Instrumental Variable Subspace Tracking Using Projection Approximation Tony Gustafsson y October 17, 1996 Abstract Subspace estimation plays an import

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1 Instrumental Variable Subspace Tracking Using Projection Approximation Tony Gustafsson y October 17, 1996 Abstract Subspace estimation plays an important role in, for example, sensor array signal processing. Recursive methods for subspace tracking, with obvious applications to non-stationary environments, have also drawn considerable interest. In this paper we present an Instrumental Variable (IV) extension of the recently developed Projection Approximation Subspace Tracking (PAST) algorithm. The IV-approach is motivated by the fact that PAST gives biased estimates when the noise vectors are not spatially white. IV-methods are well-known in the context of system identication. The basic idea of IV-methods is that the IV-vectors decorrelate the colored noise from the signal of interest, but leave the informative signal part undestroyed. The proposed algorithm is based on a (possibly over-determined) projection like unconstrained system of equations. The resulting basic algorithm has a computational complexity of 3mn + O(n 2 ) where m is the dimension of the measurement vector and n is the subspace dimension. The basic IV algorithm is also extended to a second order IV version, which is demonstrated to have better tracking properties than the basic IV-algorithm. The performance of the algorithms is demonstrated with a simulation study of time-varying sinusoids in additive colored noise. Keywords: Subspace Tracking, Sensor Array Signal Processing, Instrumental Variables, Frequency Estimation, Subspace Methods Permission to publish this abstract separately is granted This work was supported in part by the Swedish Research Council for Engineering Sciences (TFR). y Chalmers University of Technology, Department of Applied Electronics, S Gothenburg, Sweden, tonyg@ae.chalmers.se 1

2 1 Introduction Sensor array signal processing is a signal processing area that has received much attention in recent literature. Especially high-resolution subspace based methods have been studied by many researchers, see for example [9, 12]. These methods are model-based in the sense that they rely on a mathematical model of the received signal. Typical for the model-based approaches is that a serious degradation of the performance may occur if the signal model is incompatible with the 'true' signal. For example, the well-known MUSIC [13] algorithm assumes spatially white noise, i.e. the noise covariance matrix is proportional to the identity matrix. Of course, if the noise covariance is known this eect can be eliminated by pre-whitening. However, since the noise covariance generally is unknown, pre-whitening is in most cases not a realistic option. An alternative approach that does not require a known noise covariance matrix is the method of Instrumental Variables (IV), see [10] for a treatment of IV methods in the context of identifying linear time invariant dynamical systems. A combination of IV and subspace tting methods has recently been proposed as an alternative approach to array processing in spatially colored noise elds [15, 16]. Another aspect of the array processing eld that has drawn attention is the application of high-resolution frequency and direction of arrival (DOA) estimation techniques to non-stationary environments. A drawback of the traditional subspace methods in this scenario is that the singular value decomposition, SVD (or the eigenvalue decomposition, EVD), is time consuming to update. An excellent review of algorithms that tries to overcome this drawback can be found in [5]. Most of the proposed recursive methods require spatially white noise. One exception is the SWEDE algorithm [6], which allows a slightly more general structure of the noise covariance matrix. A specic example of a successful subspace tracking method is the Projection Approximation Subspace Tracking (PAST) algorithm [17]. The basic idea of PAST is that a projection-like unconstrained equation is simplied using a clever projection approximation; which leads to a Recursive Least Squares (RLS) like algorithm for tracking the signal subspace. The DOA (or frequency) estimates are then, for example, taken as the angles of the eigenvalues of a matrix obtained using the shift-invariant structure of the subspace. However, also the PAST algorithm requires spatially white noise, and tends to give biased estimates when this requirement is not fullled. This fact is the motivation of the algorithms proposed herein. The aim of this paper is to derive an IV extension of PAST. Like all other IV-methods we require that an IV-vector that is uncorrelated with the noise vector can be found. As long as this requirement is fullled, the noise vectors can be allowed to have arbitrary spatial and temporal color. A certain rank condition must also be fullled. This is since the informative part of the signal must remain undestroyed. 2

3 The IV approach leads to an unconstrained (possibly over-determined) system of equations. The solutions to this system are matrices with orthonormal columns that span the dominating (signal) subspace. Using the projection approximation idea of [17], an IV recursive subspace tracking algorithm is obtained. The resulting basic algorithm has the same computational complexity as PAST. The basic IV-algorithm does not perform any rank reduction of the sample cross covariance matrix. This limitation is overcome in all extensions of the basic algorithm. The basic IV-algorithm is for example extended to a second-order IV version, which is demonstrated to have good tracking properties. The outline of the paper is as follows: Section 2 introduces the signal model and other assumptions. Section 3 contains the derivations of the proposed IV-algorithms. These sections, more or less explicitly, focus on the DOA and frequency estimation problems. But it is important to notice that the studied data-model is not constrained to this problem. Section 4 describes a few simulation examples and Section 5 concludes the paper. The following notations are used in the paper: Matrices and vectors are represented with uppercase boldface and lowercase boldface fonts, respectively. The superscripts () H,() T denote complex conjugate transpose and transpose, respectively. The superscript () denotes complex conjugate. Further, I r denotes the r r identity matrix, E[] denotes mathematical expectation and the subspace spanned by the columns of A is denoted as R(A). The Kronecker delta function is denoted with and the rank of a matrix A is denoted with (A). A diagonal matrix consisting of the diagonal elements 1 ; : : : ; M is denoted diag( 1 ; : : : ; M ), Trfg denotes the trace operator and k k F denotes the Frobenius matrix norm. The computational complexity is in this paper dened as the number of complex multiplications, which agrees with the denition used in [5]. 2 Problem Formulation Let z(t) 2 C m1 be the observed data vector. In the array case z(t) consists of the samples of a Uniform Linear Array (ULA) with m sensors. In time series (sum of complex sinusoids) problems the vector z(t) = [z(t); : : : ; z(t + m? 1)] T (2.1) consists of m consecutive samples of an observed scalar signal. See Appendix A for a further treatment of the time series problem. In the following we assume that z(t) consists of n narrow-band plane waves impinging on an antenna array or n complex sinusoids corrupted by additive noise. Here the subspace dimension n, n < m, is assumed to be known. Hence, the following data model will be studied (see for example [17]) z(t) =?x(t) + e(t): (2.2) 3

