Adaptive algorithms for solving generalized eigenvalue signal enhancement problems

Abstract

In this paper, we investigate adaptive algorithms for solving generalized eigenvalue problems that arise in the context of signal enhancement. This problem applies in general to any setup involving vectors of signal and interference samples, including wideband temporal processing, diffuse spatial (array) processing, or any combination thereof. The algorithms attempt to solve a generalized eigenvalue (GEV) problem using only snapshots of signal and interference training vectors, and the goal is to do this with a minimum amount of data and computational resources. The algorithms considered fall into two classes: two-step approaches that first estimate the covariance matrices and then solve the GEV problem; and, stochastic gradient type algorithms that recursively update the solution in one step for each new set of data vector snapshots. The algorithms are compared on the basis of convergence rate and computational complexity.

Introduction

In many signal processing applications, we are presented with a time sequence of composite M-dimensional vectors

$$\text{x}\text{(n)=}\text{s}\text{(n)+}\text{u}\text{(n),}$$

where $\text{s}\text{(n)}$ is a linear manifold of some underlying desired signal s(n) and $\text{u}\text{(n)}$ is undesired noise or interference. If the signals are Gaussian, maximum likelihood estimators of the signal employ linear processing to maximize the output signal power while minimizing the output interference power. Here we assume that this is the case and form an instantaneous linear estimate of s(n) as

$$\text{s}\text{̂}\text{(n)=}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{x}\text{(n),}$$

where $\text{w}\text{(n)≡[w}{}_1\text{w}{}_2\text{⋯}\text{w}{}_{\mathrm{M}}\text{]}{}^{\mathrm{<mtext>T</mtext>}}$ is a “weight” vector that forms the linear estimate and superscript H denotes (Hermitian) complex conjugate; complex signals are assumed throughout for generality.

Examples of such a setup abound in the literature, where the signal and interference vectors could represent samples of a random process in the temporal domain [4] or array element signals in the spatial domain [3]. The temporal domain formulation arises in classical signal detection and estimation problems, while spatial-domain scenarios involve array processing of electromagnetic or acoustic signals.

We express the output signal and interference power as, respectively,

$$\text{P}{}_{\mathrm{s}}\text{=}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{R}{}_{\mathrm{s}}\text{w}$$

and

$$\text{P}{}_{\mathrm{u}}\text{=}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{R}{}_{\mathrm{u}}\text{w}\text{,}$$

where $\text{R}{}_{\mathrm{s}}\text{,}\text{R}{}_{\mathrm{u}}$ are the respective covariance matrices associated with the signal and interference components of $\text{x}\text{(n)}$. The intention of the adaptive algorithm is to adjust $\text{w}$ so as to make the output signal power “large” and the output interference power “small”. One objective function to realize these two goals can be formulated as the ratio

$$\text{ξ}{}_0\text{≡}\text{P}{}_{\mathrm{s}}\text{P}{}_{\mathrm{u}}\text{=}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{R}{}_{\mathrm{s}}\text{w}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{R}{}_{\mathrm{u}}\text{w}\text{.}$$

However, maximization of this ratio is often not satisfactory for a number of reasons. First of all, we need to form estimates of the signal covariance matrices, so that maximization of (5) may lead to a poorly conditioned solution that is susceptible to additive noise and other unmodeled effects. Also, in some applications, it may be desirable to “balance” the degree to which P_s is maximized and P_u is minimized. An example of this is in downlink cellular mobile communications, where $\text{w}$ is the weighting of an antenna array and the problem is to control how much emphasis should be placed on maximizing the power to the intended mobile, P_s, as compared to the resources spent on minimizing power to the co-channel mobiles, P_u [8]. In both cases, the problem is solved by “regularizing” the denominator of (5) to obtain the modified objective function

$$\text{ξ=}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{R}{}_{\mathrm{s}}\text{w}\text{w}{}^{\mathrm{<mtext>H</mtext>}}\text{(}\text{R}{}_{\mathrm{u}}\text{+α}\text{I}\text{)}\text{w}\text{,}$$

where α is a parameter whose value determines the balance of resources expended in maximizing P_s on one hand and minimizing P_u on the other. If α=0, then ξ=ξ₀ and we have the unconstrained version (5), which typically favors minimization of P_u. At the other extreme, if α→∞, then P_s is maximized without any regard to P_u. Clearly, some finite value of α will result in the best performance, as determined by the particular application.

Objective functions like , are known as Rayleigh quotients and their maximization is well known to be the solution of a generalized eigenvalue problem involving the numerator and denominator matrix pair. Thus, the solution to maximizing (6) is the eigenvector corresponding to the largest eigenvalue of the generalized eigenvalue problem

$$\text{R}{}_{\mathrm{s}}\text{w}\text{=λ(}\text{R}{}_{\mathrm{u}}\text{+α}\text{I}\text{)}\text{w}\text{.}$$

However, we do not know the exact values of the signal and interference covariance matrices, $\text{R}{}_{\mathrm{s}}$ and $\text{R}{}_{\mathrm{u}}$, and so must estimate these from the available data sequence $\text{x}\text{(n)}$. Here, we assume that the signal and interference components can be somehow isolated to provide individual training samples of signal vectors $\text{s}\text{(n)}$ and interference vectors $\text{u}\text{(n)}$. This can be accomplished, for example, if we have side information as to when either $\text{s}\text{(n)}$ alone or $\text{u}\text{(n)}$ alone are present in $\text{x}\text{(n)}$. Another way this can be provided is for example in a communications environment, whereby a known training sequence of data symbols enables the signal vector $\text{s}\text{(n)}$ to be perfectly reconstructed from the detected message data symbols and subtracted from the composite received signal to obtain the interference signal vector. Following this rationale, we take $\text{s}\text{(n)}$ and $\text{u}\text{(n)}$ as given quantities.

In this paper we use the signal and interference vectors to obtain adaptive solutions of the generalized eigenvalue problem (7). The most straightforward approach is to use these vectors to form estimates of the associated covariance matrices, which are then substituted into (7) to set up the generalized eigenvalue problem. This two-step process is the most straightforward to understand. However, it is computationally demanding, especially when the vector dimension M is large. As a first simplification we will propose a hybrid technique that simultaneously updates the covariance matrix estimates and computes an iterative solution to the optimal generalized eigenvector. To further improve numerical efficiency, we will also derive stochastic gradient algorithms that recursively update the optimal solution of $\text{w}$ in one step for each new set of data vector snapshots. In the next section, we discuss these algorithms. Then in Section 3, we introduce a simple wideband signal model and evaluate the performance of the various algorithms in Section 4. Conclusions are presented in Section 5.