Tufts, “Estimating the angles of arrival of multiple plane waves”, IEEE Trans. Aerospace Electronic Systems, Vol. 19, January 1983, pp. 134-139. c51D.A. Linebarger and D.H. Johnson, “The effect of spatial averaging on spatial correlation matrices in the presence of coherent signals.” IEEE Trans. Acoust. Speech Signal Process., Vol. 38, May 1990, pp. 880-884. VI A. Paulraj, V.U. Reddy, T.J. Shan and T. Kailath, “Performance analysis of the MUSIC algorithm with spatial smoothing in the presence of coherent sources”, 1986 MILCOM, Vol. 3, No. 3, pp. 41.5.1-41.5.5. c71M.J.D. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981.
PI B.D. Rao and K.V.S. Hari, “Performance of analysis of
Root-MUSIC”, IEEE Trans. Acoust. Speech Signal Process. Vol. 37, December 1989, pp. 1939-1949. c91B.D. Rao and K.V.S. Hari, “Effect of spatial smoothing on the performance of MUSIC and minimum norm method”, Proc IEE, Part F, Vol. 137, December 1990, pp. 449-458. Cl01K.M. Reddy and V.U. Reddy, “Further studies in spatial smoothing”, Signal Processing, Vol. 48, No. 3, February 1996, pp. 217-224. Cl11V.U. Reddy and KC. Indukumar, “Techniques in optimum weighted smoothing”, in: C.T. Leondes, ed., Stochastic Techniques in Digital Signals Processing Systems, Academic Press Series on Advances in Control and Dynamic Systems, Vol. 64, Academic Press, New York, pp. 299-351. Cl21R. Roy and T. Kailath, “ESPRIT - Estimation of signal parameters via rotational invariance techniques”, IEEE Trans. Acoust. Speech Signal Process., Vol. 37, July 1989, pp. 984-995. Cl31R.O. Schmidt, “Multiple emitter location and signal parameter estimation”, IEEE Trans. Antennas Propagat., Vol. 34, March 1986, pp. 276-280. Cl41T.J. Shan, M. Wax and T. Kailath, “On spatial smoothing for direction of arrival of estimation of coherent signals”, IEEE Trans. Acoust. Speech Signal Process., Vol. 33, August 1985, pp. 806-811. Cl51A.J. Weiss and B. Friedlander, “Performance analysis of spatial smoothing with interpolated arrays”, IEEE Trans. Acoust. Speech Signal Process., Vol. 41, May 1993, pp. 1881-1892.
PROCESSING Signal Processing 54 (1996) 261-272
Analysis of interpolated
arrays with spatial smoothing
K. Maheswara Reddy”, V.U. Reddyb,* “CASSA. Defence Research & Development, New Thippasandra, Bangalore 560 075. India bDepartment of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India
Received 15 January 1996; revised 24 June 1996
Abstract An interpolated array is a means of extending the rooting techniques and preprocessing schemes like spatial smoothing, which are available for uniform linear arrays, to arrays with arbitrary geometry. This paper analyses the effect of spatial smoothing on the performance of the interpolated arrays in the presence of interpolation errors and finite data perturbations. An attempt is made to bring out the capability of smoothing to reduce the effect of finite data perturbations and interpolation errors on the performance of the Root-MUSIC with interpolated arrays. Simplified expressions are obtained for certain special cases. Computer simulations are provided to demonstrate the usefulness of the analysis.
Ein interpoliertes Array ist ein Mittel zur Erweiterung von Rooting- und Vorverarbeitungstechniken wie rgumlicher Glgttung, die man fiir gleichfiirmige lineare Arrays zur Verfiigung hat, auf Arrays beliebiger Geometrie. Dieser Beitrag analysiert die Auswirkungen rtiumlicher Glgttung auf die Leistungsfihigkeit interpolierter Arrays beim Auftreten von Interpolationsfehlern und begrenzter Datenverfilschungen. Es wird versucht, die FHhigkeit einer Gllttung zur Reduktion der Auswirkung endlicher Daten- und Interpolationshfehler in die Leistungsfihigkeit eines ROOT-MUSICVerfahrens mit interpoliertem Array einzubringen. Fur einige Spezialfille ergeben sich vereinfachte Ausdriicke. Rechnersimulationen werden dargestellt, welche den Nutzen der Analyse demonstrieren.
Un rbseau interpolit est un moyen d’itendre aux rCseaux g gkometrie arbitraire les techniques d’extrdction de racines et les mitthodes de prC-traitement telles que le lissage spatial, qui sont disponsibles pour les rkseaux linkaires uniformes. Cet article analyse l’effet du lissage spatial sur les performances des rCseaux interpolCs en prksence d’erreurs d’interpolation et de perturbations de donn&es finies. Une tentative est faite pour mettre en relief la capacitC du lissage i rCduire l’effet des perturbations de donnCes finies et des erreurs d’interpolation sur les performances du Root-MUSIC avec des rCseaux interpolCs. Des expressions simplifites sont obtenues pour certains cas particuliers. Des simulations informatiques montrent I’utilitC de cette analyse. Keywords:
Interpolated
*Corresponding
arrays; Spatial smoothing
author. Tel.: 080 309 2278; fax: 080 334 1683; e-mail: reddy@,ece.iisc.ernet.in.