4 The model (2.2) is quite general, and the algorithms to be presented apply to this model whatever structure? may have. But, in this paper we will focus on the DOA and frequency estimation problem. Thus, the following assumptions are introduced. The m n matrix? is deterministic, but possibly time-varying, and is constructed as where? = [(! 1 ) : : : (! n )] (2.3) (! k ) = [1 e j! k : : : ej! k(m?1) ] T (2.4) is a so-called steering-vector. The un-measurable signal vector x(t) is a random n-vector with covariance matrix C x = E[x(t)x H (t)]: (2.5) The noise vector e(t) is assumed to be independent of x(t), and has zero mean and an unknown covariance matrix dened as Q e () = E[e(t)e H (t + )]: (2.6) In the time series case,! k is the angular frequency of the kth complex sinusoid. In the array case! k = 2 d sin( k) (2.7) where d is the spacing between adjacent sensors, is the wavelength and k is the DOA relative the array normal. The above relations hold when plane waves impinge on a uniform linear array (ULA). Typical for IV approaches are the following assumptions. Assume that there exists an IV vector (t) 2 C l1, l n such that A1: E[e(t) H (t)] = 0 A2: (E[x(t) H (t)]) = (C x ) = n. The assumption A2 is made in order to ensure that (?C x ) = n, which implies that R(?C x ) = R(?). This rank condition is for the time series problem treated in Appendix A. A discussion about how to nd an IV-vector that, in the array case, fullls these conditions can be found in [15]. See also the references therein. One possible approach is to consider an array that is divided into sub-arrays. Then the outputs of one of the sub-arrays can be taken as instruments. If the sub-arrays are suciently far apart, the noise in the main sub-array is uncorrelated with the IV vector. These IV-vectors are called spatial IVvectors. When a subarray is not available but the signals are temporally 4

5 correlated, an IV-vector can be obtained by delaying the sensor outputs. This approach relies on that the signal temporal correlation length is longer than that corresponding to the noise. These IV-vectors are called temporal IV-vectors. Finally, in the time series case the only choice is to use (possibly pre-ltered) delayed outputs (t) = [z(t? M); : : : z(t? M? l + 1)] T (2.8) for some user specied integer M. Hence, for the IV approach to be applicable in the time series case we require that e(t) is nitely correlated, i.e. Q e () = 0 8 > s for some s. It is also possible to consider a 'pre-ltering' of the instruments. This could for example be achieved by F (t) = F(t) (2.9) where F 2 C ll is a 'pre-ltering' matrix. A natural choice, following the general theory of IV-methods, would be to choose F = C?1=2 where C = E[(t) H (t)]. However, this possibility will not be further studied. Implicit in the denitions above is that the subspace R(?) might be slowly time-varying, i.e. R(?) = R(?(t)), and of course! k =! k (t). For notational convenience, we often suppress a notation indicating the time variation, i.e. we use C z instead of C z (t), where C z (t) = E[z(t) H (t)]. By slowly time-varying we mean that z(t) is approximately stationary, and ergodic, in a window of size 1=(1? ), where 0 < 1 is the so-called forgetting factor. The local cross covariance matrix may thus be recursively estimated with ^C z (t) = ^C z (t? 1) + z(t) H (t): (2.10) With samples z(t); t = 1; : : : we are interested in estimating the signal subspace, i.e. R(?), of z(t). With the present application, we are also interested in estimating! k. That is, our aim is to derive an ecient algorithm which estimates R(?) at time instant t, given the subspace estimate at time instant t? 1 and the sample z(t). We will also study an algorithm that additionally takes the subspace estimate at time instant t? 2 into account. 3 Derivation of the IV algorithm 3.1 Summary of the PAST-algorithm In order to motivate our IV-algorithms, a short summary of PAST [17] is given. Consider the following unconstrained criterion: V (W) = E z(t)? WWH z(t) 2 = Tr C z? 2W H C z W + W H C z WW H W (3.1) 5

6 with a matrix argument W 2 C mn, m > n, that without loss of generality is assumed to have full rank (=n). Due to the independence of x(t) and e(t) we have that C z = E[z(t)z H (t)] =?C x? H + Q e (0): (3.2) The above relation forms the basis for most subspace based methods. Let the EVD of C z be given as C z = UU H (3.3) with U = [u 1 ; : : : ; u m ], =diag( 1 ; : : : ; m ). The eigenvalues are ordered as 1 2 : : : m. If Q e (0) = 2 I m, then n > n+1 = : : : = m = 2. The dominant eigenvectors u 1 ; : : : ; u n are termed the signal eigenvectors whereas u n+1 ; : : : ; u m are the noise eigenvectors. It is easy to verify that R(U s ) = R([u 1 ; : : : ; u n ]) = R(?), i.e. the signal eigenvectors span the subspace spanned by the steering vectors. This observation motivates most subspace based algorithms. Now, if Q e () = 2 I m, the following theorem applies: Theorem 1 W is a stationary point of V (W) if and only if W = UT where U 2 C mn is orthogonal and contains any n distinct eigenvectors of C z. All stationary points of V (W) are saddle points, except when U contains the n dominant eigenvectors, U = U s. In this case V (W) attains the global minimum. Here T 2 C nn is an arbitrary unitary matrix. Proof: See [17]. 2 In order to derive a practical algorithm, replace (3.1) with V (W(t)) = tx t?k z(k)? W(t)W H (t)z(k) 2 (3.4) k=1 where, 0 < 1, is the so-called forgetting factor. Clearly, Theorem 1 is applicable also to (3.4) if C z is replaced with ^C z (t) = tx k=1 t?k z(k)z H (k): (3.5) The key idea of PAST is to replace W H (t)z(k) in (3.4) with h(k) = W H (k? 1)z(k). This projection approximation results in the criterion V (W(t)) = tx k=1 t?k kz(k)? W(t)h(k)k 2 (3.6) 6

