Eigenvalues and eigenvectors of generalized DFT, generalized DHT,DCT-IV and DST-IV matrices

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms     

Eigenvalues and eigenvectors of general gyroelectric media

Explicit and relatively simple expressions for eigenvalues and guided (propagating) eigenvectors of a general gyroelectric medium, where the preferred guided wave direction, zˆ, is parallel to the gyrotropic axis and anisotropy is confined to a plane transverse to z, are given. Some special cases of interest, namely, Hermitian, symmetric (biaxial), and uniaxial permittivity tensors, are also considered. The natural, or optic, coordinate basis is used to derive the source-free eigenvectors and to explicitly reveal the polarization states of those eigenvectors. Also under this basis, the evolution of eigenvalues and eigenvectors as off-diagonal terms of the permittivity tensor uniformly vanish, a transition from the biaxial to the uniaxial case, is discussed     

Estimation of structured covariance matrices

Covariance matrices from stationary time series are Toeplitz. Multichannel and multidimensional processes have covariance matrices of block Toeplitz form. In these cases and many other situations, one knows that the actual covariance matrix belongs to a particular subclass of covariance matrices. This paper discusses a method for estimating a covariance matrix of specified structure from vector samples of the random process. The theoretical foundation of the method is to assume that the random process is zero-mean multivariate Gaussian, and to find the maximum-likelihood covariance matrix that has the specified structure. An existence proof is given and the solution is interpreted in terms of a minimum-entropy principle. The necessary gradient conditions that must be satisfied by the maximum-likelihood solution are derived and unique and nonunique analytic solutions for some simple problems are presented. A major contribution of this paper is an iterative algorithm that solves the necessary gradient equations for moderate-sized problems with reasonable computational ease. Theoretical convergence properties of the basic algorithm are investigated and robust modifications discussed. In doing maximum-entropy spectral analysis of a sine wave in white noise from a single vector sample, this new estimation procedure causes no splitting of the spectral line in contrast to the Burg technique.     

Computation of transform domain covariance matrices

It is often of interest in applications to compute the covariance matrix of a random process transformed by a fast unitary trasform. Here, the recursive definition of fast unitary transforms [1] is used to derive recursive relations for the covariance matrices of the transformed process. These relations lead to fast methods of computation of covariance matrices and to substantial reductions of the number of arithmetic operations required.     

Robust estimation of structured covariance matrices

In the context of the narrowband array processing problem, robust methods for accurately estimating the spatial correlation matrix using a priori information about the matrix structure are developed. By minimizing the worse case asymptotic variance, robust, structured, maximum-likelihood-type estimates of the spatial correlation matrix in the presence of noises with probability density functions in the ∈-contamination and Kolmogorov classes are obtained. These estimates are robust against variations in the noise's amplitude distribution. The Kolmogorov class is demonstrated to be the natural class to use for array processing applications, and a technique is developed to determine exactly the size of this class. Performance of bearing estimation algorithms improves substantially when the robust estimates are used, especially when nonGaussian noise is present. A parametric structured estimate of the spatial correlation matrix that allows direct estimation of the arrival angles is also demonstrated     

Hypothesis testing of complex covariance matrices

Letcal ybe a mean zero complex stationary Gaussian signal process depending on a vector parametertheta prime = { theta_{1}, theta_{2}, theta_{3} }whose components represent parameters of the covariance function R(r) ofcal y. These parameters are chosen astheta_{1} = R(0), theta_{2} = |R( tau )| /R(0), theta_{3} =phase ofR( tau), and they are simply related to the parameters of the spectral density ofcal y. This paper is concerned with the determination of most powerful (MP) tests that distinguish between random signals having different covariance functions. The tests are based uponNcorrelated pairs of independent observations oncal y. Although the MP test that distinguishes betweentheta = theta_{o}and the alternative hypothesistheta = theta_{1}has been solved previously [11], the problem of identifying the random signals is often complicated by the fact that the signal powertheta_{1} = R(0)is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesislambda equiv R( tau)/R(0) = lambda_{0}and the composite alternativelambda = lambda_{1}. In addition, the uniformly MP invariant test that distinguishes between the composite hypothesestheta_{2} <_{=} | lambda_{o} |andtheta_{2} > | lambda_{0} |has also been found. In all cases, exact probability distributions have been obtained.     