262
K.M. Reddy, V. U. Reddy / Signal Processing 54 (1996) 261-272
1. Introduction Subspace-based direction-of-arrival (DOA) estimation algorithms such as MUSIC [13], ESPRIT [12] and minimum norm [4] yield high resolution and asymptotically exact DOAs. The performance of these algorithms, however, degrades when the impinging sources are correlated. In particular, when two or more impinging sources are fully coherent, the algorithms break down, and in the case of finite data, the DOA estimation performance degrades even when the sources are partially correlated. Correlation between the impinging sources arises because of either multipath propagation or due to deceptive jamming. Some preprocessing techniques such as spatial smoothing [14], weighted smoothing [l l] and redundancy averaging [S] have been proposed to alleviate the ill effects of correlation. Also, the smoothing reduces the effects of finite data pertubations thereby improving the estimation accuracy [6]. But all these techniques can only be applied to uniform linear arrays. Furthermore, the rooting techniques which are shown to yield better resolution and accuracy [S] are also applicable to uniform linear arrays only. However, in most of the real world applications, the array size and shape are often not determined based on the requirements of the direction finding system; but they are determined based on either the size and shape of the platform on which the array is to be mounted or other structures and mechanical limitations of the platform. For example, the array may have to be mounted in the nose or near the wings of the aircraft. The array geometry resulting from such physical constraints will not have a uniform linear array configuration. Also, when 360” azimuth coverage is required, we prefer a circular array to a uniform linear array. Thus, the uniform linear array configuration may not be encountered in many practical situations. It is, therefore, desirable to develop techniques which extend the preprocessing and rooting methods applicable to uniform linear arrays to the arrays of other geometry. In this paper, all the arrays other than uniform linear arrays are referred to as arbitrary arrays. In [2], Friedlander has proposed interpolated arrays for extending the use of rooting techniques
and preprocessing schemes to arbitrary arrays. This method involves developing a transformation which matches manifold of the arbitrary array to that of a chosen uniform linear array (ULA), which we refer to as the virtual ULA, within the field of view of the array. If it is not possible to match the manifolds over the entire field of view, it can be divided into smaller sectors and different transformations can be chosen for different sectors. The output of the arbitrary array, when passed through these transformations, resembles the output from the virtual ULA. It is now possible to apply rooting and smoothing techniques to the transformed data. As it is not possible to exactly match the manifolds of the arbitrary array and the virtual ULA over the entire chosen sector, there will be some interpolation error. In [ 151, Weiss and Friedlander analysed the performance of spatial smoothing with interpolated arrays. They provided theoretical expression for the mean square error (MSE) in the DOA estimate. In their analysis, however, they assumed the interpolation error to be very small and consequently its effect on the performance of the algorithm was ignored. They compared the performance of the algorithm with the Cramer-Rao bound (CRB) and also with the performance of spectral MUSIC as applied to the arbitrary array output. In this paper, an attempt is made to analyse the effect of forward spatial smoothing on the DOA estimation performance of the Root-MUSIC [l] in the presence of finite data perturbations and interpolation errors for the case of both correlated and uncorrelated sources. It is shown that in addition to reducing the correlation between the sources and the finite data perturbations [lo], the spatial smoothing also reduces the effect of interpolation errors on the performance of Root-MUSIC applied to the virtual ULA output. Simplified expressions are obtained for certain special cases. The paper is organised as follows. In Section 2, we briefly introduce the interpolated arrays and the associated spatial smoothing algorithm. In Section 3, we analyse the performance of the spatial smoothing and obtain explicit expressions for some special cases. We present the results of computer simulations in Section 4. Section 5 concludes the paper.
263
KM. Reddy, V. U. Red+ / Signal Processing 54 (1996) 261-272
2. Background and problem statement Consider an arbitrary array composed of L, sensors. We assume that the elements of the array are in the xy-plane and all the elements are omnidirectional. The steering vector of such arbitrary array can be written [3] as a, (0) = [ej2Mx,sin0+ ~1~0s 01, . . , e j2a(x,.dsm
0 + !,.,cosO)
1> T
(1)
where (x,, y,,J are the coordinates of the mth sensor in wavelength units, 8 is the angle in the xy-plane with respect to y-axis and the superscript T denotes the transpose of a vector. In the interpolated array approach [2], the field of view of the arbitrary array is divided into different sectors. For each sector, we choose a virtual ULA and match the manifold of the arbitrary array with that of the chosen ULA at a predetermined set of angles using a transformation. Let L (L d L,) be the number of elements of the virtual ULA for a given sector and d be the spacing between the elements. The steering vector of the virtual ULA is given by a(@ = [l, ePj*l, . . . ,e-j”(L-l))]T,
(2)
where IX = (2nd sin 0)/n with ,J denoting the wavelength. Let Jt’ be the number of angles at which the manifolds of the arbitrary array and the virtual ULA are to be matched in a chosen sector and assume that &i, flb2, . . . , eb, are the angles. Let Aba be the matrix obtained by stacking the steering vectors of the arbitrary array along the chosen angles A,, = [a,(e,,), . . . db,
)I.
