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Unified analysis of DOA estimation algorithms for covariance matrix transforms
SIGNAL
PROCESSING ELSEVIER
Signal Processing 55 (1996) 107-I 15
Unified analysis of DOA estimation algorithms for covariance matrix transforms ’ A. Gorokhova>*, Yu. Abramovich”,
J.F. BGhmeb
aOdessa State Polytechnic University 270044, Av.Shevchenko 1. Odessa, bRuhr University, D-4630, Boshum I, German)
Ukraine
Received 9 November 1995
Abstract In this paper, a unified asymptotic performance analysis of the subspace-based direction-of-arrival (DOA) estimation is presented for a specified class of matrix-valued transforms of the sample covariance matrix. For the general case, involving analytic transforms, under the standard assumptions of the finite length Gaussian observation, explicit expressions for the asymptotic variances of the DOA estimates are obtained. These expressions provide a classification of matrix-valued transforms according to the error convergence rate. Typical examples such as the redundancy averaging approach and the persymmetrization algorithm are examined by the introduced technique. The asymptotic relationships are validated by computer simulations. Zusammenfassuug In dieser Arbeit wird fur eine bestimmte Klasse matrixwertiger Transformationen der empirischen Kovarianzmatrix eine vereinheitlichte asymptotische Analyse der Leistungsfahigkeit der unterraum-basierten Schatzung der Einfallsrichtung (direction of arrival, DOA) durchgefiihrt. Im allgemeinen Fall erhalten wir fur analytische Transformationen und unter der iiblichen Annahme einer GauRschen Beobachtung endlicher Lange explizite Ausdriicke fiir die asymptotischen Varianzen der DOA-Schlzwerte. Diese Ausdriicke liefem eine Klassifikation matrixwertiger Transfotmationen hinsichtlich der Fehlerkonvergenzrate. Typische Beispiele wie der “Redundancy Averaging” Ansatz und der “Persymmetrization’‘-Algorithmus werden mit Hilfe der eingefuhrten Methode analysiert. Die asymptotischen Beziehungen werden durch Computersimulationen bestitigt.
Nous presentons dans cet article une analyse unifiee des performances asymptotiques de l’estimation de direction d’arrivee (DOA) basee sur les sous-espaces, pour une classe specifique de transformations matricielles de la matrice de covariance. Dans le cas general, qui implique des transformations analytiques, sous l’hypothese standard d’une observation gaussienne de longueur finie, des expressions explicites des variances asymptotiques des estimees de la DOA sont obtenues. Ces expressions foumissent une classification des transformations matricielles en fonction du taux de convergence de l’erreur. Des exemples
* Corresponding author. Address: Department Signal, Telecom Paris (ENST), 46, rue Barrault, 75634 Paris Cedex 13, France. Tel.: 33 I 458 17547; fax: 33 1 458 87935; e-mail: [email protected]. ’ This study is supported by the SASPARC project of INTAS. 0165-1684/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO165-1684(96)00123-5
A. Gorokhov et al. /Signal Processing 55 (1996) 107-115
108
typiques tels que l’approche de moyennage de la redondance et l’algorithme de persymirtrie sont examinks k I’aide de la technique prCsentCe. Les relations asymptotiques sont valid&es par des simulations sur ordinateur. Keywords:
Array processing;
Direction-of-arrival
estimation;
1. Introduction The estimation efficiency of high-resolution subspace-based methods such as MUSIC [2], MinNorm [8], ESPRIT [ 161 have been extensively investigated [25, 18, 221 when applied to the direct data covariance (DDC) estimate. The high performance of these methods and their modifications which are achievable due to the low-rank property of the direct data approach (DDA) have been recently demonstrated [25]. On the other hand, the existing rank estimation procedures [21] for the preliminary space splitting are shown to be accurate and stable enough at quite moderate sample volumes when the DDA is considered [26]. A unified asymptotic analysis of both the noise and signal subspace approaches presented in [ 1 l] is found to be sufficiently precise and enables explicit analytic comparisons to be made for most of the methods in the direct data version. However, the DDC of the observed array process cannot be directly involved in the estimation procedures of signal parameters for several array processing applications. As an example we can consider the DOA estimation problem for spatially spread sources with known spatial decorrelation parameters. For this case a preliminary focusing technique to improve the coherence of the DDC is proposed [ 151. A similar situation occurs if the transfer functions for different spatial channels are nonindentical [l]. One can also find an indirect utilization of the DDC in the subspace-based analysis for a minimum redundancy array [ 131, when this latter is used for DOA estimation of multiple uncorrelated plane waves where there are more sources than actual array elements [ 171. Some other methods improve the DOA estimation accuracy due to some preliminary modifications of the DDC prior to further subspace-based analysis. In this context we may emphasize the persymmetrization algorithm (PA) [ 141 which retrieves the ML persymmetric covariance matrix estimate assuming that the true covariance matrix has the persymmetric structure. Similarly, the redundancy averaging (RA) [12]
Performance
analysis
and spatial smoothing [23] techniques are often used to decrease the impact of the intersource correlation on the DOA estimation performance. The estimation approaches mentioned above along with many others can be represented in the most general case as a two-step procedure: (i) perform a certain matrix-valued transform of the initial DDC, (ii) apply the subspace-based DOA estimation analysis to the obtained matrix. Hence, the DOA estimation accuracy analysis for the case of a matrix-valued transform of the DDC appears to be of practical interest. Our aim in this paper is to present a unified approach to the statistical analysis of the existing subspace-based DOA estimation methods applied to the generalized matrixvalued transform of the DDC. We also examine the impact of the matrix valued transform on the convergence of the DOA estimates. As a matter of fact, the initial low-rank property of the DDC often vanishes after its modification, as in the TAM algorithm [9]. In Section 6 of this paper, we establish the sufficient condition to preserve fast convergence for the case of a low-rank DDC. This condition implies certain constraints on the utilized transform and has a clear interpretation in terms of the subspace-based analysis. Sections 7 and 8 present the results of DOA estimation performance analysis when the RA and PA techniques are exploited. Analytic relationships for both methods are verified by computer simulations.
2. Preliminaries Let us consider an M x A4 covariance matrix duced by an output of some uniform linear (ULA) of dimension M. We assume that the process is a mixture of m plane waves with e,,... , I!$,,and spatially white noise ’ of power R = ACAH + oZM, *The superscripts conjugate
T, H and # stand for transpose, and pseudoinverse.