7 which is a quadratic in W(t), and is minimized by W(t) = ^C?1 zh (t) ^C h (t): (3.7) The covariance matrices are estimated as: ^C zh (t) = ^C zh (t? 1) + z(t)h H (t) = ^C h (t) = ^C h (t? 1) + h(t)h H (t) = tx k=1 tx k=1 t?k z(k)h H (k) t?k h(k)h H (k): (3.8a) (3.8b) If the matrix inversion lemma is applied to (3.8b), an ecient RLS-like algorithm is easily derived. 3.2 IV-PAST Derivation In this section the proposed basic IV-extension of PAST is derived. The criterion (3.4) can be interpreted as (neglecting the forgetting factor): given z(k); h(k) k = 1; : : : ; t nd the least squares solution to which gives Z = [z(1); : : : ; z(t)] W(t)[h(1); : : : ; h(t)] = W(t)H (3.9) The natural IV solution to (3.9) would be W(t) = ZH H (HH H )?1 : (3.10) W(t) = Z H (H H )?1 (3.11) where = [(1); : : : ; (t)], and for simplicity we take the number of instruments as l = n (to make H H a square matrix). With a forgetting factor we nd the following IV algorithm W(t) = ^C z (t) ^C?1 h (t) (3.12) where the covariance matrices are updated according to: ^C z (t) = ^C z (t? 1) + z(t) H (t) (3.13a) ^C h (t) = ^C h (t? 1) + h(t) H (t): (3.13b) A theoretical justication of the above is as follows. Consider the solutions to V (W) = E[z(t) H (t)]? WW H E[z(t) H (t)] = 0, V (W) =?C x? WW H?C x = 0: (3.14) 7

8 The above holds under assumption A1. Provided A2 is fullled, by denition of the orthogonal projector, all solutions to (3.14) will be of the form W = U s T where R(U s ) = R(?), U s 2 C mn has orthonormal columns, and T 2 C nn is an arbitrary unitary matrix. Thus, for all solutions to (3.14) we have that WW H =? = U s U H s (3.15) is the orthogonal projector onto the space spanned by the columns of?. To derive a practical algorithm, consider the solutions to (compare with (3.14)) V (W(t)) = tx? t?k z(k) H (k)? W(t)W H (t)z(k) H (k) = 0: (3.16) k=1 The afore mentioned projection approximation, h(k) = W H (k? 1)z(k), together with (3.16) then leads to the proposed algorithm (3.12). Using the matrix inversion lemma, straightforward calculations gives the following algorithm: h(t) = W H (t? 1)z(t) (3.17a) W(t) = W(t? 1) + (z(t)? W(t? 1)h(t))K(t) (3.17b) P(t) = 1 (P(t? 1)? P(t? 1)h(t)K(t)) (3.17c) K(t) = H (t)p(t? 1) + H (t)p(t? 1)h(t) (3.17d) where P(t) = ^C?1 h (t). In the above we have assumed that initial values W(0); P(0) are given. These initial values only aect the transient behavior and are not important for the steady-state performance of the algorithm. They can for example be taken as any full-rank matrices. For faster convergence, W(0) can be taken as a subspace estimate from a conventional batch-method (using the rst samples of the signal). The computational complexity of the IV-PAST algorithm is identical to that of the original PAST algorithm, i.e. at every time instant 3mn + O(n 2 ) complex multiplications are required [17]. The original PAST-algorithm is retained if in the above, (t) is replaced with h(t). Remark 1 Due to the introduced approximations, the columns of W(t) will not be exactly orthonormal. However, simulations show that they are 'nearly' orthonormal, see Section 4. Some applications may require orthonormal columns, which may call for a re-orthogonalization scheme such as Gram- Schmidt. However, in the simulations in Section 4 no orthogonalization is performed. Under stationary signal conditions W(t) will converge to a matrix with orthonormal columns if = 1. 8

9 Remark 2 The basic IV-algorithm does not perform any rank-reduction of the sample cross covariance matrix. This means that at every time instant R( ^C z (t)) = R(W(t)), see (3.12). So, why not take W(t) = ^C z (t)? The main motivation is that the matrix ^C?1 h (t) post-multiplying in (3.12) forces the columns of W(t) to be 'nearly' orthonormal resulting in good conditioning. In scenarios with closely spaced frequencies this can be an advantage. Another possibility would be to orthogonolize the columns of ^C z (t), which would yield a complexity of O(mn 2 ), see [5]. Thus, IV-PAST can be thought of as a simple way to approximately orthogonolize the columns of ^C z (t). The basic IV-algorithm will also serve as a preview of more general rankreducing IV-approaches described in the following sections. 3.3 Extensions of the basic algorithm Rank Reduction Theorem In Section 3:2 we constrained the dimension of the IV-vector (t) to l = n. A straightforward extension, l > n, leads to the following criterion V (W(t)) = ^C z (t)? W(t)W H (t) ^C z (t) 2 F (3.18) where W(t) 2 C mn, ^C z (t) 2 C ml. This approach corresponds to what in [14, Section 8] is called the Extended IV estimate. In [14, Complement 8.5] it is shown that in the context of system identication, the accuracy of the extended estimate, as expected, increases with increasing number of instruments l (provided an optimal choice of instruments is made). Without loss of generality we assume that (W(t)) = n. With probability 1 (w.p.1) we have that ( ^C z (t)) = min(m; l) = ~n, but (C z ) = n < l. Consequently, a low-rank approximation of ^C z (t) is desired. Thus, we need the following theorem. Theorem 2 Let ^C z (t) have the singular value decomposition (SVD) ^C z (t) = ^U ^ ^V H = h ^Us ^U n i ^s 0 " H ^V s 0 ^ n ^V H n # (3.19) where ^U s 2 C mn. The remaining partitions are of appropriate dimensions. W(t) is a stationary point of (3.18) if and only if W(t) = ^UT, where ^U denotes any n left singular vectors of ^U and T 2 C nn denotes an arbitrary unitary matrix. At each stationary point V (W(t)) equals the sum of the squares of the singular values whose corresponding singular vectors are not involved in ^U. 9