(3)
Similarly, let Ab be the matrix obtained by stacking the steering vectors of the virtual ULA along the chosen angles d46= [a(e,,), . . . ,a(eb, )I.
The approximation in (5) is because of the inconsistency of the equations which may arise in trying to match the two different manifolds at as many angles as possible. Solving (5) using least-squares technique, we get B = AbAg = AbAya (A,,Ara)- ‘,
where (.)” represents the pseudo inverse and (.)” represents the complex conjugate transpose. This transformation can now be applied to the output of the arbitrary array to obtain the data which would resemble the output from the virtual ULA. We can then apply the techniques available for the ULA to the transformed data. Assume that M narrowband sources impinge on the arbitrary array. Let us assume that the sources are in the same plane as the elements of the array and that the DOAs of these sources are 8i, e2, . . , eM. The array output vector can then be written as x,(t) = &s(t) - n(r),
(7)
where s(t) is the signal vector, n(t) is the complex circularly Gaussian-distributed noise vector which is assumed to be spatially white and uncorrelated with s(t), and A, is the matrix of direction vectors of the arbitrary array A, = [aa(
a,(e,), . . , dedi.
(8)
The array covariance matrix for the arbitrary array can be expressed as R, = E[x,(t)x;(t)]
= A,SA,H + 0’1,
(9)
where S denotes the signal covariance matrix, c2 is the variance of the additive noise, I is an identity matrix and E(.) denotes the expectation operator. Applying the transformation B to the covariance matrix R,, we obtain the transformed covariance matrix as R, = BR,B”
(4)
(6)
= BA,SA;B”
-I- a2BBH.
(10)
Then, the desired transformation B which matches the manifold of the arbitrary array to that of the virtual ULA is given by
Noting that the matrix B is chosen for matching of the manifold of the arbitrary array to that of the virtual ULA, we can write the transformed matrix Rb as
Ab ‘v BA,,,.
R,, 2: ASAH + 02BBH,
(5)
(11)
264
KM
Reddy, V. U. Reddy / Signal Processing
where A is the matrix of direction vectors for the virtual ULA, ‘4 = [a(&), ... ,4&f)l.
(12)
Note that the approximation in (11) appears because of interpolation errors. We can now apply the prepocessing and rooting techniques to the matrix Rtl. In spatial smoothing, we divide the ULA into K overlapping subarrays. In the present case of interpolated arrays, the ULA is a virtual ULA and hence the subarrays are also virtual. The spatially smoothed covariance matrix, denoted by Rbs, is the average of the virtual subarray covariances and can be obtained from the covariance matrix Rb by computing (for forward smoothing) (13) where Z, = [er, el+l, . . . , ef +LO_r] with el denoting the Ith column of the identity matrix of size L and L,, (L, > M) is the size of the virtual subarray. Note from (11) and (13) that the noise part of the covariance matrix Rbs is not in the form of aZZ.We, therefore, apply prewhitening to Rbs. The prewhitening matrix is given by
-l/2 1
(14)
The smoothed covariance matrix obtained after prewhitening can be expressed as R,, = Z&.&Z& = Rnw;
i Z;BA,SA;BHZ,Rfw z-1
+ 02Z.
(15)
In MUSIC [ 131; we perform the eigen decomposition of the covariance matrix R,,. The minimum eigenvalue of R,, is a2 and its multiplicity is L, - M. The eigenvectors corresponding to the minimum eigenvalue are called noise eigenvectors and they are orthogonal to the matrix R,,A, where A, is the direction matrix of the virtual subarray. The MUSIC exploits this property and estimates the DOAs of the sources by searching for the nulls (minima in the present case) of the spectrum D,(O) = u:!(e)R,H,E,E,HR,,v,(e),
(16)
54 (1996) 261-272
where E, is the noise subspace of R,, and u,(0) is the normalized steering vector for the subarray of the virtual ULA. Because of the structure of uf(0), D,(8) given in (16) can be written as a polynomial. We can thus apply Root-MUSIC to R,,. The paper analyses the performance of the RootMUSIC with smoothing applied to the transformed data in the presence of correlated sources, finite data perturbations and interpolation errors.
3. Performance
of the Root-MUSIC for interpolated arrays
smoothing
with
The transformation B matches, in the leastsquares sense, the manifold of the arbitrary array to that of the virtual ULA in a given sector. Consequently, there exists some deviation between the transformed arbitrary array manifold and the virtual ULA manifold. This deviation, referred to as the interpolation error, is given by Aa (e)
= a(e) - Baa(e).
(17)
Because of this interpolation error, the transformed arbitrary array direction vector matrix deviates from that of the virtual ULA. This deviation can be expressed as AA=A-BA,.
(18)
As a result, the noise subspace of the covariance matrix R,, is not perfectly orthogonal to the direction vectors of the virtual ULA and this introduces errors in the DOA estimates obtained by the RootMUSIC algorithm. Substituting for BA, from (18) in (15), we get R,, = Rnwf i Z; [ASAH + AASAAH l-1 - ASAAH]Z,Rfw
- AASAH
+ a2Z.