R proarray output DOA (T: (2.1)
transpose
A. Gorokhov et al. 1 Signal Processing 55 (1996)
where C is an m x m intersource covariance matrix, A is an M x m matrix of array response vectors being complex sinusoids with the frequencies o(&) = 2n(d/&)sin9,, n = l,...,m:
(2.3)
where S= [Sl,. ,S,] and A= diag {ni}y=r. We are considering a K-element actual antenna array with an arbitrary geometry and an output signal X(t) E (CKhaving the complex multivariable Gaussian distribution: X(t) N CN(K, 0, B) [24]. The empirical spatial covariance matrix of the process X(t) can be traditionally obtained on the basis of N independent observations X(t), t = 1,. . . , N, as follows: d = ;
g X(t)X(t>H. r-l
in 0 with the null-spectrum
~(0) = a(0)“filVfia(0).
(3.1) the
(2.2)
d is the interelement distance of the considered ULA, 0 is the DOA with respect to the array broadside and E., is the central wavelength of the examined narrowband process. Let 1.12 . . 3 i., be the m largest eigenvalues of the matrix R, im > CJ, with the corresponding set of eigenvectors SI , . . . , S,,, and Il a projector onto the orthogonal (noise) subspace. Then one can write down the eigendecomposition of R as follows: R=SAffolI,
problem
Alternatively, they can be obtained by retrieving roots of the polynomial
A = [a(O, ). . . ,a(&,)], a(e) = [,-io(@(M- I)@,. . , eiu(Q)(M--l1121 T,
minimization function:
109
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(2.4)
Suppose now that there exists some consistent estimate R of the covariance matrix R obtained by a certain transform .Y of the sample covariance matrix & i.e., k = Y(b). Consequently, the DOA parameters 81,. . , 0, can also be consistently estimated by applying the subspace-based DOA estimation algorithms to li.
3. Asymptotic DOA estimation error We need a common description of the existing subspace-based estimators. Most of them can be classified as either signal or orthogonal subspace methods. The latter ones essentially exploit some estimate I? of the orthogonal subspace projector Il so that the empirical DOA parameters are given by solving a
@)
= [Z-(M--lf/2 ,.,.,= (M-i)12]T,
(3.2)
where w is some non-negative weighting matrix. Some more detailed descriptions of the estimators can be found in [ 1 I]. We just notice here that fi is the empirical counterpart of a certain weighting matrix W. In the most trivial cases, one has W = ZM (the standard MUSIC) or W = {6,~6,~}$, (MinNorm). Both spectral and root versions are shown [ 111 to be asymptotically equivalent. Let us introduce the estimation errors Ail p I? -- ll andAB, a t?,-e,,n= l,...,m.Thensomestraightforward calculations similar to those given in [l l] provide the asymptotic DOA: A& = -
‘“C
2ndcos8,
Cs{a(B,,)“Afi
WIZa,(~,)}
a1(&,)HllW17Ha~(&,)
+o(IlAfill~)> a,(e)
= [ - i(M _ l)e-iw(H)(M-‘)i2,, $(A4 - l)e
(3.3) .,
io(O)(M- I ),12T 3 ’
where II IIF is the Frobenious norm and !R{.}, S{_.} stand here and later for the real and imaginary part. Notice that (3.3) holds for both spectral (3.1) and root (3.2) criteria. Meanwhile the variety of signal subspace methods like ESPRIT, state-space realization (SSR) [9] and matrix-pencil method [5] are identical even in the noisy case [ 191. The equivalent DOA estimates can be found as described in [ 111, via the eigenvalues of a matrix p = i’#,!?‘, where ,!? is a sample counterpart of S, 1 (T) denotes the submatrix obtained from the original one by excluding its last (first) row. The DOA estimation errors, similar to [ 111, are given by
110
A. Gorokhov et al. /Signal
where e, = [S,]Er and 6in is a Kronecker delta. The results cited above [ 1 l] allow us to use: - the asymptotic expression (3.3) for both the spectral and root orthogonal subspace estimators with the arbitrary weighting; _ the asymptotic expression (3.4) for the variety of signal subspace based methods; Since the actual analysis concerns the asymptotic error values, we only examine first order terms of Taylor series throughout this paper and use the ‘=’ mark to reflect the equality of modulo to higher-order terms. According to the relationships (3.3), (3.4), one needs the asymptotic representation of Al? and II&?. The appropriate expressions in AR p R - R
can be conventionally obtained through the asymptotic eigenprojector perturbation technique [7]. Denoting Ro A R - old,one has ri = Ii’ - R!ARll - IIARR; + o(]]A&),
(3.5)
Processing 55 (1996)
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transform Y. This latter requires some unified asymptotic description of the mapping AB -+ AR.
4. Generalized matrix-valued transform be a transform mapping the space of Hermitian K x K matrices into the space of Hermitian M x M matrices. We consider here a class XB of such transforms, continuously differentiable in some neighbourhood of B E XK. Let F : ZK -+ SM
Lemma 1. A given transform T : XK 4 XM verifies ~7 E XB for some B E XK ifs T(b)
- 9-(B) = Y#?
- B) + o( I@ - BllF), (4.1)
where _%‘B is a linear transform .2$ : CKxK + CM xM dejined as follows:
ii?=~~-fi=~SSH-t~~Aih +IIARR;
+ o( I]A&).
Taking into account the asymptotic ZI(S,!? + &!?) Il&n
(3.6)
identity IIZ??
=
= II&Srr we find from (3.6):
= IIARR;.
(3.7)
Plugging (3.5) into (3.3) and (3.7) into (3.4), one can easily find the following unified expression: 2nd
CO&,
Y,,
’
(3.8)
where CI, and /I,, are M-component complex vectors and yn are positive scalars defined as follows: - Orthogonal subspace methods: %I = R$+%J, yn =
Zij
=
[6ip6jq]:,,=,,
i, j = 1,. . . ,K.
(4.2)
Consequently, once a given transform F is continuously differentiable in some neighborhood of B E XK, its perturbation at the point Y(B) can be asymptotically represented as a certain linear transform of the argument perturbation. Such a transform can be described by the partial derivatives of its elements with respect to the elements of the argument matrix according to (4.2). One useful property is the following. Lemma 2. For any 5
E XB the associated 2~ veri-
fin = nwn%(e,),
al(e,)HnwnHal(e,).
(3.9)
kf’BE .sK.
(4.3)
- Signal subspace methods: c1n = n ,iw@)& ( Pn = R;a(&),
_ &
AL#H~,, >
yn = 1.
(3.10)
We finally need to express the perturbation AR in terms of the observation covariance matrix perturbation AB k B - B for the rather general case of the
We should emphasize that many practically used transforms, like the already cited RA and PA, are linear themselves. Evidently in this case the initial transform F coincides with the associated linear transform _% E 9, the latter does not depend upon the value B since the partial derivatives in (4.2) are
constant.
A. Gorokhov et al. /Signal
5. Asymptotic distribution of DOA estimates
to if a B
Processing 55 (1996)
(l/N),uipj.
