Allan O. Steinhardt

Adaptive array detection of uncertain rank one waveforms

Adaptive array detection of known (within a complex scaling) rank one space time waveforms in unknown spatial noise has received considerable attention. The two published solutions are the adaptive matched filter, and the GLRT. We expand on this work to consider the case of rank one waveforms that are uncertain, i.e., only partially known. More precisely, we model the space time steering vector as the Kronecker product of two vectors, each of which is unknown but is known to lie in a known subspace. Applications for such a model include detection in the presence of multipath and spectral or polarization diversity in both radar processing and wireless communication. Using the principle of invariance, we construct detectors based on the maximal invariant. We show that the SNR required to achieve a given detection probability (for a given false alarm rate) is only weakly impacted by waveform uncertainty. Thus, our detector approaches the performance of earlier detectors, which entail known waveforms.

Stabilized hyperbolic Householder transformations

A modification of the hyperbolic Householder scheme is introduced which is demonstrably stable theoretically (according to an established stability criterion) and which exhibits superior numerical behavior in simulations. The modified transform scheme effects downdating by applying conventional orthonormal, rather than hyperbolic, Householder transformations to the data. The latter have preferable numerical properties. However, the construction of these orthonormal operators itself requires hyperbolic computations. Thus, the proposed method is, in some sense, half hyperbolic and half orthonormal. There is no computational penalty incurred with these stabilized hyperbolic Householder transforms; they enjoy an operation count identical to their conventional counterparts. >

Existence of the hyperbolic singular value decomposition

Abstract The hyperbolic singular value decomposition is defined on a general n × m matrix and an m × m signature matrix pair. It is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products. Such differences arise in signal processing applications in the context of the covariance differencing. The hyperbolic SVD applies in problems where the conventional SVD cannot be employed. The existence of the hyperbolic singular value decomposition is here extended to the most general case, where neither the general matrix nor the matrix product is assumed full rank.

The PDF of adaptive beamforming weights

The author derives a closed-form expression for the marginal probability density function (PDF) for the weight vector coefficients in a minimum-variance distortionless response (MVDR) adaptive beamformer, when the snapshots are independently identically distributed (IID) normal and the weights are computed via sample matrix inversion. The marginal PDF allows one to determine the dynamic range required to avoid saturation (with a specified degree of probability) in digital and/or analog implementation of beamforming weights. >

Matrix Downdating Techniques For Signal Processing

We are concerned with a problem of finding the triangular (Banachiewicz-Cholesky) factor of the covariance matrix after deleting observations from the corresponding linear least squares equations. Such a problem, often referred to as downdating, arises in classical signal processing as well as in various other broad ares of computing. Examples include recursive least squares estimation and filtering with a sliding rectangular window in adaptive signal processing, outlier suppression and robust regression in statistics, and the modification of Hessian matrices in the numerical solution of non-linear equations. Formally the problem can be described as follows: Given an n xn upper triangular matrix L and an n-dimensional vector x such that LTL - xxT > 0 find an n xn lower triangular matrix L such that LLT = LLT - XXT We will look at the following issues relevant to the downdating problem: - stability - rank-1 downdating algorithms - generalization to modifications of a higher rank

The hyperbolic singular value decomposition and applications

A new generalization of singular value decomposition (SVD), the hyperbolic SVD, is advanced, and its existence is established under mild restrictions. Two algorithms for effecting this decomposition are discussed. The new decomposition has applications in downdating in problems where the solution depends on the eigenstructure of the normal equations and in the covariance differencing algorithm for bearing estimation in sensor arrays. Numerical examples demonstrate that, like its conventional counterpart, the hyperbolic SVD exhibits superior numerical behavior relative to explicit formation and solution of the normal equations. (However, unlike ordinary SVD, it is applicable to eigenanalysis of covariances arising from a difference of outer products).<<ETX>>

The total P/sub FA/ of the multiwindow harmonic detector and its application to real data

The authors study the multiwindow spectral analysis method as it applies to the detection of sinusoidal signals. They examine the probability of false alarm P/sub FA/. The total P/sub FA/ (sinusoidal frequency unknown) is shown analytically to be bounded below by the order statistics (minimum) of BM/K independent identically distributed (i.i.d.) beta variates, where M is the length of the data record used in the detection, K the number of windows, and B the width of the frequency band of interest. Simulation results indicate a much larger bound, the minimum of BM i.i.d. beta variates. It is shown that for real signals, the assumptions made in the derivation of the detector break down at frequencies close to zero and to half the sampling frequency. >

Multiwindow method for spectrum estimation and sinusoid detection in an array environment

In many applications (radar, communication, plasma physics) the signal of interest is sinusoidal and is hidden in non-white noise. The multi-window method of spectrum estimation gives a constant false alarm test for the presence of a sinusoid in a time series if the noise can be assumed Gaussian. In this paper we generalize the method to an array and study the resulting test. Expressions for the probabilities of detection and of false alarm are obtained analytically and receiver operation characteristic curves are computed for a particular scenario under different conditions.

Corrections to 'Harmonic retrieval of analog signals'

In the above mentioned paper (ibid., vol.36, p.134-6, Jan. 1988) the author represented the signal mu (t) in baseband form, and he now argues that this is an error since it results in unbounded power estimates p/sub i/. It is necessary to treat mu (t) directly as a bandpass signal. This changes the support of omega in eq. (1) and the region of integration in eq. (2) and (4), and a new version of eq. (6) results. >