(19)
Eq. (19) can be broken into R,, = R, + AR,,,
(20)
where R, = i
i
R,,ZFASAHZIRfw
+ a2Z
I-1 = R,,AfSfA~R~w
+ a2Z
(21)
KM.
AR,,f = - 4 i R,,ZT I-1
Reddy, V.U. Red& J Signal Processing
[AASAH
+ ASAAH] Z,R;‘,.
(22)
Note that RF represents the smoothed covariance matrix of the virtual ULA (computed directly from the ULA configuration) with prewhitening. In expressing ARbf by (22) we implicitly assumed that the interpolation errors are small and neglected the terms containing more than one AA. In (21), S, and A, represent the smoothed signal covariance matrix and the direction matrix for the subarray of the virtual ULA, respectively. In practice, we estimate the covariance matrix R,, from finite number of snapshots. This also introduces certain errors in the covariance matrix. The finite data covariance matrix can be expressed as ffbs=~hf+hRF=~ft~F+~bS,
265
54 (1996) 261-272
the covariance matrix R,, ufl (0) is the derivative of uf (0) w.r.t. 0 and Re(.) represents the real part of (.). The above equations hold when the SNR or the number of snapshots is relatively high, and when the interpolation errors are small. The first and the second terms in (24) are due to finite data perturbations and the third term is due to interpolation errors. Thus, the overall MSE in the DOA estimate is a simple addition of two parts, one due to finite data perturbations and the other due to interpolation errors. As the number of snapshots tends to infinity, we get the MSE due to interpolation. The MSE in the DOA estimate due to interpolation errors only is deterministic and is given by
CRe(~“A% B)l"
E(AeZ) =“’ = [UT, ((ji)Rfwpn~n,ufl
((j,)]” ’
(29)
3.1. MSE due to jnite data perturbations
(23)
where ARr represents the perturbation due to finite data. Note that the perturbation due to finite data is random while the deviation in the covariance matrix due to interpolation errors, AR,,, is deterministic. Following the steps given in [9], we can obtain the mean square error (MSE) in ith DOA estimate due to finite data perturbations and interpolation errors as E [A($]
If the interpolation (24) can be simplified
errors as
are very small,
then
Let us consider the case with no smoothing, i.e., K = 1. The expression (30) can then be simplified as E(A0;) CT’ [(S-‘)ii NL
+ aZ(S-‘(AHR~~R~,A)-‘S-‘)ii] 2$(~JR!P~R,wU,
(oil
-’ (31)
(27) H
H
Rs, = R .wA,S,A,R nw,
(28)
whereR,, = Ryp = Eb,(t)y,H(t)],y,(t) is the output vector of the pth virtual subarray obtained after prewhitening. N is the number of snapshots, P, is the projection matrix on to the noise subspace of
where u1 (0) is the first derivative of u(0) w.r.t. f1,u(e) is the normalized steering vector of the virtual ULA and (.)ii represents the ith diagonal element of (.). Note from (31) that the denominator contains L and the terms (U’:(ei)R~~P,R”,U,(Bi)) and (AHR:wR,,,A) are also dependent on L, the size of the virtual ULA. Thus, as L decreases, the MSE performance of the Root-MUSIC deteriorates. Further, the value of L cannot be more than L,, the number of elements in the arbitrary array, as this choice makes the covariance matrix R,, to be singular. Hence, care must be exercised while selecting the number of elements in the virtual ULA.
266
KM
Reddy, V.U. Reddy 1 Signal Processing 54 (1996) 261-272
Let us consider the case with smoothing, i.e., K > 1. From (30), it can be shown that (see [lo] for the steps involved) E[A@] d
where Y = diag[e-j”l, . . . ,e-joM] and (.)* denotes complex conjugate of (.). Combining (35) and (34) and noting that S, !I’ are diagonal matrices, we get aHAR& = - f CX~R,,~: ZTAA !l’*(‘- ” I=1
G2 [ST ‘)ii + G’(Sy 1(A~RF;‘wR”wAf)- ‘Sy l)ii] NLo
x
2vfH1(ei)R~~wPrPnwUf1(ei)
(32) We note that the equality in (32) holds if the number of subarrays is one, i.e., when there is no smoothing. We also note that by replacing the source covariance matrix S by the smoothed source covariance matrix S, (which accounts for the reduction in the source correlations due to smoothing), L by Lo, A by A, and v1 (QJ by ufl (0,) (which account for the reduction in the array size because of smoothing) in (31), we obtain the RHS of (32). Thus, if the spatial smoothing were to reduce only the source correlations, then the MSE with smoothing should have been given by the RHS of (32). But, as seen from (32), the MSE can be smaller. This suggests that the smoothing also reduces the noise perturbations due to finite data, in addition to reducing the correlation among the impinging signals.
SAY RfWR$uf(Oi).
(36)
From (28), R$ can be written as R$ = &&(A~Rj;lyR,J/)-l xS;l
(A~RfWRwR.,AJIA~RH IlW.