The asymptotic interpretation of Lemma 1 allows establish a linear relationship between AA and Ai the latter is a slight perturbation of B. Let d be sample covariance matrix defined by (2.4), and E #OK satisfies the condition = R.
*
(5.1) X
According to (4.1) and (5.1), the error Al estimator & 4 Y(B) is given by Aff = _%“(A& + o( IlA&).
111
The absence
of correlation
of the
/i; 1 = (2nd cos &)2 2N
l$lpi/Lj(l +6~)~~II’gl’ + o(N-'), .’ Yn
n = l,...,m.
(5.7)
The relationships (5.6)-(5.7) and (3.8)-(3.9) will be used directly for performance evaluation and convergence analysis. This main practical result can however be enhanced by the complete statistical description of DOA estimates 3
Theorem 1. Let 8 be dejned
by (2.4) and
w$xJ;,q=,: “2
5
(5.6)
(5.2)
4 i,
the
where
The relationship (5.2) could be directly substituted into (3.8). However, further evaluation of the empirical DOA variances leads to quite complicated expressions which are difficult to interpret and are not efficient for computational purposes. An alternative approach allows us to essentially simplify the task by working directly in the eigenbasis of B. First of all, we consider the eigendecomposition B = UC@, where CT = [lJi,...,U~l is a K x K unitary matrix of eigenvectors with the corresponding eigenvalues ~1 3 . . > pK SO that c = diag {pi}y!i. One c-an now transfer to the eigenbasispf B by defining A A U” AB U, or dually AB = UAIJ”. Substituting this result into (5.2) we have Ab =
between
entries of A, i.e. the components of (5.3) obviously simplifies the expressions for the empirical DOA variances. Final results are obtained from (5.4) with minor additional calculations:
var’8’] X(B)
107-115
Aij9B(Uiq).
(5.3)
j-1
‘pq = (2&)2
1
COS ep COS t$ 2N
K
Further substitution into (3.8) provides a final expression for DOA estimation errors:
-
x iFjpipj( 1 + 6ij)-‘!R{TzTy}
+ o(N-‘) (5.8)
and 9~ is dejined by F according to (4.2).
(5.4) Notice that A can be treated as the DDC estimator of the diagonal covariance matrix C, i.e., A = ;
$ Y(t)Y(t)H r-l
- c,
Y(t) = U%(t).
(5.5)
According to the standard relationship for the fourthorder moments of Gaussian variables, the elements * of A are mutually uncorrelated and var
6. Convergence analysis In this section, we consider the dependence of DOA estimation accuracy upon the structural properties of the covariance matrix B and the transform Y-. In the classical direct data version the considerable success of the subspace-based techniques is due to the rather high DOA estimation quality, especially in the case 3 All proofs are omitted here because are available upon request.
of space limitation,
they
112
A. Gorokhov et al. /Signal
of weak additive noise. This result, examined in numerous publications, can be interpreted as the trivial case of our problem with Y(B) = B. In this case, we obviously have R = B and Y(B) = B. Consequently, M = K, ;li = pi, Si = Ui, i = 1,. . . ,m and from (3.8)-(3.9) Pm+1 = . . . = ,UM = (T. It follows that the vectors CI, and /In always belong to the alternative spaces. More precisely, CI, E E,(B), LX,,E E,(B),
fi,, E E,(B) /InE E,(B)
similarly,
107-115
spaces correspondingly. We also define a set gk such that for all B E gk dim( %!s ) = k. Similarily to the above mentioned trivial example, the fast convergence condition requires that rt = 0, i, j 6m. However according to (6.1), we have the following s@icient condition: -YX Q q
V,(B)
E G(B),
&(B)H-%(Ui)$r) for the orthogonal subspace methods, for the signal subspace methods,
p,H_%‘(Vi q)a,,
E K(B)H, (6.2)
%(Uiq)&(B) (6.1)
where E,(B) is the signal subspace of F(B) (i.e. its dominant m-dimensional subspace), and E,(B) its orthogonal complement to CM. According to (5.1), we have for i, j
and,
Processing 55 (1996)
= 0. Consequently,
r! = 0 for i, j d m. Substituting this result into (5.6), one can see, that all the pipj components of the summation (or equivalently &1j) vanish for i, j
XB which are capable of preserving fast convergence due to the low-rank property of the DDC. In fact, a wide class of such transforms can be deduced from (5.6) and (5.7). To conserve the generality of results, we decompose all the eigenvalues of B into two subsets: &i’,(B) P {pi}%1 and An(B) P {/ti}fc=k+l SO that pk+t = . . . = pK = po. The associated subspaces are given by 42,(B) 4 span { Ut, . . . , Uk} and a,(B) =A span{Uk+l,..., UK}. If the matrix B is such that pk >> ~0, we say, that B is a low-rank and address 42!,(B) and an(B) as its signal and orthogonal sub-
E K(B),
En(B)H2’B(Uiq)
E En(B)H.
One should notice that a more precise description can be hardly established for the most general case. This leads us to consider the following set of matrix-valued transforms, for which we verify the fast convergence condition. These transforms are addressed as robust transforms throughout this paper.
Definition 1. A given 5 ciated 3~ satisfies (6.2).
E XB is robust if the asso-
The defined robustness property depends upon the value of B and thus can be interpreted as a local robustness. In practice, we prefer to deal with the absolutely robust transforms, i.e. the transforms which are robust for any B of our potential interest. Fortunately, the local robustness has a clear interpretation which helps to understand its nature and often simplifies classifying the transforms. To present this result, we need an additional technical constraint: for any B E i& there exists a scalar EO > 0 such that Y&%: [l-&l
3s : 9-(&B) = c&(B).
(6.3)
This asymptotic invariance to the scaling factor is certainly met in practice, usually in a more general sense, i.e. for any E > 0. Now the following property holds.
Theorem 2. A given F E X, satisfying (6.3) is locally robust if and only if there exists E > 0 such that for any 6B E SK verifying llBBl/~ < E, B+ 6B E _@?kand es(B+ 6B) = e,(B): E,(B + 6B) = E,(B). It is now easy to see that the robustness of some transform Y in the sense of fast convergence is equivalent to the asymptotic robustness of the subspace mapping,
113
A. Gorokhov et al. /Signal Processing 55 (3996) 107-115
i.e. the resulting signal subspace E, of the transform
locally depends only upon the signal subspace +YSof the argument.
7. Redundancy averaging Fig. 1. RMSE versus signal-to-noise
This approach estimates the covariance matrix by reconstructing R from the partially sampled covariante lags corresponding to each inter-element distance. The empirical covariance lags are obtained by averaging the entries of B having the same interelement distance:
Z?,,= &j-i),
i
liH = A’,
M-T
where k(r)
= c Bpp+r, p=l
0 d z < M.
(7.1)
This method is usually utilized for the intersource correlation compensation [ l&6] and a kind of initialization procedure for the ML Toeplitz covariance matrix estimation [lo]. It is also known as a structure-forcing technique in localization of independent sources, see [41. It is possible to show that the RA does not verify the robustness condition of Theorem 2 in the absolute sense. To prove this fact, one can consider a simple case of m = k = 2 and B satisfying (2.1) with some diagonal matrix C. Evidently, E,(B) = span {A}. Meanwhile, it is shown [6] that even for a slight variation of B: B’ = AC’A” + all such that C{, = Cir, Ck, = C22, C{, = Czr = E, any arbitrarily small E, causes E,(B’) # span {A}. Consequently, E,(B’) # E,(B) while U,(g) = U,(B). Since the RA is not robust, it can lead to a certain performance degradation. To demonstrate this phenomenon, we present here the asymptotic performance evaluation in terms of the empirical DOA root mean square error (RMSE) and validation by computer simulations. According to (7.1) the transform YEA is linear. The corresponding 6p coincides with ru and is given due to (4.2) by
Substituting (7.2) into (5.6) - (5.7) we calculate the RMSE of the empirical values of o(B) for the RA algorithm. Similar calculations are performed for the DDA with the same signal environment. The empirical DOA RMSE values are plotted here and later for both MUSIC and ESPRIT estimators. Solid lines correspond to the examined transform while dashed ones reflect the DDA performance. Verifications of the analytical results are provided for the Root-MUSIC version of the examined transform with 32 trails and are dotted in all figures. We consider here a ten element ULA with two independent plane waves. Fig. 1 displays DOA RMSE versus SNR for the number of snapshots N = 100 and bearings w(&) = 0.27c, a(&) = 0.3~. In contrast to the DDA, one can observe a kind of suturation phenomenon typical for nonrobust transforms, i.e. as the SNR is increased above a certain threshold value, the performance only improves slightly.
8. Persymmetrization
K =M.
(7.2)
approach
In contrast to the previous example we now consider another transform capable of preserving the lowrank property of the initial DDC. This transform was originally proposed as a kind of structure-forcing technique for symmetric linear arrays [ 141, providing an ML covariance matrix estimate in the Gaussian case. The PA is given by matrix-valued transform defined by the exchange matrix JM = {fip+q_~_r}EqX, :
(8.1) This transform appears to be linear with respect to the elements of & 9 = YEA, and can be described according to (4.2) and (8.1) as R = L.&*(B) with %‘A(&)
i,,j = 1,. . , K,
ratio: T: theory, A4: modelling.
=
M=K,
$
{hpdjq
i,j=l,...,
+
&M+l-qfi!M+l.-P};y~,
M.
,
(8.2)
A. Gorokhov et al. /Signal
114
It is now possible to establish the robustness of the PA approach when the signal subspace of the true covariance matrix is spanned by some persymmetric subspace (i.e. the signal subspace of some persymmetric matrix). Lemma 3. The PA is robust if the true value of the argument matrix B E @k admits some eigendecom-
Processing 55 (1996)
107-115
DOA RMSE, I rad
Signal-twmse
Fig. 2. RMSE versus signal-to-noise
ratio, dB
ratio: T: theory, M: modelling.
position
B= VCF, where V is an M x M unitary matrix verifying JM V
= v. This sufficient condition is verified for the general model (2.1) with non-perfectly correlated signals since the signal subspace of B coincides with the column space of A in this case. Notice that the robustness condition is satisfied if the contributing sources are not perfectly correlated, i.e. the intersource covariance matrix C has full rank. On the other hand, the PA cannot be robust in the singular case at least when rank (B) < A4 but rank(R) = M. Such a situation may occur when a total number m of plane waves reaches M. However a noticeable accuracy improvement is possible due to the PA approach if the contributing sources are highly correlated or even in the case of perfect correlation when m < M [20]. Let us now focus on the case of independent sources. Here, the essential advantage of PA has been shown for spatial filtering applications [3]. The conclusion appears to be less encouraging in the subspace based DOA estimation context, where we have the following asymptotic result. Theofern-3. Let b be dejined by (2.4) and the covariance of DOA estimates obcov{&, e,}s tained via the transform F-, i,j = 1,. . . , m. Then the classical noise subspace estimator (W = IM) and the signal subspace estimators verifv the following equality: COV{di,
Gj},$&
= COV{f?;,
dj}yDDA.
numerical example. We consider the same parameters for the array and environment as in the previous simulations. Theoretic RMSE values of the PA (or equivalently DDA) versus the SNR are shown in Fig. 2. These curves are slightly different from their empirical counterparts plotted here for the PA.
9. Summary
A generalized approach to the asymptotic analysis of the subspace-based DOA estimation accuracy is given for a specified class of arbitrary matrix-valued transforms of the Gaussian direct data covariance matrix. The analytic expressions obtained require the first-order derivatives of the examined transform and can be easily utilized especially when the matrixvalued transform is linear with respect to each element of its argument. On the other hand, these closed-form expressions allow us to identify the convergence properties of the transform when its argument verifies the so-called low-rank condition. The analytic approach is then applied to the redundancy averaging and the persymmetric approach - the two opposite examples with regard to the convergence rate. Moreover, the latter method is shown to be equivalent to the classical direct data approach for most subspacebased techniques. The achieved theoretical results are validated by computer simulations.
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(8.3)
We finally conclude that DOA estimates provided by the DDA and the PA techniques are statistically equivalent in the asymptotic domain. Some verification for this analytic conclusion is provided by the following
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1987, pp. 1517-1523. analysis of root-music”, IEEE Trans. Acoust. Speech Signal Process.. Vol. 37, December 1989, pp. 1939-1949. [I91 B. Rao and K. Hari, “Performance analysis of ESPRIT and TAM in determining the direction of arrival of plane waves in noise”, IEEE Trans. Acoust. Speech Signal Process., Vol. 37, December 1989, pp. 1990-1995. WI B. Rao and K. Hari, “Weighted subspace methods and spatial smoothing”, IEEE Trans. Signal Process., Vol. 41, February 1993, pp. 788-803. [211 J. Rissanen, “Modeling by shortest data description”. Automatica, Vol. 14, 1978, pp. 465471. WI R. Roy, A. Paulraj and T. Kailath, “Comparative performance of ESPRIT and MUSIC for direction-of-arrival estimation”, CH2461-2187 IEEE, pp. 580-584. ~231T.J. Shan and T. Kailath, ‘Adaptive beamforming for coherent signal and interference”, IEEE Trans. Acoust. Speech Signal Process., Vol. 33, December 1989, pp. 5277536. and Y. Fujikoshi, Modern ~241M. Siotani, T. Hayakawa Multivariate Statistical Analysis: A Graduate Course and Handbook (Amer. Sci. Press, Ohio, 1985). v51 P. Stoica and K. Sharman, “Maximum likelihood methods for direction of arrival estimation”, IEEE Trans. Acousr. Speech Signal Process., Vol. 38, July 1990, pp. 1132-l 143. WI M. Wax and T. Kailath, “Detection of signals by information theoretic criteria”, IEEE Trans. Acoust. Speech Signal Process., Vol. 33, No. 2, April 1985, pp. 387-392.