10 All stationary points of V (W(t)) are saddle points except when ^U = ^U s. In this case V (W(t)) attains the global minimum and V (W(t))j W(t)= ^U st = ~nx k=n+1 ^ 2 k : (3.20) where ^ k are the singular values of ^C z (t). Note that for this choice, W(t)W H (t) ^C z (t) = ^U s ^ s ^V H s, which in the sense of the Frobenius norm is the best possible rank n approximation of ^C z (t). Proof: For notational convenience, let C = ^C z (t); ~C = CC H and W = W(t). Evaluate (3.18) as V (W) = Trn? C? WWH C? C? WW H C H o = Trn ~C + W H ~CWW H W? 2W H ~CW o : (3.21) The theorem is then proved by the proof of Theorem 1, using that C = ^U ^ ^V H, ~ C = ^U ^ 2 ^U H :2 (3.22) There are several possible ways to derive practical algorithms based on Theorem 2. However, we constrain ourselves to two dierent approaches, namely the previously studied projection approximation and a gradient based method. 3.4 EIV-PAST Once again the following approximation is applied: W H (t) ^C z (t) = W H (t) which gives the criterion tx k=1 t?k z(k) H (k) tx k=1 t?k W H (k? 1)z(k) H (k) = ^C h (t) {z } h(k) (3.23) V (W(t)) = ^C z (t)? W(t) ^C h (t) 2 : (3.24) F The minimizing argument of (3.24) is given by W(t) = ^C z (t) ^C y h(t) (3.25) where (:) y denotes the Moore-Penrose pseudo-inverse. This approach will in most cases improve the accuracy of the estimates. For example, in Section 10

11 4 we will see that the tracking capabilities are much more 'well-behaved' in this case. However, the main disadvantage is that this scheme leads to an increased computational complexity. In Appendix B we give an ecient recursive updating formula of (3.25). We give the algorithm without derivation. This is since the calculations to a great extent parallels the calculations in [14, Complement C9.1]. See also [7] for further reading of Extended IVtechniques. An operation count of the algorithm given in Appendix B gives a complexity of 3ml + O(mn). Note that the matrix inversion that arises in (B.1b) is of size (2 2), so it is a simple matter to invert it. Remark 3 One point of view worth mentioning is that the recursive update of the inverse of the 'covariance' matrix in this case involves positive denite matrices, see Appendix B. The update formulas for P(t) do not guarantee its positive (semi) deniteness. In ill-conditioned situations this may lead to numerical instability. The simplest approach to handle this problem is to evaluate only the upper right triangular part of P(t), see (B.1j). The remaining elements of P(t) are obtained using that P(t) = P H (t), i.e. P(t) is Hermitian. This approach guarantees that P(t) is at least positive semidenite. See also [11, Section 6.5] for an overview of regularization techniques, i.e. forcing P(t) to be non-singular. In [7] it is suggested that the so-called SRIF-algorithm [2, Section V.2] is used to update the 'system identication' extended IV-algorithm. However, to our knowledge, the SRIF-algorithm is not easily extended to the present case. Remark 4 It could be interesting to compare the performance of an algorithm based on the above projection approximation with a 'substitution' based approximation, i.e. W(t)W H (t) W(t)W H (t? 1). The latter approximation would give the following subspace estimate W(t) = ^C z (t) W H (t? 1) ^C z (t) y : (3.26) Our practical experience is that the 'substitution' approximation oers better tracking performance than the EIV-PAST algorithm. This is since the substitution approach remembers only the previous estimate, whereas EIV- PAST in principle remembers all previous estimates. Under stationary signal conditions the dierence is however insignicant. The drawback with the substitution approach is that there is no easy update formula for (3.26). Hence, a considerably increased complexity is obtained. 3.5 Gradient method, IVg A gradient based approach to minimize (3.18) can be devised as W k+1 (t) = W k (t)? k V 0 (W k (t)); k = 0; 1; : : : (3.27) 11

12 where k > 0 are step-sizes. Here subscript () k denotes the kth iteration of the minimization procedure and V 0 (W k (t)) W(t)=W k (t) : (3.28) The gradient can be evaluated as (see [3] for a denition of the derivative of a scalar function with respect to a matrix): V 0 (W(t)) = h?2 ~C(t) + ~C(t)W(t)W H (t) + W(t)W H (t) ~C(t)i W(t) (3.29) where ~ C(t) = ^C z (t) ^C H z(t). Using the approximation W H (t)w(t) I n the following gradient based algorithm is found: W 0 (t) = W(t? 1) (3.30a) R k (t) = W k (t) H ^C z (t) (3.30b) W k+1 (t) = W k (t)? k Wk (t)r k (t)? ^C z (t) Rk (t) H (3.30c) where the covariance matrix for example is updated as in (3.13a). At every iteration, 3mnl + O(nl) complex multiplications are required. A number of comments are now in place. Our experience is that, at every time instant, one iteration of (3.30c) is enough. The complexity is however still too large for many real-time applications. Since V (W(t)) has no local minima, global convergence is guaranteed if one nds the signal subspace by minimizing V (W(t)) with iterative methods. At convergence W(t) does not contain the left singular vectors of the estimated covariance matrix. But, what really matters is that W(t) contains an orthonormal basis for the signal subspace. Gradient based optimization methods are known to be rather ineective close to the minimum of the criterion. A Newton based method would probably be more eective. But there are several drawbacks with a Newton scheme. The major drawback is that the complexity would increase. Another drawback is that the Hessian of the criterion (3.18) will be ill-conditioned since there is no unique minimizing argument of (3.18). This problem can be overcome with regularization. By monitoring the quantity? I m? W(t)W H (t) ^C z 2 F ~nx k=n+1 ^ 2 k ; (3.31) 12