(37)
Substituting this in (36) and noting that S = S, for the uncorrelated sources, we can simplify (36) to get 1 ff%Rbf/ZI= - K&O
ctHRnw,il #-Aa
ej”i’z- ‘), (38)
where Lo is the size of the virtual subarray and is given by Lo = L - K + 1. Let Aa, be the effective interpolation error vector (with smoothing) in the direction ei, Aar(Bi) = f i ZTAa(Bi)ej@ic’- ‘).
(39)
I-1
Eq. (38) can now be written as 3.2. Asymptotic performance uncorrelated
for the case of
CruAR&? = - &
sources
In this section, we try to bring out the asymptotic performance of the Root-MUSIC with smoothing for the interpolated arrays when the sources are uncorrelated (S and S, are diagonal). Substituting for fi from (27) in the numerator of (29), we obtain mHARbf/I= u~AR~,-R,#~R,,,v~(~~).
(33)
Substituting for AR,, from (22) and noting that P, is orthogonal to R,,A,, we get “HA%fP 1 = - K CX~R,,~5 Z~AASAHZIRfWR,#fR,,vf(Oi). I=1
(34)
From the definition of A and Z,, we can express AHZl = y*“- “&!,
(35)
uHR,,,,,Aaf(Bi).
(40)
Substituting (40) in (29), we get the expression for the asymptotic performance of the Root-MUSIC with spatial smoothing in the presence of uncorelated sources as A@
=’
CRe(~HR,,A~,#J)12
LO u~l(ei)R~~,P,Rnwvr1(ei)12'
(41)
Note from (41) that the MSE in the ith DOA estimate (which is square of the error in the present case) is dependent only on the effective interpolation error in the direction of the ith source. In Appendix A we have shown that the norm of the effective interpolation error vector reduces with smoothing. Hence, we can expect the smoothing to improve the asymptotic performance of the RootMUSIC by reducing the effect of interpolation errors.
KM
3.3. Asymptotic performance correlated sources
Reddy. V. U Reddv 1 Signal Processing 54 (1996) 261-272
4. Discussion of simulation results
for the case of
We now attempt to arrive at the asymptotic performance of the Root-MUSIC with smoothing when the sources are correlated. Following the steps of the previous section, we can show that c?‘AR,/j = - &
AfHIR,,P,,R,,, 0!
5 .ZTAASY*“-“Sj’
,
I=1
(42) where A,,
is given by (43)
If we consider the case of no smoothing, K = 1, then (42) can be simplified as
i.e.,
(x~AR~SB = - 1 JL
(44)
CIHR,wAa(
267
Substituting (44) in (29), we obtain the asymptotic MSE in the presence of correlated sources which is same as (41) with K = 1. When K > 1, simplification of (42) leads to unwieldy expressions even for a two-source case and hence is not presented here. However, using simulations, we show that the asymptotic performance improves with smoothing for correlated sources also. We conclude this section with the following remark. While designing the transformation matrix B, it is possible to incorporate the location inaccuracies of the array elements and the differences among these elements. Whenever the calibration data is available, the data can be used while constructing the matrix Aha given in (3). The resulting transformation B and the smoothing then reduce the effects of location inaccuracies and the use of non-identical elements on the performance of the algorithm. Noting that the Root-MUSIC and spatial smoothing (in their original forms) can only be applied to linear arrays with identical elements, the interpolated arrays can be viewed as a way to extend the use of rooting and preprocessing techniques to non-ideal arrays also. This appears to be one of the major advantages of using the interpolated arrays.
We conducted simulations to test the usefulness of the analysis of the previous section. In our simulations, the arbitrary array was taken as a semicircular array. The semicircular array was assumed to lie in the xy-plane with the first element at the origin and the last element on the x-axis at a distance equal to the diameter of the array. The rest of the elements were placed at equal distance along the semi-circle. The semicircle is towards the positive jj-axis. The virtual ULA was chosen to be along the x-axis such that the positions of the end elements of the ULA match with those of the semicircular array. The angle % was measured w.r.t. the y-axis. In the simulations and numerical evaluation, a semicircular array consisting of 25 isotropic elements with half wavelength inter-element spacing (straight line distance between the elements) was assumed. The number of snapshots was fixed at 100. We considered two-source equi-power scenarios in our simulations. We generated the snapshot data with the noise vector n(t) consisting of zero mean, unit variance independent complex circularly Gaussian random variables and the signal vector consisting of zero mean complex circularly Gaussian random variables whose variances were fixed to give the desired signal powers. In the case of uncorrelated signals, the random variables comprising the signal vector were generated independently. In the case of correlated signals. the second signal was generated as SZ(L)= Psi(t) + JY&(t),
(45)
where sl(t) is the signal from the first source with variance 0,’ and sj(t) is a zero mean, complex circularly Gaussian distributed random variable with variance a:. The signal s3(t) is uncorrelated with the signal s1 (t). The snapshots were made independently by changing the seeds randomly from snapshot to snapshot. The simulation results were obtained by averaging over 100 independent Monte Carlo runs. We may also point out that in any Monte Carlo run if the DOA estimation error is more than l”, then we treated that run as the one in which the algorithm failed to estimate the DOA.
