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SECOND-ORDER VERSUS FOURTH-ORDER MUSIC ALGORITHMS : AN ASYMPTOTICAL STATISTICAL ANALYSIS Eric Moulines, Jean-François Cardoso Télécom Paris - CNRS URA 820 - GdR TdSI 46 rue Barrault, 75634 Paris Cedex 13, France. E-mail : [email protected] ABSTRACT Direction finding techniques are usually based on the 2ndorder statistics of the received data. In this paper, we propose a MUSIC-like direction finding algorithm which uses a matrix-valued statistic based on the contraction of the 4thorder cumulant tensor of the array data (4-2 MUSIC). We then derive, in a unified framework, the asymptotic covariance of estimation errors for both 2nd-order and 4th-order based MUSICs, and show that the 4th-order method can perform equally well or even better than the 2nd-order method, even when the noise spatial structure is known. geometry, see [9]). Another approach is to use a matrix-valued statistics based on Hybrid Non-Linear Moments : in the ULA case, an appealling statistic is the cross-covariance matrix between the signal and its coordinate-wise distortion by a zero-memory non linearity [7,10]. Given y the complex random vector at the array output, we propose to estimate : R̃ = E( y ∗ My y y ∗ ) and R = E(yy ∗ ) and to form the following matrix-valued statistics, depending on M : 1. INTRODUCTION Q(M) = R̃ − R M R − R Tr( MR ) Current array processing techniques are based on the second-order statistics of the received signals. In many situations, in particular in digital communications, received signals are non-Gaussian so that they contain valuable statistical information in their moments of order greater than two. In these circumstances, it makes sense to develop array processing techniques that exploit higher-order information [1-7]. Of particular interest are the higher-order cumulants, which enjoy the very appealing property of being asymptotically insensitive to Gaussian noise : it is thus neither necessary to know, to model, nor to estimate the noise covariance, as long as the noise is Gaussian, which is a reasonnable assumption for most communications media [8]. Note that M is an arbitrary hermitian matrix. We demonstrate in the following that Q (M) is the result of the contraction of the 4th-order array output cumulant tensor with the matrix M. This approach offers two distinct advantages over [1,7] : first, Q(M) provides directly the signal subspace even when signals are correlated and for any array geometry (at least when M is definite positive) ; in contrast, the methods in [1,7] are designed for the ULA case and independent sources. Second, the "natural" sample estimates of Q (M) are hermitian : no further processing is needed for eigendecomposition. Finally, there is an extra degree of freedom in the choice of weighting matrix M : it can be shown that M acts much like a spatial beamformer, that can focus on specific look directions. Here, we consider one particular array processing problem : high-resolution direction finding. The most popular methods for estimating the direction-of-arrivals (DOAs) of signals impinging on an array of sensors are based on the concept of the signal subspace. This subspace is obtained, in the standard MUSIC technique, from the eigendecomposition of the sample covariance matrix. Besides the derivation of a new method, we present, in a unified framework, the asymptotic covariance of estimation errors for both 2nd- and 4th-order MUSIC, and we show that, in some practical situations, 4th-order based method can perform equally well or even better than 2nd-order method, even when the noise spatial structure is known. Some methods have been recently proposed [3-4] that exploit the full set of 4th-order cumulants. These approaches offer some distinctive advantages over 2nd-order MUSIC but the price to pay is a cumulant estimation cost that grows as m 4 (m being the number of sensors), while covariance estimation cost grows as m 2 . Hence, it is interesting to consider cumulant