13 the user have a possibility to detect changes in the subspace dimension. The above relation holds approximately also for the EIV-PAST estimates A second order IV-algorithm, 2IV-PAST In [4] a second order recursive algorithm for adaptive signal processing (and system identication) is proposed. An extension of ordinary PAST to a second order recursive PAST can be found in [1]. This algorithm has the same computational complexity as PAST, but additionally computer memory is required. In [1] it is demonstrated that in some cases the second order algorithm oers better tracking performance than PAST. Inspired by this, we will derive a second order IV version of PAST. Introduce a hypothetical subspace W hyp and dene an 'equation error' together with the additional signal where h(t) = W H z(t) and W = U s T. Dene w(t) = z(t)? W hyp h(t) (3.32) v(t) = z(t)? Wh(t) (3.33) C w = E[w(t) H (t)] = E[(Wh(t)?W hyp h(t)+v(t)) H (t)] = (W?W hyp )C h : (3.34) The last equality in (3.34) follows since E[v(t) H (t)] = E[(z(t)? WW H z(t)) H (t)] =?? E[e(t)H (t)] = 0 (3.35) by assumption A1. So far both W and W hyp are assumed to be known deterministic matrices. Additionally, W is assumed to be an orthogonal matrix spanning the true signal subspace. However, we will use the above relationship (3.34) to dene an estimator of W as, where the hypothetical subspace W hyp is taken as the previous estimate W hyp = W(t? 1): W(t) = W(t? 1) + m ^C w (t) ^C y h(t) (3.36) where the estimates of the covariance matrices are dened by ^C h (t) = (1? ) ^C h (t? 1) + h(t) H (t) (3.37a) ^C w (t) = (1? ) ^C w (t? 1) + w(t) H (t) (3.37b) and ; m are step-sizes. The extra scaling factor in (3:38) is introduced in order to conform with the frame-work of [1]. Strictly speaking, this choice of W hyp invalidates the calculations in (3.34). So, once again we emphasize that (3.34) is merely used to dene our estimator. 13

14 The step-size m in (3.36) leads to an additional smoothing of the subspace estimate. It oers the possibility of increasing the updating speed in (3.37a,b) without getting too noisy estimates in (3.36). To gain some insight of the estimation principle, consider a xed hypothetical subspace matrix. It is then easy to show that the resulting estimates can be written as, neglecting the scaling factor in (3.37a,b): W(t) = (1? m )W hyp + m W EP (t) (3.38) where W EP (t) denotes the subspace estimate of the EIV-PAST algorithm. Consequently, for a xed W hyp and if m = 1, the EIV-PAST algorithm is retained. The second order algorithm consequently encompasses the EIV- PAST algorithm. The algorithm is now specied by (3.36),(3.37a) and (3.37b). An 4ml + O(mn) algorithm that is more well suited for practical implementation can be found in Appendix C. In order to gain some insight, the algorithm in Appendix C can for the special case l = n (i.e. no rank reduction) be simplied to the following expressions: h(t) = W H (t? 1)z(t) (3.39a) W(t)(t) = W(t? 1) + (W(t? 1)? W(t? 2))? I n? h(t) H (t)p(t) P(t) = + m (z(t)? W(t? 1)h(t)) H (t)p(t) (3.39b) 1 1? P(t? 1)? P(t? 1)h(t)H (t)p(t? 1) 1? + H (t)p(t? 1)h(t)! (3.39c) where P(t) = ^C?1 h (t). It is assumed that initial values are given. From the algorithm above we see that the current estimate is a combination of both W(t? 1) and W(t? 2), which motivates the name 'second-order'. This relationship can also be seen in Appendix C, see (C.7). Finally, following the discussion in [1], a reasonable choice of the stepsizes are as follows: m = 4 : (3.40) Hence, the selection of the step-sizes is reduced to a one-dimensional problem. The somewhat ad-hoc motivation of this choice is that the associated Ordinary Dierential Equation (ODE), see for example [11, Section 4] for a treatment of the theory of associated ODE's, then at convergence has a real-valued double pole. Strictly speaking, the ODE discussion is valid only for the linear regression problem, i.e. ARX systems [10]. The motivation of the second-order algorithm is rather ad-hoc. One way to understand the algorithm is that a time-varying ltering of the estimates delivered by the Extended IV-algorithm is performed. But, we select the step-sizes of the second-order algorithm so that the underlying Extended 14

15 IV-algorithm is run with a larger step-size than normal use of the Extended IV-algorithm would admit. The estimates of the second order algorithm is to our knowledge not the minimizing argument of any 'typical' criterion such as (3.4). However, from practical experience we are convinced that the second order approach oers many advantages. For example, it will with simulations be demonstrated that a faster response to sudden changes can be achieved, without degrading the stationary performance. Finally, Remark 3 applies also to the 2IV-PAST algorithm. 3.6 Summary In the present section several IV-based subspace tracking algorithms have been proposed. We conclude the section with a summary of the main features of the algorithms, see Table 1, and with a note on sliding windows. All of the IV-based tracking algorithms are based on exponentially weighted criteria. These criteria correspond to covariance estimates that also are exponentially weighted, i.e. ^C z (t) = ^C z (t? 1) + z(t) H (t): (3.41) Another possibility to update the covariance estimates is to use a so-called sliding window. Then the covariance estimates are taken as ^C z (t) = ^C z (t? 1) + z(t) H (t)? z(t? L) H (t? L) (3.42) where the user dened integer L is the window length. This approach will not be pursued any further. However, in [17], it is via simulations demonstrated that the tracking performance in some cases are improved with the sliding window approach. 4 Examples In this section the performance of PAST, IV-PAST, EIV-PAST, IVg and 2IV-PAST will be demonstrated. It is not the aim of the present paper to give an exhaustive comparison of dierent subspace-tracking algorithms. We restrict ourselves to compare the IV-algorithms only with the original PAST-algorithm. We will demonstrate that the IV-based methods reduce bias caused by colored noise. The IV-based algorithms will also be compared with the truncated SVD of ^C z. In a sense, given ^C z the truncated SVD is the best possible way to nd the subspace estimate. It is therefore interesting to compare the SVD-based estimates with those of our IV-algorithms. All approaches nd the frequency estimates using the ESPRIT-approach, i.e. the angles of the eigenvalues of W y 2:m (t)w 1:m?1(t), where W i:j denotes rows i to j of W. Other approaches for nding the frequencies may be considered. One advantage with the ESPRIT-approach is that the columns 15