KM.
268
Reddy, V.U. Reddy 1 Signal Processing
54 (1996) 261-272
sector. This sector was selected from - 60” to 60” and JV was chosen as 50. Fig. 1 shows the norm of the effective interpolation error vector (defined in (39)) as a function of the angle in the chosen sector for K = 1 and K = 4. It is seen from this figure that the effective interpolation error vector norm reduces as the number of subarrays increases. This reduction in the interpolation errors improves the performance of the Root-MUSIC especially when the errors are not very small. Table 1 shows the asymptotic performance of the Root-MUSIC with smoothing applied to transformed data as a function of the number of subarrays in the presence of uncorrelated sources. The DOAs of the sources (55’ and 60”) were chosen such that the interpolation errors are maximum (see Fig. 1). From the manifold of the semicircular array and the transformation B, R,, was computed and the Root-MUSIC with smoothing was applied to this matrix to determine the asymptotic performance. We refer to this result as the one obtained from the algorithm in our Tables 1 and 2. The theoretical values of the MSE (which is square of the error in the present case) were obtained by numerically evaluating (41). Note from the results
Thus, the simulation results presented here also contain the number of times the algorithm failed to estimate the DOAs. The search for the minima in the case of spectral MUSIC was conducted in steps of 0.005”. For all simulations in this section, SNR referes to the signal to noise ratio without the array gain. The particulars of the signal scenario, number of subarrays and the number of snapshots used are given in the captions. For obtaining the transformation B, the angles at which the manifold of the arbiteb,, eb2, . . . , t!9b.y., rary array is to be matched with that of the virtual ULA, can be placed uniformly over the sector or can be chosen using the Chebyshev distribution. When the angles are uniformly placed, the norm of the interpolation error vector keeps increasing as we move towards the ends of the sector. If they are chosen using the Chebyshev distribution [7], the norm remains within a given limit throughout the sector. The number of angles chosen for matching (J+‘) in the sector must always be greater than or equal to the number of elements in the arbitrary array to obtain a transformation B which gives least interpolation error. In the simulations, the angles were placed uniformly over the sector and the sources were assumed to belong to that single
-15
-45’ -60
I -40
-20
0
20
40
60
angle in degrees Fig. 1. Norm
of the effective interpolation
error vector
with smoothing
(La = 25, L = 25, sector = - 60” to 60”, A” = 50).
KM
Table 1 Asymptotic performance of the Root-MUSIC with smoothing applied to the transformed data for uncorrelated two-source scenario (L, = 25, L = 25, DOAs = 55” and 60”, SNR = 6 dB, p = 0.0, sector = - 60” to 60”, ,V” = 50)
Number of subarrays
From
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.5474 0.4323 0.1374 0.3708 0.6230 0.7539 0.8177 0.7559 0.5686 0.3285 0.1054 0.2367 0.3707 0.1432
algorithm
Evaluation
Spectral MUSIC applied to arbitrary array output
Root-MUSIC with smoothing applied to transformed data
x x x x x x x x x x x x x
0.5132 0.9318 0.3389 0.5825 0.7297 0.8135 0.8482 0.7664 0.5709 0.3298 0.1075 0.2803 0.3584 0.1420
No. of subarrays
MSE in DOA estimate
(K)
(deg’)
No. of failures
MSE in DOA estimates
lo-’ 10-Z IO-’ 10-l 10-Z lo-’ 1om2 lo-* lo-* 1o-2 10m4 1om3 10-l
x x x x x x x x x x x x x
1 2 3 4 5 6 7 8 9
0.0924
2
of (41)
lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-’ lo-* lo-’ 1o-2 1O-4 10m3 1O-2
Table 2 Asymptotic performance of the Root-MUSIC with smoothing applied to the transformed data for correlated two-source scenario (L, = 25, L = 25, DOAs = 55” and 60”, SNR = 6 dB, p = 0.9, sector = - 60” to 60”, ,Y’ = 50)
Number of subarrays
Table 3 Finite data performance of the Root-MUSIC with smoothing applied to the transformed data for closely spaced sources (L, = 25, L=25, N = 1000, DOAs =O” and 3”, p -0.0, SNR = 0 dB, ,V = 50, sector = - 60” to 60’)
Asymptotic performance of the Root-MUSIC with smoothing (A@ in deg’)
(K)
Asymptotic performance of the Root-MUSIC with smoothing (A@ in deg2)
(K)
From algorithm
Evaluation
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.5379 0.4054 x 0.1505 x 0.3828 x 0.6086 x 0.7165 x 0.7786 x 0.7892 x 0.7204 x 0.5825 x 0.3659 x 0.1268 x 0.1435 x 0.1154x
0.4296 0.1222 x 0.4948 x 0.6601 x 0.7003 x 0.7429 x 0.7736 x 0.7664 x 0.6901 x 0.5527 x 0.3424 x 0.1132x 0.3744 x 0.1254x
lo-* lo-* lo-’ 10-2 lo-’ lo-’ IO-’ lo-’ 1O-2 lo-’ lo-* 1o-4 10-Z
269
Reddy, V. U. Red& / Signal Processing 54 (1996) 261-272
of (29)
10-i lo-’ 1o-2 1o-2 lo-’ lo-’ lo-’ lo-* lo-’ 1O-2 lo-’ 1o-5 1O-2
of the table that the MSE in the DOA estimate is maximum when there is no smoothing and is much less with smoothing. Note also that the predicted results using (41) match well with the simulation
(dcg)’
No. of failures
0.0192 0.0216 0.0303 0.0412 0.0536 0.0632 0.0737 0.0876 0.1011
0 0 0 0 0 0 0 0 0
