The Wiener filter is analyzed for stationary complex Gaussian signals from an information-theoretic point of view. A dual-port analysis of the Wiener filter leads to a decomposition based on orthogonal projections and results in a new multistage method for implementing the Wiener filter using a nested chain of scalar Wiener filters. This new representation of the Wiener filter provides the capability to perform an information-theoretic analysis of previous, basis-dependent, reduced-rank Wiener filters. This analysis demonstrates that the recently introduced cross-spectral metric is optimal in the sense that it maximizes mutual information between the observed and desired processes. A new reduced-rank Wiener filter is developed based on this new structure which evolves a basis using successive projections of the desired signal onto orthogonal, lower dimensional subspaces. The performance is evaluated using a comparative computer analysis model and it is demonstrated that the low-complexity multistage reduced-rank Wiener filter is capable of outperforming the more complex eigendecomposition-based methods.

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and the noise is MVN with mean ScjJ and covariance R = a 2 R o:where J-L = 0 under H o and J-L > 0 under H 1 • This is the standard detection problem wherein the polarity of the signal x is assumed known. Near the end of Section V we replacewhere polarity is unknown.We shall assume that the signal x obeys the linear subspaceThe detection problems to be studied in this paper may be described as follows. We are given N samples from a real, scalar time series {y(n), n = 0,1, 0 0 " N -I} which are assembled into the N-dimensional measurement vectorBased on these data, we must decide between two possible hypotheses regarding how the data was generated. The null hypothesis H o says that the data consist of noise tI only. The alternative hypothesis H 1 says that the data consist of a sum of signal J-LX and noise tI; that is, (2.2) (2.1)to include subspace interferences. These problems involve unknown parameters in the mean and covariance of a multivariate normal (MVN) distribution. For each problem in the class, we establish invariances for the GLR and find that they are identical to the natural invariances for the problem. We show that a monotone function of the GLRT equals one of the uniformly most powerful invariant (UMP-invariant) tests derived in [1]. This means that the GLRT is itself UMPinvariant. In addition to tying up the theories of invariance and the GLRT, our results generalize and extend previous work on these problems published in [1]-[6]. We begin our development by establishing the invariances of the GLRT in the MVN problem. We then specialize our results for structured means in order to derive UMP-invariant GLRT detectors for matched subspace filtering in subspace interference. The GLRT produces an UMP-invariant detector, which is CFAR if the noise variance is unknown. As we shall find, the optimum detector may be interpreted as a null steering or interference rejecting processor followed by a matched subspace detector.

Prony analysis is an emerging methodology that extends Fourier analysis by directly estimating the frequency. damping. strength. and relative phase of modal components present in a given signal. The ability to extract such information from transient stability program simulations and from large-scale system tests or disturbances would be quite valuable to power system engineers. This paper reports early results in the application of this method to stability program output. It also includes benchmarks against known models and a brief mathematical summary.

Abstract-In this paper, we use the theory of generalized likelihood ratio tests (GLRTs) to adapt the matched subspace detectors (MSDs) of [1] and [2] to unknown noise covariance matrices. In so doing, we produce adaptive MSDs that may be applied to signal detection for radar, sonar, and data communication. We call the resulting detectors adaptive subspace detectors (ASDs). These include Kelly's GLRT and the adaptive cosine estimator (ACE) of [6] and [19] for scenarios in which the scaling of the test data may deviate from that of the training data. We then present a unified analysis of the statistical behavior of the entire class of ASDs, obtaining statistically identical decompositions in which each ASD is simply decomposed into the nonadaptive matched filter, the nonadaptive cosine or -statistic, and three other statistically independent random variables that account for the performance-degrading effects of limited training data.

Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probability distribution that is invariant under rotation in the complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of complexvalued data: (1) utilizing the complete statistical characterization of complex-valued random signals; and (2) the optimization of real-valued cost functions with respect to complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some T.

Oblique projection operators are used to project measurements onto a low-rank subspace along a direction that is oblique to the subspace. They may be used to enhance signals while nulling interferences. In this paper, we give several basic results for oblique projections, including formulas for constructing oblique projections with desired range and null space. We analyze the algebra and geometry of oblique projections in order to understand their properties. We then show how oblique projections can be used to separate signals from structured noise (such as impulse noise), damped or undamped interfering sinusoids (such as power line interference), and narrow-band noise. In some of the problems we address, the oblique projection provides an alternative way to implement an already known solution. Expressing these solutions as oblique projections brings geometrical insight to the study of the solution. The geometry of oblique projections enables us to compute performance in terms of angles between signal and noise subspaces. As a special case of removing impulse noise, we can use oblique projections to interpolate missing data samples. In array processing, oblique projections can be used to simultaneously steer beams and nulls. In communications, oblique projections can be used to remove intersymbol interference.

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