16 of W(t) are not required to be orthonormal. For all algorithms, the following initial values were used: P(0) = I n W(0) = [I n 0 T (m?n)n ]T : (4.1) However, the transient will typically not be shown. Both real-valued and complex-valued data will be considered. Earlier we have stated that the columns of the subspace estimates are 'nearly' orthonormal. To quantify 'nearly', the following measure of deviation from orthonormality will be studied W H (t)w(t)? I n F : (4.2) The simulations are focused on the frequency estimation problem. Further, unless otherwise stated, only one iteration of (3.27) is performed. Example 1 Real-valued data Consider the scalar signal z(t) = 2X j=1 a j cos (2f j (t) + ' j ) + e(t) (4.3) where a 1 = a 2 = p 2. The random phases ' j are independent and uniformly distributed in (?; ). Thus, with this setting n = 4, and we chose m = 8 which gives The IV-vector is chosen as z(t) = [z(t); : : : ; z(t + 7)] T : (4.4) (t) = [z(t? M); : : : ; z(t? M? l + 1)] T (4.5) with M = 11. The number of instruments l is for IV-PAST l = 4. The other IV-algorithms use l = m = 8. The SVD-based subspace estimate is obtained from the n principal left singular vectors of with l = 8. The noise is given by ^C z (t) = ^C z (t? 1) + z(t) H (t): (4.6) e(t) = 1 "(t); (4.7) 1? 0:8q?1 where q?1 is the delay operator and "(t) is white Gaussian noise. Note that for this noise process condition A1 is violated E[e(t) H ] 6= 0: (4.8) 16

17 We will show that we still can achieve an improvement compared to the original PAST-algorithm. The forgetting factors were for all rst order algorithms chosen as = 0:97. It is not straightforward to transform this value to a step-size for 2IV-PAST, so that a fair comparison is obtained. One possible way is to choose the step-size so that the mean square error (MSE) of the estimated frequencies under stationary signal conditions are equal for 2IV-PAST and EIV-PAST. This approach was also used to determine the step-size of IVg. It turned out that = 0:21 (!) and = 0: approximately gave the same MSE as did EIV-PAST. Finally, dene the Signal to Noise ratio as 'signal power/noise power'. Our rst example serves as an illustration of the bias reduction oered by the IV-approach, see Figure 1. For reasons of simplicity, only PAST and EIV-PAST are compared. From Figure 1 we conclude that the IV-approach clearly reduces the bias. This feature is shared with all of our proposed algorithms. Thus, in the following further comparisons with PAST will be omitted. Next we focus on the tracking performance. Consider a sudden step change in a frequency. Figures 2,3 and 4 shows the outcome of one realization of the frequency tracks, deviations from orthonormality and the subspace angles between the SVD and the EIV-PAST estimates. We dene the subspace angle as the largest principal angle [8, Section 12]. A number of comments are in place. In Figure 2 we see the advantage with a rankreducing algorithm. A much more 'well-behaved' response to the change is achieved. Another advantage with the rank-reducing approaches is that the estimates of the constant frequency are less aected by the change. Further, a smoother behavior during the stationary part is achieved. Note also the fast, and smooth, response of 2IV-PAST. From Figure 3 we conclude that all algorithms give nearly orthonormal columns. The deviation is larger during the transient phase. Next we study the subspace angle between the SVD and the EIV-PAST subspace estimates, see Figure 4. Note that the estimates of EIV-PAST in this sense are very close to the left singular vectors of ^C z. It is only during the transient phase the dierence is noticeable. Since EIV-PAST remembers previous subspace estimates, the response is slightly delayed compared to the SVD-estimates. Next we consider an example with a frequency modulated frequency. Let f 1 (t) = 0:3t and f 2 (t) = 0:1t + 0: sin t : (4.9) In the following we omit the results of the basic IV-algorithm. This is since its tracking capabilities are rather poor. The outcome of one realization of the frequency tracks together with the true instantaneous frequency (d=dt f 2 (t)) are shown in Figure 5. For reasons of simplicity, only ^f 2 (t) are shown. Neither the SVD or the EIV-PAST estimates behave smoothly when the frequency changes rapidly. This is since ^C z is updated too slowly to cope with the changes in the signal. Note that it is almost impossible to 17

18 distinguish the SVD and EIV-PAST freqeuncy estimates! Note also the smooth behavior of 2IV-PAST. IVg is based on the same ^C z as EIV-PAST. However, the step-size makes the frequency tracks smoother, and more delayed than those of EIV-PAST. But, this is accomplished at a higher computational cost than both EIV-PAST and 2IV-PAST. Example 2 Complex-valued data In this example complex-valued data will be considered. Consider the scalar signal z(t) = 2X j=1 e j(2f j(t)+' j ) + e(t) (4.10) where the real and imaginary part of e(t) are Gaussian and independent of each other, and each component of e(t) is generated as in (4.7). Consider the example in Figure 6, where SNR=15 db, = 0:97; = 0:21 and = 0:0003. The number of instruments were was chosen as l = 5, and also m = 5. It turned out that the step-size of the IVg-algorithm for stability reasons had to be chosen smaller than in the previous examples. So, the comparison in Figure 6 is perhaps not 'fair'. The step-size is chosen so that the stationary performance is better for IVg, hence the smoothness of the frequency tracks. Once again we see how close the SVD-estimates are to the EIV-PAST estimates. Both approaches fail to work in the region where the frequencies cross. Hence, the behavior is due to the ill-conditioning of ^C z (t). Since 2IV-PAST is run with a larger updating speed, the problem is locally magnied for the second-order algorithm. These problems suggest that in a scenarios with closely spaced frequencies, regularization may be necessary, see Remark 3. Alternatively, the subspace dimension should be adjusted when the frequencies are too close. Our last example concerns the convergence properties. For reasons of simplicity we study only EIV- PAST. With (4.10) and f 1 (t) =?0:1t; f 2 (t) =?0:3t, the deviations from orthonormality and the subspace angle to the true subspace, will for dierent be compared. The noise realizations are identical in all cases, and the transient eects due to (4.1) are in this case shown. See Figure 7 and Figure 8. The signal to noise ratio was chosen to SNR=15 db. From Figure 7 and Figure 8 we conclude that for = 1, the subspace estimate tends to a matrix with orthonormal columns whose range is close to the true subspace. 5 Conclusions In this paper several Instrumental Variable generalizations of the subspace tracking algorithm PAST have been proposed. The presented algorithms are able to track time-varying subspaces in spatially and/or temporally colored 18