[I81 B. Rao and K. Hari, “Performance
PROCESSING ELSEVIER
Signal Processing 55 (1996) 107-I 15
Unified analysis of DOA estimation algorithms for covariance matrix transforms ’ A. Gorokhova>*, Yu. Abramovich”,
J.F. BGhmeb
aOdessa State Polytechnic University 270044, Av.Shevchenko 1. Odessa, bRuhr University, D-4630, Boshum I, German)
Ukraine
Received 9 November 1995
Abstract In this paper, a unified asymptotic performance analysis of the subspace-based direction-of-arrival (DOA) estimation is presented for a specified class of matrix-valued transforms of the sample covariance matrix. For the general case, involving analytic transforms, under the standard assumptions of the finite length Gaussian observation, explicit expressions for the asymptotic variances of the DOA estimates are obtained. These expressions provide a classification of matrix-valued transforms according to the error convergence rate. Typical examples such as the redundancy averaging approach and the persymmetrization algorithm are examined by the introduced technique. The asymptotic relationships are validated by computer simulations. Zusammenfassuug In dieser Arbeit wird fur eine bestimmte Klasse matrixwertiger Transformationen der empirischen Kovarianzmatrix eine vereinheitlichte asymptotische Analyse der Leistungsfahigkeit der unterraum-basierten Schatzung der Einfallsrichtung (direction of arrival, DOA) durchgefiihrt. Im allgemeinen Fall erhalten wir fur analytische Transformationen und unter der iiblichen Annahme einer GauRschen Beobachtung endlicher Lange explizite Ausdriicke fiir die asymptotischen Varianzen der DOA-Schlzwerte. Diese Ausdriicke liefem eine Klassifikation matrixwertiger Transfotmationen hinsichtlich der Fehlerkonvergenzrate. Typische Beispiele wie der “Redundancy Averaging” Ansatz und der “Persymmetrization’‘-Algorithmus werden mit Hilfe der eingefuhrten Methode analysiert. Die asymptotischen Beziehungen werden durch Computersimulationen bestitigt.
Nous presentons dans cet article une analyse unifiee des performances asymptotiques de l’estimation de direction d’arrivee (DOA) basee sur les sous-espaces, pour une classe specifique de transformations matricielles de la matrice de covariance. Dans le cas general, qui implique des transformations analytiques, sous l’hypothese standard d’une observation gaussienne de longueur finie, des expressions explicites des variances asymptotiques des estimees de la DOA sont obtenues. Ces expressions foumissent une classification des transformations matricielles en fonction du taux de convergence de l’erreur. Des exemples
* Corresponding author. Address: Department Signal, Telecom Paris (ENST), 46, rue Barrault, 75634 Paris Cedex 13, France. Tel.: 33 I 458 17547; fax: 33 1 458 87935; e-mail: [email protected]. ’ This study is supported by the SASPARC project of INTAS. 0165-1684/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO165-1684(96)00123-5
A. Gorokhov et al. /Signal Processing 55 (1996) 107-115
108
typiques tels que l’approche de moyennage de la redondance et l’algorithme de persymirtrie sont examinks k I’aide de la technique prCsentCe. Les relations asymptotiques sont valid&es par des simulations sur ordinateur. Keywords:
Array processing;
Direction-of-arrival
estimation;
1. Introduction The estimation efficiency of high-resolution subspace-based methods such as MUSIC [2], MinNorm [8], ESPRIT [ 161 have been extensively investigated [25, 18, 221 when applied to the direct data covariance (DDC) estimate. The high performance of these methods and their modifications which are achievable due to the low-rank property of the direct data approach (DDA) have been recently demonstrated [25]. On the other hand, the existing rank estimation procedures [21] for the preliminary space splitting are shown to be accurate and stable enough at quite moderate sample volumes when the DDA is considered [26]. A unified asymptotic analysis of both the noise and signal subspace approaches presented in [ 1 l] is found to be sufficiently precise and enables explicit analytic comparisons to be made for most of the methods in the direct data version. However, the DDC of the observed array process cannot be directly involved in the estimation procedures of signal parameters for several array processing applications. As an example we can consider the DOA estimation problem for spatially spread sources with known spatial decorrelation parameters. For this case a preliminary focusing technique to improve the coherence of the DDC is proposed [ 151. A similar situation occurs if the transfer functions for different spatial channels are nonindentical [l]. One can also find an indirect utilization of the DDC in the subspace-based analysis for a minimum redundancy array [ 131, when this latter is used for DOA estimation of multiple uncorrelated plane waves where there are more sources than actual array elements [ 171. Some other methods improve the DOA estimation accuracy due to some preliminary modifications of the DDC prior to further subspace-based analysis. In this context we may emphasize the persymmetrization algorithm (PA) [ 141 which retrieves the ML persymmetric covariance matrix estimate assuming that the true covariance matrix has the persymmetric structure. Similarly, the redundancy averaging (RA) [12]
Performance
analysis
and spatial smoothing [23] techniques are often used to decrease the impact of the intersource correlation on the DOA estimation performance. The estimation approaches mentioned above along with many others can be represented in the most general case as a two-step procedure: (i) perform a certain matrix-valued transform of the initial DDC, (ii) apply the subspace-based DOA estimation analysis to the obtained matrix. Hence, the DOA estimation accuracy analysis for the case of a matrix-valued transform of the DDC appears to be of practical interest. Our aim in this paper is to present a unified approach to the statistical analysis of the existing subspace-based DOA estimation methods applied to the generalized matrixvalued transform of the DDC. We also examine the impact of the matrix valued transform on the convergence of the DOA estimates. As a matter of fact, the initial low-rank property of the DDC often vanishes after its modification, as in the TAM algorithm [9]. In Section 6 of this paper, we establish the sufficient condition to preserve fast convergence for the case of a low-rank DDC. This condition implies certain constraints on the utilized transform and has a clear interpretation in terms of the subspace-based analysis. Sections 7 and 8 present the results of DOA estimation performance analysis when the RA and PA techniques are exploited. Analytic relationships for both methods are verified by computer simulations.