NAR estimators of spatial covariance matrices for adaptive arraydetection

The theory of noise-alone-reference (NAR) power estimation is extended to the estimation of spatial covariance matrices. A NAR covariance estimator insensitive to signal presence is derived. The SNR (signal-to-noise ratio) loss incurred by using this estimator is independent of the input SNR and is less than that encountered with the maximum likelihood covariance estimator given that the same number of uncorrelated snapshots is available to both estimators. The analysis assumes first a deterministic signal. The results are extended and generalized to signals with unknown parameters or random signals. For the random signal case, generalized and quasi-NAR covariance estimators are presented     

Eigenvalue analysis of spatial covariance matrices for correlated signals

A comparative analysis is presented of the signal-subspace eigenvalues of a conventional covariance matrix and a covariance matrix after a spatial smoothing preprocessing procedure is performed. This comparison clearly demonstrates the advantages and disadvantages of the spatial smoothing technique.<>     

On the generalized eigenvectors of a class of moment matrices

An investigation is made of the eigenstructure of a class of lower triangular moment matrices that arose in the context of finding the forced response of IIR filters to typical excitations. It is found that the Jordan matrix can have at most two types of Jordan blocks. The modal matrix is shown to have a peculiar structure where the progenitors in the column partitions corresponding to the Jordan blocks have a certain pattern     

Penalized maximum-likelihood estimation of covariance matrices withlinear structure

In this paper, a space-alternating generalized expectation-maximization (SAGE) algorithm is presented for the numerical computation of maximum-likelihood (ML) and penalized ML (PML) estimates of the parameters of covariance matrices with linear structure for complex Gaussian processes. By using a less informative hidden-data space and a sequential parameter-update scheme, a SAGE-based algorithm is derived for which convergence of the likelihood is demonstrated to be significantly faster than that of an EM-based algorithm that has been previously proposed. In addition, the SAGE procedure is shown to easily accommodate penalty functions, and a SAGE-based algorithm is derived and demonstrated for forming PML estimates with a quadratic smoothness penalty     

A maximal invariant framework for adaptive detection withstructured and unstructured covariance matrices

We introduce a framework for exploring array detection problems in a reduced dimensional space by exploiting the theory of invariance in hypothesis testing. This involves calculating a low-dimensional basis set of functions called the maximal invariant, the statistics of which are often tractable to obtain, thereby making analysis feasible and facilitating the search for tests with some optimality property. Using this approach, we obtain a locally most powerful invariant test for the unstructured covariance case and show that all invariant tests can be expressed in terms of the previously published Kelly's generalized likelihood ratio (GLRT) and Robey's adaptive matched filter (AMF) test statistics. Applying this framework to structured covariance matrices, corresponding to stochastic interferers in a known subspace, for which the GLRT is unavailable, we obtain the maximal invariant and propose several new invariant detectors that are shown to perform as well or better than existing ad-hoc detectors. These invariant tests are unaffected by most nuisance parameters, hence the variation in the level of performance is sharply reduced. This framework facilitates the search for such tests even when the usual GLRT is unavailable     

Adaptive eigendecomposition of data covariance matrices based onfirst-order perturbations

In this paper, new algorithms for adaptive eigendecomposition of time-varying data covariance matrices are presented. The algorithms are based on a first-order perturbation analysis of the rank-one update for covariance matrix estimates with exponential windows. Different assumptions on the eigenvalue structure lead to three distinct algorithms with varying degrees of complexity. A stabilization technique is presented and both issues of initialization and computational complexity are discussed. Computer simulations indicate that the new algorithms can achieve the same performance as a direct approach in which the exact eigendecomposition of the updated sample covariance matrix is obtained at each iteration. Previous algorithms with similar performance require O(LM²) complex operations per iteration, where L and M respectively denote the data vector and signal-subspace dimensions, and involve either some form of Gram-Schmidt orthogonalization or a nonlinear eigenvalue search. The new algorithms have parallel structures, sequential operation counts of order O(LM) or less, and do not involve any of the above steps. One particular algorithm can be used to update the complete signal-subspace eigenstructure in 5LM complex operations. This represents an order of magnitude improvement in computational complexity over existing algorithms with similar performance. Finally, a simplified local convergence analysis of one of the algorithms shows that it is stable and converges in the mean to the true eigendecomposition. The convergence is geometrical and is characterized by a single time constant     