results. Small fluctuations in the values with the number of subarrays is because of the change in the virtual subarray size as the number of subarrays increases. Table 2 gives the asymptotic performance in the presence of correlated sources. Note from the results that smoothing improves the asymptotic performance in this case too. However, the improvement is slightly less than that with the uncorrelated sources. Table 3 shows the simulation results obtained for the Root-MUSIC with smoothing for two closely spaced uncorrelated sources when the SNR is 0 dB. The DOAs of the sources were chosen such that the interpolation errors in the corresponding directions are negligibly small. It can be observed from the table that the spectral MUSIC applied to arbitrary array output fails in some runs because the algorithm could not resolve the sources. However, the Root-MUSIC is able to resolve the sources very well and gives the best performance when the number of subarrays is one. As the number of subarrays increases, the MSE performance of the RootMUSIC degrades. This is because the sources are closely spaced and the aperture plays a dominant role in determining the MSE performance. Note, however, that the MSE performance of the
270
KM. Reddy, V.U. Reddy / Signal Processing
Table 4 Finite data performance of the Root-MUSIC with smoothing applied to the transformed data for the correlated sources (L, = 25, L = 25, N = 100, DOAs = 0” and 15”, p = 0.9, SNR = - 6 dB, sector = - 60” to 60” JV = 50)
No. of subarrays (K) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Spectral MUSIC applied to arbitrary array output
Root-MUSIC with smoothing applied to transformed data
MSE in DOA estimate (deg’)
No. of failures
MSE in DOA estimates (deg)’
No. of failures
0.2496
0
0.2431 0.1404 0.0845 0.0430 0.0442 0.0412 0.0419 0.0444 0.0441 0.0489 0.0503 0.0506 0.0553 0.0572 0.0622 0.0722 0.1021 0.1698
31 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Root-MUSIC with smoothing is better than that with spectral MUSIC as long as the subarray size is not too small compared to the source spacing. Table 4 shows the simulation results for the case of two widely spaced correlated sources with SNR of - 6 dB. The correlation between the sources is kept at 0.9. The table shows the MSE in the DOA estimate for the source at 15”. Observe from the results that the Root-MUSIC fails to estimate the DOAs in some of the runs, particularly when the number of subarrays is 1. This is because of a spurious root (within the unit circle) which has an amplitude greater than that around 15”. The spurious root occurs especially at low SNRs. Because of low SNR and N being the same as in the earlier case, the finite data errors are more in the present case. Until these finite data perturbations and interpolation errors are brought down to a lower level by
54 (1996) 261-272
smoothing, Root-MUSIC can encounter spurious roots leading to some failures. It is seen from the table that the MSE performance of Root-MUSIC improves as the number of subarrays increase. The performance is optimum when K = 6. We note that the MSE performance of the Root-MUSIC with smoothing applied to transformed data is approximately six times better than that with spectral MUSIC applied to arbitrary array output. From the results of Table 4, it may also be noted that as K becomes too large, the error in the DOA estimate increases. This is because the aperture of the subarray becomes small compared to the source spacing when K is made very large. For example, when K = 15, the beam width for the subarray is approximately 20” while the source spacing is 15” (The spacing between the elements in the virtual ULA is 0.321 approximately.) In order to get the best performance with smoothing, the value of K should be chosen so that the beam width of the subarray is less than the source spacing. Fig. 2 shows the MSE in the DOA estimate obtained by applying the Root-MUSIC with smoothing for a correlated two-source scenario for a relatively high SNR. The figure also gives the plot of numerical values obtained from the bound in (32). The difference between the bound and the simulation results can be attributed, to a large extent, to the capability of smoothing to reduce the finite data perturbations. Note that the more the number of subarrays, the higher is the reduction in the finite data perturbations. The slight deterioration in the performance for larger values of K (K > 16) is because the aperture of the subarray becomes small compared to the source spacing.
5. Concluding remarks In this paper, we have analysed the performance of the interpolated arrays with spatial smoothing in the presence of correlated sources, finite data perturbations and the interpolation errors. In particular, an attempt is made to bring out the capability of smoothing to reduce the effect of noise perturbations due to finite data and the effect of interpolation errors on the performance of the Root-MUSIC applied to the transformed data. Simplified
KM
Reddy, V.U. Reddy / Signal Processing
x
271
54 (1996) 261-272
simulation
al
$ 0.06 m 2 0.05 : 0.04 B 0.03
t 0.02
,,
x
x
x 0.01
x
I-\0'
0
x
x
*
.x
x
x
x
x
x
x
x
x -4
2
4
6
6
10 12 number of subarrays
I
14
16
18
20
Fig. 2. Performance of the Root-MUSIC with smoothing applied to the transformed data as a function of the number (La = 25. L = 25, DOAs = 0’ and 15”, SNR = 0 dB, p = 0.9, N = 100, sector = - 60’ to 60’) .)k” = 50).
expressions are derived for the asymptotic performance for certain special cases. Computer simulations are provided to support the theoretical assertions.