2. Preliminaries Let us consider an M x A4 covariance matrix duced by an output of some uniform linear (ULA) of dimension M. We assume that the process is a mixture of m plane waves with e,,... , I!$,,and spatially white noise ’ of power R = ACAH + oZM, *The superscripts conjugate
T, H and # stand for transpose, and pseudoinverse.
R proarray output DOA (T: (2.1)
transpose
A. Gorokhov et al. 1 Signal Processing 55 (1996)
where C is an m x m intersource covariance matrix, A is an M x m matrix of array response vectors being complex sinusoids with the frequencies o(&) = 2n(d/&)sin9,, n = l,...,m:
(2.3)
where S= [Sl,. ,S,] and A= diag {ni}y=r. We are considering a K-element actual antenna array with an arbitrary geometry and an output signal X(t) E (CKhaving the complex multivariable Gaussian distribution: X(t) N CN(K, 0, B) [24]. The empirical spatial covariance matrix of the process X(t) can be traditionally obtained on the basis of N independent observations X(t), t = 1,. . . , N, as follows: d = ;
g X(t)X(t>H. r-l
in 0 with the null-spectrum
~(0) = a(0)“filVfia(0).
(3.1) the
(2.2)
d is the interelement distance of the considered ULA, 0 is the DOA with respect to the array broadside and E., is the central wavelength of the examined narrowband process. Let 1.12 . . 3 i., be the m largest eigenvalues of the matrix R, im > CJ, with the corresponding set of eigenvectors SI , . . . , S,,, and Il a projector onto the orthogonal (noise) subspace. Then one can write down the eigendecomposition of R as follows: R=SAffolI,
problem
Alternatively, they can be obtained by retrieving roots of the polynomial
A = [a(O, ). . . ,a(&,)], a(e) = [,-io(@(M- I)@,. . , eiu(Q)(M--l1121 T,
minimization function:
109
107-115
(2.4)
Suppose now that there exists some consistent estimate R of the covariance matrix R obtained by a certain transform .Y of the sample covariance matrix & i.e., k = Y(b). Consequently, the DOA parameters 81,. . , 0, can also be consistently estimated by applying the subspace-based DOA estimation algorithms to li.
3. Asymptotic DOA estimation error We need a common description of the existing subspace-based estimators. Most of them can be classified as either signal or orthogonal subspace methods. The latter ones essentially exploit some estimate I? of the orthogonal subspace projector Il so that the empirical DOA parameters are given by solving a
@)
= [Z-(M--lf/2 ,.,.,= (M-i)12]T,
(3.2)
where w is some non-negative weighting matrix. Some more detailed descriptions of the estimators can be found in [ 1 I]. We just notice here that fi is the empirical counterpart of a certain weighting matrix W. In the most trivial cases, one has W = ZM (the standard MUSIC) or W = {6,~6,~}$, (MinNorm). Both spectral and root versions are shown [ 111 to be asymptotically equivalent. Let us introduce the estimation errors Ail p I? -- ll andAB, a t?,-e,,n= l,...,m.Thensomestraightforward calculations similar to those given in [l l] provide the asymptotic DOA: A& = -
‘“C
2ndcos8,
Cs{a(B,,)“Afi
WIZa,(~,)}
a1(&,)HllW17Ha~(&,)
+o(IlAfill~)> a,(e)
= [ - i(M _ l)e-iw(H)(M-‘)i2,, $(A4 - l)e
(3.3) .,
io(O)(M- I ),12T 3 ’
where II IIF is the Frobenious norm and !R{.}, S{_.} stand here and later for the real and imaginary part. Notice that (3.3) holds for both spectral (3.1) and root (3.2) criteria. Meanwhile the variety of signal subspace methods like ESPRIT, state-space realization (SSR) [9] and matrix-pencil method [5] are identical even in the noisy case [ 191. The equivalent DOA estimates can be found as described in [ 111, via the eigenvalues of a matrix p = i’#,!?‘, where ,!? is a sample counterpart of S, 1 (T) denotes the submatrix obtained from the original one by excluding its last (first) row. The DOA estimation errors, similar to [ 111, are given by
110
A. Gorokhov et al. /Signal
where e, = [S,]Er and 6in is a Kronecker delta. The results cited above [ 1 l] allow us to use: - the asymptotic expression (3.3) for both the spectral and root orthogonal subspace estimators with the arbitrary weighting; _ the asymptotic expression (3.4) for the variety of signal subspace based methods; Since the actual analysis concerns the asymptotic error values, we only examine first order terms of Taylor series throughout this paper and use the ‘=’ mark to reflect the equality of modulo to higher-order terms. According to the relationships (3.3), (3.4), one needs the asymptotic representation of Al? and II&?. The appropriate expressions in AR p R - R
can be conventionally obtained through the asymptotic eigenprojector perturbation technique [7]. Denoting Ro A R - old,one has ri = Ii’ - R!ARll - IIARR; + o(]]A&),
(3.5)
Processing 55 (1996)
107-115
transform Y. This latter requires some unified asymptotic description of the mapping AB -+ AR.
4. Generalized matrix-valued transform be a transform mapping the space of Hermitian K x K matrices into the space of Hermitian M x M matrices. We consider here a class XB of such transforms, continuously differentiable in some neighbourhood of B E XK. Let F : ZK -+ SM
Lemma 1. A given transform T : XK 4 XM verifies ~7 E XB for some B E XK ifs T(b)
- 9-(B) = Y#?
- B) + o( I@ - BllF), (4.1)
where _%‘B is a linear transform .2$ : CKxK + CM xM dejined as follows:
ii?=~~-fi=~SSH-t~~Aih +IIARR;
+ o( I]A&).
Taking into account the asymptotic ZI(S,!? + &!?) Il&n
(3.6)
identity IIZ??
=
= II&Srr we find from (3.6):
= IIARR;.
(3.7)
Plugging (3.5) into (3.3) and (3.7) into (3.4), one can easily find the following unified expression: 2nd
CO&,
Y,,
’
(3.8)
where CI, and /I,, are M-component complex vectors and yn are positive scalars defined as follows: - Orthogonal subspace methods: %I = R$+%J, yn =
Zij
=
[6ip6jq]:,,=,,
i, j = 1,. . . ,K.
(4.2)
Consequently, once a given transform F is continuously differentiable in some neighborhood of B E XK, its perturbation at the point Y(B) can be asymptotically represented as a certain linear transform of the argument perturbation. Such a transform can be described by the partial derivatives of its elements with respect to the elements of the argument matrix according to (4.2). One useful property is the following. Lemma 2. For any 5
E XB the associated 2~ veri-
fin = nwn%(e,),
al(e,)HnwnHal(e,).