based m×m matrix-valued statistics. Such statistics are, of course, required to provide consistent estimates of the true signal subspace. Several approaches have been proposed. The most natural is to use a m 2 subset (or slice) of higher-order cumulants : the selection of an appropriate slice depends on the symmetry of the problem. In the Uniform Linear Array case (ULA), the 4th-order diagonal cumulant slice is a reasonable choice [1] (for extensions to arbitrary array The paper is organized as follows. Section 2 formulates the problem and establishes model assumptions. Section 3 gives the expression of the contracted 4th-order cumulant tensor and shows how it can be used to obtain the DOA’s. Section 4 investigates the asymptotic distribution of the resulting bearing estimates and compares it, in the single source case, with the asymptotic distribution of 2-MUSIC. 2. PROBLEM FORMULATION Assume an array of m sensors, n narrowband far-field emitters and define a ( θ ) ∈ C m to be the array response for a narrowband emitter at DOA θ. The array output y ( t ) ∈ C m is given by y(t) = As ( t ) + n ( t ) 3. ESTIMATION METHODS where s ( t ) = [ s 1 (t),..., sn (t) ]T and . . . A = [ a ( θ1 ), , a ( θn ) ]. The signal emitted by the p-th source, sp (t), is a stationary non-Gaussian complex random variable. Signals emitted at different times, by the same source or by different sources, are assumed to be statistically independent. The additive n (t) is a complex Gaussian noise, with arbitrary spatial structure. The array output is sampled at N distinct time instants. Based on these measurements y ( t ) ,1 ≤ t ≤ N , the problem of interest is to determine the DOAs θp , 1 ≤ p ≤ n . As explained in the introduction, we focus here on a class of algorithms, exploiting only 4th-order cumulants only, while keeping the same complexity as the standard 2nd-order methods. The main idea is to contract the AOQ tensor. Let M ij be a (1,1) hermitian tensor, that is, a linear self-adjoint operator on C m . The contraction consists in summing over a pair of indices, which yields In the following, we assume that signals and noise are complex circular, which means, omitting the dependence on t : E ( si si . . . si s ∗i . . . s ∗i ) = 0 p≠q 1 2 p p +1 p +q Due to inherent symmetries in symbol constellation, such a condition is approximately fulfilled by digital communication signals of PSK type (at least when the signals are independent). Circularity implies that odd-order statistical moments of the array output are zero. Therefore, we will concern ourselves only with the even-order moments, and, in particular the 4th-order moments. These are given by µ ij lk = E ( yi y j y k yl ) where y j (resp. y j ) denotes the j-th component of y (resp. y ∗ ). Under the circularity assumption, 4th-order cumulants are expressed in terms of 4th- and 2nd-order moments as : Q ij lk = Cum ( yi , y j , y k , yl ) Q ij lk = µ ij lk − µi k µl j − µi j µl k The set of four index quantities Q ij lk defines a (2,2)-tensor [9]. This tensor is referred to as the Array Output Quadricovariance (AOQ). In the same manner, we can also define the source signal quadricovariance (SSQ) : K βα δδ = Cum ( s α , s β , s γ , s δ ) We finally drop the explicit dependence of a on θ ; the concise notation a αi (respectively a− iα ) will now stand for the ith component of the directional vector associated with the DOA θα , i.e a(θα ) (respectively a ∗ (θα )). Using these conventions, we directly get, thanks to cumulant multilinearity and additivity properties Q ij lk = a αi a− βj a− kγ a δl K βα γδ whith no noise contribution thanks to the Gaussian hypothesis. We have used here the standard summation convention : summation over any index repeated once as a superscript and once as a subscript is implicit. The range of summation is not stated explicitely but is implied by the position of the indices. In principle, both the 2nd-order and the 4th-order cumulants can be used to estimate the DOAs. However, the use of