19 noise elds. One requirement is that we must be able to nd an IV-vector that is uncorrelated with the noise vector. Additionally, a certain rank condition must be fullled. The original algorithm (PAST) is a special case of the basic IV algorithm proposed herein (put (t) = h(t)). The basic IV algorithm has the same computational complexity as PAST. Also, a least squares (Extended IV), a gradient based algorithm and a secondorder version of the basic IV algorithm were proposed. The performance of the algorithms was illustrated in a simulation study. The conclusions are that an IV approach in our examples improves the results when the noise is not spatially white. The main features of the basic IV-algorithm is that it has low complexity and is able to track rapid changes. However, the basic IV-algorithm does not perform any rank-reduction of the sample crosscovariance matrix. The Extended IV algorithm requires more computations but a much more 'well-behaved' response, due to the rank-reduction, to sudden changes was achieved. The second order algorithm oers better trade-o between tracking and stationary performance. This is due to the fact that we can increase the updating speed of certain covariance estimates without getting too noisy estimates. A drawback of our algorithms is that they have no 'built-in' rank-detection. However, a quantity that can be used to detect changes in the subspace dimension was given. Even though all simulation results indicate global convergence of the IV-algorithms, there has been no theoretical analysis performed. Hence, remaining issues for future studies include rank estimation and convergence analysis. A Appendix In this section a further treatment of the time series problem under stationary signal conditions is given. Consider the complex valued scalar signal z(t) = nx k=1 a k e j(! kt+' k ) + e(t) (A.1) with real valued amplitudes a k. The additive noise term e(t) represents a possibly non-white disturbance. The random phases ' k are assumed to be mutually independent and uniformly distributed in (?; ). Introduce the vector of stacked samples z m (t) = [z(t); : : : ; z(t + m? 1)] T : (A.2) Then the following relation is easily veried z m (t) =? m x(t) + e m (t) (A.3) with obvious denition of e m (t). The deterministic matrix? m is dened by? m = [(! 1 ); : : : ; (! n )] (A.4) 19

20 where the so-called steering vectors are given by (! k ) = [1; e j! k ; : : : ; e j! k (m?1) ] T. In? m, subscript 'm' denotes the length of the steering vectors. The unmeasurable signal vector x(t), assumed to be independent of the noise, is given by x(t) = [a 1 e j(! 1t+' 1 ) ; : : : ; a n e j(!nt+'n) ] T (A.5) with a covariance matrix C x = E[x(t)x H (t)] = diag(a 2 1 ; : : : ; a2 n): (A.6) Choose the IV-vector as (t) = [z(t? M); : : : ; z(t? M? l + 1)] T (A.7) where the user-dened integer M is chosen so that E[e m (t) H (t)] = 0. The number of instruments is chosen so that l n. Now it easy to verify that (t) =? l x(t? M) + e l (t? M) (A.8) and x(t? M) = diag(e?j! 1M ; : : : ; e?j!nm )x(t) = Mx(t) (A.9) so that E[x(t) H (t)] = E[x(t)x H (t)]m? H l = diag(a 2 1e j! 1M ; : : : ; a 2 ne j!nm )? H l = D? H l : (A.10) The above relation follows since x(t) is independent of the noise by assumption. Thus, if a 2 k 6= 0 and the frequencies! k are distinct (i.e.! k 6=! l unless k = l) the rank-condition A2 is satised for all M if l n. This is since a diagonal matrix is of full rank if the diagonal entries are nonzero, and? l, which is a so-called Vandermonde matrix, is of full rank if the frequencies are distinct. However, M must still be chosen so that A1 is satised. B Appendix In this appendix we give, without derivation, the recursive update formulas for the Extended Instrumental Variable subspace tracking algorithm. We omit the calculations since they to a large extent parallel those in [14, Com- 20

21 plement C9.1]. W(t) = W(t? 1) + (v(t)? W(t? 1)(t)) K(t) (B.1a) K(t) =? 2 (t) + H (t)p(t? 1)(t)?1 H (t)p(t? 1) (B.1b) (t) = [w(t) h(t)] (B.1c) w(t) = ^C h (t? 1)(t) (B.1d)? 2 (t) = H (t)(t) (B.1e) 0 v(t) = h ^Cz (t? 1)(t) z(t)i (B.1f) ^C h (t) = ^C h (t? 1) + h(t) H (t) (B.1g) ^C z (t) = ^C z (t? 1) + z(t) H (t) (B.1h) h(t) = W H (t? 1)z(t) (B.1i) P(t) = 1 (P(t? 1)? P(t? 1)(t)K(t)) (B.1j) 2 where P(t) = ^Ch (t) ^C H h(t)?1. C Appendix In this appendix an algorithm that is more computationally ecient than (3.36) is given. The dening relationships are for easy reference repeated below: h(t) = W H (t? 1)z(t) (C.1a) w(t) = z(t)? W(t? 1)h(t) (C.1b) ^C h (t) = (1? ) ^C h (t? 1) + h(t) H (t) (C.1c) ^C w (t) = (1? ) ^C w (t? 1) + w(t) H (t) (C.1d) W(t) = W(t? 1) + m ^C w (t) ^C y h(t): (C.1e) For notational convenience, let R(t) = ^C h (t) and r(t) = ^C w (t). Assume that R(t) has full rank. Then R y (t) = R? H (t) R(t)R H?1 (t) {z } P(t) : (C.2) 21