(3.9)
kf’BE .sK.
(4.3)
- Signal subspace methods: c1n = n ,iw@)& ( Pn = R;a(&),
_ &
AL#H~,, >
yn = 1.
(3.10)
We finally need to express the perturbation AR in terms of the observation covariance matrix perturbation AB k B - B for the rather general case of the
We should emphasize that many practically used transforms, like the already cited RA and PA, are linear themselves. Evidently in this case the initial transform F coincides with the associated linear transform _% E 9, the latter does not depend upon the value B since the partial derivatives in (4.2) are
constant.
A. Gorokhov et al. /Signal
5. Asymptotic distribution of DOA estimates
to if a B
Processing 55 (1996)
(l/N),uipj.
The asymptotic interpretation of Lemma 1 allows establish a linear relationship between AA and Ai the latter is a slight perturbation of B. Let d be sample covariance matrix defined by (2.4), and E #OK satisfies the condition = R.
*
(5.1) X
According to (4.1) and (5.1), the error Al estimator & 4 Y(B) is given by Aff = _%“(A& + o( IlA&).
111
The absence
of correlation
of the
/i; 1 = (2nd cos &)2 2N
l$lpi/Lj(l +6~)~~II’gl’ + o(N-'), .’ Yn
n = l,...,m.
(5.7)
The relationships (5.6)-(5.7) and (3.8)-(3.9) will be used directly for performance evaluation and convergence analysis. This main practical result can however be enhanced by the complete statistical description of DOA estimates 3
Theorem 1. Let 8 be dejned
by (2.4) and
w$xJ;,q=,: “2
5
(5.6)
(5.2)
4 i,
the
where
The relationship (5.2) could be directly substituted into (3.8). However, further evaluation of the empirical DOA variances leads to quite complicated expressions which are difficult to interpret and are not efficient for computational purposes. An alternative approach allows us to essentially simplify the task by working directly in the eigenbasis of B. First of all, we consider the eigendecomposition B = UC@, where CT = [lJi,...,U~l is a K x K unitary matrix of eigenvectors with the corresponding eigenvalues ~1 3 . . > pK SO that c = diag {pi}y!i. One c-an now transfer to the eigenbasispf B by defining A A U” AB U, or dually AB = UAIJ”. Substituting this result into (5.2) we have Ab =
between
entries of A, i.e. the components of (5.3) obviously simplifies the expressions for the empirical DOA variances. Final results are obtained from (5.4) with minor additional calculations:
var’8’] X(B)
107-115
Aij9B(Uiq).
(5.3)
j-1
‘pq = (2&)2
1
COS ep COS t$ 2N
K
Further substitution into (3.8) provides a final expression for DOA estimation errors:
-
x iFjpipj( 1 + 6ij)-‘!R{TzTy}
+ o(N-‘) (5.8)
and 9~ is dejined by F according to (4.2).
(5.4) Notice that A can be treated as the DDC estimator of the diagonal covariance matrix C, i.e., A = ;
$ Y(t)Y(t)H r-l
- c,
Y(t) = U%(t).
(5.5)
According to the standard relationship for the fourthorder moments of Gaussian variables, the elements * of A are mutually uncorrelated and var
6. Convergence analysis In this section, we consider the dependence of DOA estimation accuracy upon the structural properties of the covariance matrix B and the transform Y-. In the classical direct data version the considerable success of the subspace-based techniques is due to the rather high DOA estimation quality, especially in the case 3 All proofs are omitted here because are available upon request.
of space limitation,
they
112
A. Gorokhov et al. /Signal
of weak additive noise. This result, examined in numerous publications, can be interpreted as the trivial case of our problem with Y(B) = B. In this case, we obviously have R = B and Y(B) = B. Consequently, M = K, ;li = pi, Si = Ui, i = 1,. . . ,m and from (3.8)-(3.9) Pm+1 = . . . = ,UM = (T. It follows that the vectors CI, and /In always belong to the alternative spaces. More precisely, CI, E E,(B), LX,,E E,(B),
fi,, E E,(B) /InE E,(B)
similarly,
107-115
spaces correspondingly. We also define a set gk such that for all B E gk dim( %!s ) = k. Similarily to the above mentioned trivial example, the fast convergence condition requires that rt = 0, i, j 6m. However according to (6.1), we have the following s@icient condition: -YX Q q
V,(B)
E G(B),
&(B)H-%(Ui)$r) for the orthogonal subspace methods, for the signal subspace methods,
p,H_%‘(Vi q)a,,
E K(B)H, (6.2)
%(Uiq)&(B) (6.1)
where E,(B) is the signal subspace of F(B) (i.e. its dominant m-dimensional subspace), and E,(B) its orthogonal complement to CM. According to (5.1), we have for i, j
and,
Processing 55 (1996)
= 0. Consequently,
r! = 0 for i, j d m. Substituting this result into (5.6), one can see, that all the pipj components of the summation (or equivalently &1j) vanish for i, j
XB which are capable of preserving fast convergence due to the low-rank property of the DDC. In fact, a wide class of such transforms can be deduced from (5.6) and (5.7). To conserve the generality of results, we decompose all the eigenvalues of B into two subsets: &i’,(B) P {pi}%1 and An(B) P {/ti}fc=k+l SO that pk+t = . . . = pK = po. The associated subspaces are given by 42,(B) 4 span { Ut, . . . , Uk} and a,(B) =A span{Uk+l,..., UK}. If the matrix B is such that pk >> ~0, we say, that B is a low-rank and address 42!,(B) and an(B) as its signal and orthogonal sub-
E K(B),
En(B)H2’B(Uiq)
E En(B)H.
One should notice that a more precise description can be hardly established for the most general case. This leads us to consider the following set of matrix-valued transforms, for which we verify the fast convergence condition. These transforms are addressed as robust transforms throughout this paper.
Definition 1. A given 5 ciated 3~ satisfies (6.2).
E XB is robust if the asso-
The defined robustness property depends upon the value of B and thus can be interpreted as a local robustness. In practice, we prefer to deal with the absolutely robust transforms, i.e. the transforms which are robust for any B of our potential interest. Fortunately, the local robustness has a clear interpretation which helps to understand its nature and often simplifies classifying the transforms. To present this result, we need an additional technical constraint: for any B E i& there exists a scalar EO > 0 such that Y&%: [l-&l
3s : 9-(&B) = c&(B).
(6.3)
This asymptotic invariance to the scaling factor is certainly met in practice, usually in a more general sense, i.e. for any E > 0. Now the following property holds.