the 2nd-order moments requires to know, to model or to estimate the noise covariance. As long as the noise is Gaussian, the use of 4th-order cumulants eliminates this need. jk l Q(M) ij ∆ = Qi l Mk Since Q is a tensor, the contraction has a simple algebraic interpretation. Let F denote the space of the linear operators on C m ; the space of (2,2) tensors can be canonically associated with the space of linear operators on F. The (1,1) tensor Q(M) resulting from the contraction, is the image by the AOQ of the operator M (cf appendix). From the above expressions we get a αi a− βj ( a− kγ M lk a δl K βα γδ ) = a αi a− βj K βα γδ ( A H M A) δγ In "index-free" formalism, this last equality can be more concisely written Q (M) = A C A H where C is the contraction of the SSQ tensor on the matrix A H M A, i.e C = K (A H M A). The last relation is strikingly similar to the one obtained at 2nd-order with Q (M) in place of the array output covariance matrix and C in place of the signal covariance. Since only 4th-order cumulants are used, the properties of the measurement noise do not appear in the expression of the exact statistics. Q (M) is rank defective, with a rank equal to the rank of C, and its range is spanned by the steering vectors. Hence Q (M) can serve as a basis for standard DOA estimation methods. From now on, we will concern ourselves only with MUSIC-type algorithms : a MUSIC algorithm based on Q (M) will be, for obvious reasons, referred to as 4-2-MUSIC. Note that M is a "free" parameter which can be adjusted to increase DOA performances (see next section). In absence of any information on DOA and on spatial correlation of the noise, a neutral choice is : M = I. In presence of known (or even unknown) coherent interferences, M can be chosen to "steer nulls" in the direction of the interferers, while ensuring that signals received from the desired directions pass unattenuated. Finally, M could be replaced by a consistent estimate of an appropriate statistic. An interesting choice, the inverse of signal covariance, is discussed below. 4. PERFORMANCE ANALYSIS In this section, we sketch the derivation of the statistical performances of 4-2-MUSIC DOA estimation (details will be published elsewhere) and compare these performances with those achieved by 2-MUSIC. In order to avoid index invasion, we use in the following a few special notations that are defined in appendix. We base our analysis on the following continuity theorem : Let the sample statistic ŜN be a strongly consistent estimate of N ( ŜN − S ) is asymptotically S, i.e. ŜN → S (a.s) and √ Gaussian with finite asymptotic covariance Σ( Ŝ ) ∆ = lim ( N Cov( ŜN − S ) ) < ∞ N→∞ V̂N ∆ = g ( ŜN ) is a strongly consistent estimate of then V∆ = g ( S ), provided that g (.) satisfies some regularity conditions. Moreover, lim N Cov( V̂N − V ) ) = g ′ ( S ) Σ( Ŝ ) g ′ ( S )H N→∞ In the above formulation, we have not specified the algebraic nature of ŜN (and S) because the continuity theorem holds regardless ofthe nature of ŜN : a scalar, a vector, or even a linear operator on an Hilbert space. To assess the statistical perfomances of MUSIC-type methods, we apply the continuity theorem twice : we first relate the covariance matrix of DOA estimates θ̂ to the covariance of Π̂ , the estimate of Π which is the orthogonal projector onto the noise-subspace. Second, we calculate the covariance of Π̂ from the covariance of the sample statistics. The MUSIC estimates are obtained by searching the single variable localisation function | Π̂ a(θ) | 2 = Tr( Π̂ a (θ)a (θ)∗ ) = Π̂ ∗ a (θ)a (θ)∗ for its minima. The DOA estimates θ̂ are thus related to the projector onto the sample noise subspace Π̂ by a set of n equations ∗ ∂S( θ̂ ) Π̂ = 0 ∂θi i = 1, . . . , n H where S(θ) ∆ = A (θ) A (θ) . This set of n implicit equations determines θ̂ as a function of Π̂ . The first derivative of this function with respect to each θi can be shown to be the (1,1)tensor ∂ θ̂ i ∂ Π̂ = − h(θi )−1 ∂S(θ) ∗ ∂θi ∗ 2 2 where h ( θ ) ∆ = Π ∂ S/∂θ . Provided that Π̂ is a consistent estimate of Π, it follows directly from the continuity theorem and the above equation that θ̂ are consistent estimates of the true parameter θ, with asymptotic covariance given by : −1 Σ( θ̂i , θ̂ j ) = h ( θi ) h ( θ j ∂S ( θ ) )−1 ∂θi ∗ ∂S ( θ ) Σ( Π̂ ) ∂θ j This expression is a MUSIC "built-in" feature : it is a consequence of the way the method exploits the information provided by the sample noise projector. To determine the asymptotic covariance Σ( Π̂ ), we use the following lemma, which relates the perturbation of a projector on an invariant subspace to the perturbation of the operator itself [12]. Lemma : Let B be an hermitian operator and λ0 be an eigenvalue of B. The orthogonal projector Π( B ) onto the invariant subspace associated with λ0 is an (infinitely) differentiable function of B. Its first derivative, denoted Π′(B) is the (2,2)-hermitian tensor : Π′( B ) = [ Π , Γ ]2 where Γ is # Γ∆ = ( λ0 I − B ) the pseudo-inverse of ( λ0 I − B ) : We now assume, for a fair comparizon with 2-MUSIC, that the noise is spatially white with power σ. For 2-MUSIC, the sample noise projector Π̂ is estimated from the eigendecomposition of the sample covariance matrix R̂. Since R̂ is a consistent estimator of array output covariance R, Π̂ is a consistent estimator of Π, which corresponds, in this case, to the invariant subspace of R associated with the eigenvalue σ. We can therefore apply directly the above lemma, with Γ = (σI − R)# . The continuity theorem then yields : Σ2 ( Π̂ ) = [ Π , Γ ]2 Σ( R̂ ) [ Π , Γ ]2 The last point is to evaluate the asymptotic covariance of the sample covariance R̂. This result can be directly derived, in the Gaussian case, from the central moments of the Wishart distribution. It can be established as well in the non-Gaussian case, from standard formula on cumulants of ordinary kstatistics [11] : Σ( R̂ ) = Q + R ⊗2 R. Combining these two results, together with ΠR = RΠ = σΠ and Q (Π)=0, we get : Σ2 ( Π̂ ) = σ [ Π , Γ R Γ ]2 (1) The covariance of the DOAs can finally be obtained from the above expressions. Note that this formula goes beyond "traditionnal" performance analysis results, since we have assumed a non-Gaussian model for the signals [15]. It proves an important result (see also [14]) : the exact joint pdf’s of the sources has no influence on the performance of 2-MUSIC (the only quantity of interest is the source covariance matrix). The covariance of the projector for 4-2-MUSIC is denoted Σ4−2 (Π) and can be derived in a similar manner ; the sample projector is a consistent estimator of the orthogonal projector onto the noise subspace Π, which corresponds to the kernel of Q (M). The projector perturbation formula can therefore be applied with Γ=Q (M)# , the pseudo-inverse of Q (M). All the difficulty lies in the derivation of the covariance of the contracted quadricovariance [11]. It can be demonstrated that : Σ4−2 ( Π ) = σ [ Π , Γ L Γ ]2 + [ ΓRΓ , ΠRMRMRΠ ]2 + [ ΠRMRΓ , ΓRMRΠ ]1 (2) where L is a (1,1)-tensor depending on the cumulants of the signals up to order six jkn m l L ij = H ilm Mn Mk kn l j jk m n l nk l m j jk m l n + Q lm M k M m n R i + 2Q il M k R m M n + Q il M k M n R m + Q ml M n M k R i j k m n k l k l n m j + Ri Rl Mk Rm Mn Rk + Ri Mk Rl Mn Rm and where the "hexacovariance" (!) H is of course defined by jkn ∆ j k n H ilm = Cum(yi ,y ,yl ,y ,ym ,y ). In its basic version, 4-2-MUSIC uses the identity as a weighting matrix : M = I and eq. (2) takes the simpler form Σ4−2 ( Π ) = σ [ Π , Γ L̃ Γ ]2 j L̃ i (3) jab ja b ab j = H iab + R ij Q ab ab + 2 Q ib R a + Q ib R a jb a j a b a b j + Q ab R i + R i R b R a + R i R a R b + σ2 R ij Expression (3) involves the (pseudo)-inverse of the AOQ, Γ, twice. This suggests that, provided the kurtosis are large enough, 4-2-MUSIC methods can perform better than the 2MUSIC method, even when the noise cross-spectrum matrix is known. An identical conclusion is drawn in [5]. This can be checked analytically in the case of a single