Combining (A.4) and (39) the expression for the effective interpolation error vector (with smoothing) can be written as Auf (0) = +fH
Appendix A
Aa,p (Q,
where the vector
In this appendix, we show that effective interpolation error vector smoothing, i.e., we show
the norm of reduces with
AafH(@Au,(O) d AaH (0)Aa (0). Let the interpolation
f is defined as
f = [I, e-jw, . .. ,e-jw(X-l)]T, From
(‘4.6)
(A.5) we get
Aa7 (@Au, (0) = $ Applying
Aa; (0) ffHAa,,
Schwartz
by Au(e) = [A a, (0) . . , Aa,(0)lT.
(A.5)
(H).
(AI)
error vector Au (0) be given
AuH (@Au, (0) d (A.2)
&f
inequality
(A.7)
to (A.7) we obtain
HfAu:p (fWa,,
(f4.
From (A.6) (A.4) and the definition can be simplified to obtain
From (A.2), we get
of subarrays
(.A.8) of Z,, 04.8)
AaH (@Au (0) A+ (@Au, (0) = ClAai (@I2 +
...
Let the vector Aas,
+ lAur.(0)12].
(4.3) d;
be given by
Au,~ (0) = [AaT (t?)Z, , . . , AaT (0)ZK]T.
(A.4)
[[Au1 (@I’ + 2lA~,(8)1~ +
~IA~L-IV)I~
+
...
+ IAaL(Q)121.
(A.9)
272
KM
Reddy, V. U. Reddy / Signal Processing 54 (1996) 261-272
From (A.9) and (A.3), we arrive at the inequality in (A.l). The equality holds when K = 1.
References
Cl1 A.J. Barabell, “Improving the resolution performance of eigen structure based direction finding algorithms”, Proc. Internat. Co& Acoust. Speech Signal Process. 1983,
pp. 336-339. PI B. Friedlander, “The Root-MUSIC algorithm for direction finding with interpolated arrays”, Signal Processing, Vol. 30, No. 1, January 1993, pp. 15-29. c31B. Friedlander and A.J. Weiss, “Direction finding for wideband signals using an interpolated array”, IEEE Trans. Signal Process., Vol. 41, April 1993, pp. 1684-1634. c41R. Kumaresan and D.W. Tufts, “Estimating the angles of arrival of multiple plane waves”, IEEE Trans. Aerospace Electronic Systems, Vol. 19, January 1983, pp. 134-139. c51D.A. Linebarger and D.H. Johnson, “The effect of spatial averaging on spatial correlation matrices in the presence of coherent signals.” IEEE Trans. Acoust. Speech Signal Process., Vol. 38, May 1990, pp. 880-884. VI A. Paulraj, V.U. Reddy, T.J. Shan and T. Kailath, “Performance analysis of the MUSIC algorithm with spatial smoothing in the presence of coherent sources”, 1986 MILCOM, Vol. 3, No. 3, pp. 41.5.1-41.5.5. c71M.J.D. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981.
PI B.D. Rao and K.V.S. Hari, “Performance of analysis of
Root-MUSIC”, IEEE Trans. Acoust. Speech Signal Process. Vol. 37, December 1989, pp. 1939-1949. c91B.D. Rao and K.V.S. Hari, “Effect of spatial smoothing on the performance of MUSIC and minimum norm method”, Proc IEE, Part F, Vol. 137, December 1990, pp. 449-458. Cl01K.M. Reddy and V.U. Reddy, “Further studies in spatial smoothing”, Signal Processing, Vol. 48, No. 3, February 1996, pp. 217-224. Cl11V.U. Reddy and KC. Indukumar, “Techniques in optimum weighted smoothing”, in: C.T. Leondes, ed., Stochastic Techniques in Digital Signals Processing Systems, Academic Press Series on Advances in Control and Dynamic Systems, Vol. 64, Academic Press, New York, pp. 299-351. Cl21R. Roy and T. Kailath, “ESPRIT - Estimation of signal parameters via rotational invariance techniques”, IEEE Trans. Acoust. Speech Signal Process., Vol. 37, July 1989, pp. 984-995. Cl31R.O. Schmidt, “Multiple emitter location and signal parameter estimation”, IEEE Trans. Antennas Propagat., Vol. 34, March 1986, pp. 276-280. Cl41T.J. Shan, M. Wax and T. Kailath, “On spatial smoothing for direction of arrival of estimation of coherent signals”, IEEE Trans. Acoust. Speech Signal Process., Vol. 33, August 1985, pp. 806-811. Cl51A.J. Weiss and B. Friedlander, “Performance analysis of spatial smoothing with interpolated arrays”, IEEE Trans. Acoust. Speech Signal Process., Vol. 41, May 1993, pp. 1881-1892.