Theorem 2. A given F E X, satisfying (6.3) is locally robust if and only if there exists E > 0 such that for any 6B E SK verifying llBBl/~ < E, B+ 6B E _@?kand es(B+ 6B) = e,(B): E,(B + 6B) = E,(B). It is now easy to see that the robustness of some transform Y in the sense of fast convergence is equivalent to the asymptotic robustness of the subspace mapping,
113
A. Gorokhov et al. /Signal Processing 55 (3996) 107-115
i.e. the resulting signal subspace E, of the transform
locally depends only upon the signal subspace +YSof the argument.
7. Redundancy averaging Fig. 1. RMSE versus signal-to-noise
This approach estimates the covariance matrix by reconstructing R from the partially sampled covariante lags corresponding to each inter-element distance. The empirical covariance lags are obtained by averaging the entries of B having the same interelement distance:
Z?,,= &j-i),
i
liH = A’,
M-T
where k(r)
= c Bpp+r, p=l
0 d z < M.
(7.1)
This method is usually utilized for the intersource correlation compensation [ l&6] and a kind of initialization procedure for the ML Toeplitz covariance matrix estimation [lo]. It is also known as a structure-forcing technique in localization of independent sources, see [41. It is possible to show that the RA does not verify the robustness condition of Theorem 2 in the absolute sense. To prove this fact, one can consider a simple case of m = k = 2 and B satisfying (2.1) with some diagonal matrix C. Evidently, E,(B) = span {A}. Meanwhile, it is shown [6] that even for a slight variation of B: B’ = AC’A” + all such that C{, = Cir, Ck, = C22, C{, = Czr = E, any arbitrarily small E, causes E,(B’) # span {A}. Consequently, E,(B’) # E,(B) while U,(g) = U,(B). Since the RA is not robust, it can lead to a certain performance degradation. To demonstrate this phenomenon, we present here the asymptotic performance evaluation in terms of the empirical DOA root mean square error (RMSE) and validation by computer simulations. According to (7.1) the transform YEA is linear. The corresponding 6p coincides with ru and is given due to (4.2) by
Substituting (7.2) into (5.6) - (5.7) we calculate the RMSE of the empirical values of o(B) for the RA algorithm. Similar calculations are performed for the DDA with the same signal environment. The empirical DOA RMSE values are plotted here and later for both MUSIC and ESPRIT estimators. Solid lines correspond to the examined transform while dashed ones reflect the DDA performance. Verifications of the analytical results are provided for the Root-MUSIC version of the examined transform with 32 trails and are dotted in all figures. We consider here a ten element ULA with two independent plane waves. Fig. 1 displays DOA RMSE versus SNR for the number of snapshots N = 100 and bearings w(&) = 0.27c, a(&) = 0.3~. In contrast to the DDA, one can observe a kind of suturation phenomenon typical for nonrobust transforms, i.e. as the SNR is increased above a certain threshold value, the performance only improves slightly.
8. Persymmetrization
K =M.
(7.2)
approach
In contrast to the previous example we now consider another transform capable of preserving the lowrank property of the initial DDC. This transform was originally proposed as a kind of structure-forcing technique for symmetric linear arrays [ 141, providing an ML covariance matrix estimate in the Gaussian case. The PA is given by matrix-valued transform defined by the exchange matrix JM = {fip+q_~_r}EqX, :
(8.1) This transform appears to be linear with respect to the elements of & 9 = YEA, and can be described according to (4.2) and (8.1) as R = L.&*(B) with %‘A(&)
i,,j = 1,. . , K,
ratio: T: theory, A4: modelling.
=
M=K,
$
{hpdjq
i,j=l,...,
+
&M+l-qfi!M+l.-P};y~,
M.
,
(8.2)
A. Gorokhov et al. /Signal
114
It is now possible to establish the robustness of the PA approach when the signal subspace of the true covariance matrix is spanned by some persymmetric subspace (i.e. the signal subspace of some persymmetric matrix). Lemma 3. The PA is robust if the true value of the argument matrix B E @k admits some eigendecom-
Processing 55 (1996)
107-115
DOA RMSE, I rad
Signal-twmse
Fig. 2. RMSE versus signal-to-noise
ratio, dB
ratio: T: theory, M: modelling.
position
B= VCF, where V is an M x M unitary matrix verifying JM V
= v. This sufficient condition is verified for the general model (2.1) with non-perfectly correlated signals since the signal subspace of B coincides with the column space of A in this case. Notice that the robustness condition is satisfied if the contributing sources are not perfectly correlated, i.e. the intersource covariance matrix C has full rank. On the other hand, the PA cannot be robust in the singular case at least when rank (B) < A4 but rank(R) = M. Such a situation may occur when a total number m of plane waves reaches M. However a noticeable accuracy improvement is possible due to the PA approach if the contributing sources are highly correlated or even in the case of perfect correlation when m < M [20]. Let us now focus on the case of independent sources. Here, the essential advantage of PA has been shown for spatial filtering applications [3]. The conclusion appears to be less encouraging in the subspace based DOA estimation context, where we have the following asymptotic result. Theofern-3. Let b be dejined by (2.4) and the covariance of DOA estimates obcov{&, e,}s tained via the transform F-, i,j = 1,. . . , m. Then the classical noise subspace estimator (W = IM) and the signal subspace estimators verifv the following equality: COV{di,
Gj},$&
= COV{f?;,
dj}yDDA.
numerical example. We consider the same parameters for the array and environment as in the previous simulations. Theoretic RMSE values of the PA (or equivalently DDA) versus the SNR are shown in Fig. 2. These curves are slightly different from their empirical counterparts plotted here for the PA.
9. Summary
A generalized approach to the asymptotic analysis of the subspace-based DOA estimation accuracy is given for a specified class of arbitrary matrix-valued transforms of the Gaussian direct data covariance matrix. The analytic expressions obtained require the first-order derivatives of the examined transform and can be easily utilized especially when the matrixvalued transform is linear with respect to each element of its argument. On the other hand, these closed-form expressions allow us to identify the convergence properties of the transform when its argument verifies the so-called low-rank condition. The analytic approach is then applied to the redundancy averaging and the persymmetric approach - the two opposite examples with regard to the convergence rate. Moreover, the latter method is shown to be equivalent to the classical direct data approach for most subspacebased techniques. The achieved theoretical results are validated by computer simulations.
References
(8.3)
We finally conclude that DOA estimates provided by the DDA and the PA techniques are statistically equivalent in the asymptotic domain. Some verification for this analytic conclusion is provided by the following
[I] Y. Abramovich, V. Kachur and V. Mikhailukov, “The effectiveness of spatial interference compensation in systems with nonidentical reception channels”, Sov. J. Corn. Techno/. Electron., Vol. 35, No. 2, 1990, pp. 96104. (English translation of radiotekhnica i electronica, No. 6, 1989, pp. 11961205).
A. Gorokhov
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