source with nonvanishing r-th order cumulant k (r) . Denote k̃ (r) the standardized r-th order cumulant, i.e. k̃ (r) = k (r) /k r(2)/2 , and ρ the Signal-to-Noise Ratio (SNR), ρ = k (2) /σ. Let us first choose M = I. The asymptotic variances for 2-MUSIC and 4-2MUSIC are respectively equal to Σ2 ( θ̂ ) = Σ4−2 ( θ̂ ) = a0 a1 a2 a3 −1 +ρ 1 ρ h (θ) a + a ρ−1 + a ρ−2 + a ρ−3 3 012 2 ρ h (θ)k̃ (4) = 2 + 5 k̃ (4) + k̃ (6) = 6 + 5 k̃ (4) = m +6 = m +2 For ULAs, the geometrical factor h ( θ ) is shown to be inversely proportional to the number of sensors m, so that the asymptotic covariance of 4-2-MUSIC does decrease as m is increased. To get more quantitative information, let take a somewhat idealized model of a PSK modulation : assume that the source is a complex random variable with constant modulus and a phase uniformly distributed over [ −π , π ]. The first standardized cumulants of this random variable are respectively equal to : k̃ (4) = −1, and k̃ (6) = 4. If the SNR is large enough so that higher order terms can be neglected, it appears that the performances of 4-2-MUSIC and 2-MUSIC are identical. The remarkable result is that 4-2-MUSIC achieves this performance without any information on the Noise Covariance Matrix (NCM). On the other hand, even at this large SNR limit, errors in the estimation of the NCM can severely affect performances of 2-MUSIC. We now illustrate the benefit that could be drawn from the use of an appropriate spatial filtering matrix M. Suppose that M is chosen to be the inverse of the source autocorrelation matrix : −1/2 in the single source case, M = R −1 a (θ)a (θ)∗ . Of S = ( k (2) ) course, RS is not known exactly ; however, it can be demonstrated that the large sample properties of the estimates are not affected RS is replaced by a consistent (initial) estimate R̂S , so that the performance analysis remains valid [13]. The asymptotic covariance is then modified in its higher order terms which become independent of the number of sensors m : a 2 = 6 , a 3 = 2. This formula suggests that an appropriate spatial filtering allows us to significantly decrease the variance at low SNRs. CONCLUSION A new array processing method has been introduced in this paper based on the contracted quadricovariance of the array output data. It offers a computational cost comparable to 2ndorder MUSIC while eliminating the need to model additive Gaussian noise. Performance analysis demonstrates that 4-2MUSIC can yield as good or even better bearing estimates than 2-MUSIC with known noise covariance matrix, provided that source kurtosis is large enough. These algorithms are of practical interest for communication signals such as PSK modulation. APPENDIX Let ( ei , i = 1,m ) be an orthonormal basis of E = C m . The vector space of all linear operators on E is denoted by F : elements of F are sometimes called "matrices" (in this paper, matrices are referred to as (1,1)-tensors). We denote A ij the entries of the matrix associated with an operator A. The adjoint of A is denoted A H ; its associated matrix is the transpose-conjugate of the matrix associated with A. F is a H Hilbert space, with the inner product A ∗ B ∆ = Tr ( B A ). Similarly, the vector space G of all linear operators on F can be identied with the space of (2,2)-tensors. We now define two "tensor products" : ⊗1 and ⊗2 . They combine any pair of (1,1)-tensors, A and B in F, to yield a (2,2)-tensor, according to the defining rules : ( A ⊗1 B ) M ( A ⊗2 B ) M ∆ A B ∗ M = A Tr(MB H ) = ∆ A M BH = The first tensor product is a (2,2)-tensor, similar to the classical vector outer product. The second tensor product essentially behaves as the Kroenecker product. In index notation, the coordinates of these tensors on the canonical basis ( ei e ∗j ) ⊗1 ( ek e ∗l ) are ( A ⊗1 B ) ij lk = A ij (B H ) kl ( A ⊗2 B ) ij lk = A ki (B H ) lj Finally, we define two symmetrized "tensor products" : [ A , B ]1,2 ∆ = A ⊗1,2 B + B⊗1,2 A REFERENCES [1] R. 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