Fast frequency allocation in WDM systems with unequally spaced channels

A new method for allocating the unequally space channels (USCs) in WDM systems to minimise the crosstalk caused by four-wave mixing is presented. The proposed method has the advantages of easy computer implementation, fast generation of USC sets, and flexible change of system parameters. The results show that a larger-size set can be easily constructed from the smaller-size set to facilitate engineering applications. The optimal/suboptimal USC sets are also obtained.     

Inferring the eigenvalues of covariance matrices from limited,noisy data

The eigenvalue spectrum of covariance matrices is of central importance to a number of data analysis techniques. Usually, the sample covariance matrix is constructed from a limited number of noisy samples. We describe a method of inferring the true eigenvalue spectrum from the sample spectrum. Results of Silverstein (1986), which characterize the eigenvalue spectrum of the noise covariance matrix, and inequalities between the eigenvalues of Hermitian matrices are used to infer probability densities for the eigenvalues of the noise-free covariance matrix, using Bayesian inference. Posterior densities for each eigenvalue are obtained, which yield error estimates. The evidence framework gives estimates of the noise variance and permits model order selection by estimating the rank of the covariance matrix. The method is illustrated with numerical examples     

Block time and frequency domain modified covariance algorithms forspectral analysis

Block modified covariance algorithms are proposed for autoregressive parametric spectral estimation. First, the authors develop the block modified covariance algorithm (BMCA) which can be implemented either in the time or in the frequency domain-with the latter being more efficient in high-order cases. A block algorithm is also developed for the energy weighted combined forward and backward prediction. This algorithm is called energy weighted BMCA (EWBMCA) and its performance is analogous to that of the weighted covariance method proposed by Nikias and Scott (1983). Time-varying convergence factors, designed to minimize the error energy from one iteration to the next, are given for both algorithms. In addition, three updating schemes are presented, namely block-by-block, sample-by-sample, and sample-by-sample with time-scale separation. The performance of the proposed algorithms is examined with stationary and nonstationary narrowband and broadband processes, and also with sinusoids in noise. Lastly, the authors discuss the computational complexity of the proposed algorithms and give performance comparisons to existing modified covariance algorithms     

The inversion of covariance matrices by finite Fourier transforms (Corresp.)

A matrix series is derived that converges to the inverse of a covariance matrix. The members of the series are derived from a circular matrix that can be inverted by taking finite Fourier transforms. An example of the method is presented.     

Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices

A technique is proposed for generating initial orthonormal eigenvectors of the discrete Fourier transform matrix F by the singular-value decomposition of its orthogonal projection matrices on its eigenspaces and efficiently computable expressions for those matrices are derived. In order to generate Hermite-Gaussian-like orthonormal eigenvectors of F given the initial ones, a new method called the sequential orthogonal procrustes algorithm (SOPA) is presented based on the sequential generation of the columns of a unitary matrix rather than the batch evaluation of that matrix as in the OPA. It is proved that for any of the SOPA, the OPA, or the Gram-Schmidt algorithm (GSA) the output Hermite-Gaussian-like orthonormal eigenvectors are invariant under the change of the input initial orthonormal eigenvectors.     

A complete factorization of paraunitary matrices with pairwisemirror-image symmetry in the frequency domain

The problem of designing orthonormal (paraunitary) filter banks has been addressed in the past. Several structures have been reported for implementing such systems. One of the structures reported imposes a pairwise mirror-image symmetry constraint on the frequency responses of the analysis (and synthesis) filters around π/2. This structure requires fewer multipliers, and the design time is correspondingly less than most other structures. The filters designed also have much better attenuation. We characterize the polyphase matrix of the above filters in terms of a matrix equation. We then prove that the structure reported in a paper by Nguyen and Vaidyanathan (1988), with minor modifications, is complete. This means that every polyphase matrix whose filters satisfy the mirror-image property can be factorized in terms of the proposed structure