22 From Appendix B we have the following updating formula for P(t) P(t) = 1 (P(t? 1)? (1? ) 2 (C.3a) P(t? 1)(t) (1? ) 2 (t) + H (t)p(t? 1)(t)?1 H (t)p(t? 1))? (1? ) 2 (t) = H (t)(t) 1? 1? 0 (t) = [R(t? 1)(t) (1? )h(t)] : (C.3b) (C.3c) Note that the matrix inverse appearing in (C.3a) is of size (2 2). Hence, it is a simple matter to invert it. Now, r(t)r H (t) = (1? ) 2 r(t? 1)R H (t? 1) + 2 j(t)j 2 w(t)h H (t) + (1? )w(t) H (t)r H (t? 1) + (1? )r(t? 1)(t)h(C.4) H (t): The rst term on the right hand side of the above equation can be rewritten as (1? ) 2 r(t? 1)R H (t? 1) = (1? ) 2 r(t? 1)R H (t? 1)P(t? 1)P?1 (t? 1) = (1? ) 2 (W(t? 1)? W(t? 2)) P?1 (t? 1)(C.5) m which gives m (1? ) 2 r(t? 1)R H (t? 1) = (W(t? 1)? W(t? 2)) (I n? (t) (1? ) 2 (t) + H (t)p(t? 1)(t)?1 H (t)p(t? 1)):(C.6) We have not been able to nd any further major simplications of (C.1e). Hence, the 4ml + O(mn) second order IV-algorithm reads as W(t) = W(t? 1) + References (W(t? 1)? W(t? 2)) (I n? (t) (1? ) 2 (t) + H (t)p(t? 1)(t)?1 H (t)p(t? 1)) + m (1? ) w(t) H (t)r H (t? 1) + r(t? 1)(t)h H (t) + m 2 j(t)j 2 w(t)h H (t): [1] A. Andersson and H. Broman. \A second order recursive algorithm with applications to adaptive ltering and subspace tracking". Technical Report CTH-TE-30, Chalmers University of Technology, Gothenburg, Sweden, May Submitted to IEEE Trans SP. (C.7) 22

23 [2] G.J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic, New York, [3] J Brewer. Kronecker products and matrix calculus in system theory. IEEE Transactions on Circuits and Systems, CAS-25:772{781, Sep [4] H. Broman and A. Andersson. \Parameter Estimation: A Second Order Recursive Least Squares Algorithm". In Proc. SYSID'94, 10th IFAC Symposium on System Identication, pages 447{452, Copenhagen, Jul [5] P. Comon and G. H. Golub. \Tracking a few Extreme Singular Values and Vectors in Signal Processing". Proceedings of the IEEE, 78:1327{ 1343, Aug [6] A. Eriksson. \Algorithms and performance analysis for spatial and spectral parameter estimation". PhD thesis, Uppsala Universisy, Uppsala, Sweden, [7] B. Friedlander. \The overdetermined recursive intrumental variable method". IEEE Trans. Automatic Control, AC-29:353{356, Feb [8] G.H. Golub and C.F. VanLoan. Matrix Computations, 2 nd edition. Johns Hopkins University Press, Baltimore, MD., [9] H. Krim and M. Viberg. \Sensor array processing: Two decades later". Technical report CTH-TE-28. Department of Applied Electronics, Chalmers University of Technology, Jan [10] L. Ljung. System Identication: Theory for the User. Prentice-Hall, Englewood Clis, NJ, [11] L. Ljung and T. Soderstrom. \Theory and Practice of Recursive Identication". The MIT Press, Cambridge, Massachusetts, [12] R. H. Roy. \ESPRIT, Estimation of Signal Parameters via Rotational Invariance Techniques". PhD thesis, Stanford Univ., Stanford, CA, Aug [13] R.O. Schmidt. \Multiple Emitter Location and Signal Parameter Estimation". In Proc. RADC Spectrum Estimation Workshop, pages 243{ 258, Rome, NY, [14] T. Soderstrom and P. Stoica. System Identication. Prentice-Hall, London, U.K.,

24 [15] P. Stoica, M. Viberg, M. Wong, and Q. Wu. \A unied Instrumental Variable Approach to Direction nding in colored noise elds". Technical report, Chalmers University of Technology, CTH-TE-32, Gothenburg, Sweden, Jul To appear in Digital Signal Processing Handbook, CRC Press,1996. [16] M. Viberg, P. Stoica, and B. Ottersten. \Array Processing in Correlated Noise Fields Based on Instrumental Variables and Subspace Fitting". IEEE Trans. SP, May [17] B. Yang. \Projection Approximation Subspace Tracking". IEEE Trans. SP, 43(1):95{107, Jan

25 Figure and table captions Table 1 Main features of the IV-algorithms Figure 1 One realization of the frequency estimates. SNR=5 db, e(t) = 1 1?0:8q?1 "(t). = 0:97 Figure 2 One realization of the frequency estimates. SNR=8 db, e(t) = 1 1?0:8q?1 "(t). = 0:97, = 0:21, = 0: Figure 3 Deviation from orthonormality for a step-change. SNR=8 db, e(t) = 1 1?0:8q?1 "(t). = 0:97, = 0:21, = 0: Figure 4 Subspace angle between SVD and EIV-PAST subspace estimates. SNR=8 db, e(t) = 1 1?0:8q?1 "(t). = 0:97, = 0:21, = 0: Figure 5 Frequency estimates for a frequency modulated frequency. SNR=10 db, e(t) = 1 1?0:8q?1 "(t). = 0:97, = 0:21, = 0: Figure 6 Frequency estimates for complex linear FM. SNR=15 db, = 0:97, = 0:21, = 0:00003 Figure 7 Deviation from Orthonormality for EIV-PAST. SNR=15 db, Solid: = 1, Dashed: = 0:97, Dotted: = 0:95 Figure 8 Subspace Angle between EIV-PAST estimate and true Subspace. SNR=15 db, Solid: = 1, Dashed: = 0:97, Dotted: = 0:95 25

26 Algorithm Complexity Rank Reduction Projection Approximation IV-PAST 3mn + O(n 2 ) No Yes EIV-PAST 3ml + O(mn) Yes Yes IVg 3mnl + O(nl) Yes No 2IV-PAST 4ml + O(mn) Yes Yes Table 1: Frequency (Normalized) PAST: EIV PAST: Sample No. Figure 1: Frequency (Normalized) EIV PAST: IV PAST: 2IV PAST: IVg: SVD: Sample No. Figure 2: 26

27 EIV PAST: IV PAST: 2IV PAST: IVg: Sample No Figure 3: Degrees Sample No. Figure 4: 27

28 0.16 Frequency (Normalized) EIV PAST: 2IV PAST: IVg: SVD: Sample No. Figure 5: Frequency (Normalized) EIV PAST: 2IV PAST: IVg: SVD: Sample No. Figure 6: 28

29 10 2 Deviation from Orthonormality Sample No. Figure 7: Degrees Sample No Figure 8: 29