Benjamin Friedlander

Matched subspace detectors

We formulate a general class of problems for detecting subspace signals in subspace interference and broadband noise. We derive the generalized likelihood ratio (GLR) for each problem in the class. We then establish the invariances for the GLR and argue that these are the natural invariances for the problem. In each case, the GLR is a maximal invariant statistic, and the distribution of the maximal invariant statistic is monotone. This means that the GLR test (GLRT) is the uniformly most powerful invariant detector. We illustrate the utility of this finding by solving a number of problems for detecting subspace signals in subspace interference and broadband noise. In each case we give the distribution for the detector and compute performance curves. >

Lattice filters for adaptive processing

This paper presents a tutorial review of lattice structures and their use for adaptive prediction of time series. Lattice filters associated with stationary covariance sequences and their properties are discussed. The least squares prediction problem is defined for the given data case, and it is shown that many of the currently used lattice methods are actually approximations to the stationary least squares solution. The recently developed class of adaptive least squares lattice algorithms are described in detail, both in their unnormalized and normalized forms. The performance of the adaptive least squares lattice algorithm is compared to that of some gradient adaptive methods. Lattice forms for ARMA processes, for joint process estimation, and for the sliding-window covariance case are presented. The use of lattice structures for efficient factorization of covariance matrices and solution of Toeplitz sets of equations is briefly discussed.

Direction finding in the presence of mutual coupling

An eigenstructure-based method for direction finding in the presence of sensor mutual coupling, gain, and phase uncertainties is presented. The method provides estimates of the directions-of-arrival (DOA) of all the radiating sources as well as calibration of the gain and phase of each sensor and the mutual coupling in the receiving array. The proposed algorithm is able to calibrate the array parameters without prior knowledge of the array manifold, using only signals of opportunity and avoiding the need for deploying auxiliary sources at known locations. The algorithm is described in detail, and its behavior is illustrated by numerical examples. >

The discrete polynomial-phase transform

The discrete polynomial-phase transform (DPT) is a new tool for analyzing constant-amplitude polynomial-phase signals. The main properties of the DPT are its ability to identify the degree of the phase polynomial and to estimate its coefficients. The transform is robust to deviations from the ideal signal model, such as slowly-varying amplitude, additive noise and nonpolynomial phase. The authors define the DPT, derive its basic properties, and use it to develop computationally efficient estimation and detection algorithms. A statistical accuracy analysis of the estimated parameters is also presented. >

Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements

The problem of estimating the frequencies, phases, and amplitudes of sinusoidal signals is considered. A simplified maximum-likelihood Gauss-Newton algorithm which provides asymptotically efficient estimates of these parameters is proposed. Initial estimates for this algorithm are obtained by a variation of the overdetermined Yule-Walker method and periodogram-based procedure. Use of the maximum-likelihood Gauss-Newton algorithm is not, however, limited to this particular initialization method. Some other possibilities to get suitable initial estimates are briefly discussed. An analytical and numerical study of the shape of the likelihood function associated with the sinusoids-in-noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure. Some numerical examples are presented to illustrate the performance of the proposed estimation procedure. Comparison to the performance corresponding to the Cramer-Rao lower bound is also presented, using a simple expression for the asymptotic Cramer-Rao bound covariance matrix derived in the paper. >

A passive localization algorithm and its accuracy analysis

The problem of estimating source location from noisy measurements of range differences (RDs) is considered. A localization technique based on solving a set of linear equations is presented and its accuracy properties are analyzed. An optimal weighting matrix for the least squares estimator is derived. The analytical expressions for the variance and bias of the estimator are validated by Monte-Carlo simulation. The problem of estimating source velocity given measurements of range differences and range-rate differences is briefly considered, and a linear equation technique is derived.

Direction finding algorithms based on high-order statistics

Two direction finding algorithms are presented for nonGaussian signals, which are based on the fourth-order cumulants of the data received by the array. The first algorithm is similar to MUSIC, while the second is asymptotically minimum variance in a certain sense. The first algorithm requires singular value decomposition of the cumulant matrix, while the second is based on nonlinear minimization of a certain cost function. The performance of the minimum variance algorithm can be assessed by analytical means, at least for the case of discrete probability distributions of the source signals and spatially uncorrelated Gaussian noise. The numerical experiments performed seem to confirm the insensitivity of these algorithms to the (Gaussian) noise parameters. >

Analysis of the asymptotic relative efficiency of the MUSIC algorithm

An analytical performance evaluation of the errors of the direction-of-arrival estimates obtained by the MUSIC algorithm for uncorrelated sources is provided. Explicit asymptotic formulas are derived for the means and the covariance of the estimates. The covariances are then compared to the Cramer-Rao lower bound. It is shown that for a single course, the MUSIC algorithm is asymptotically efficient. For multiple sources, the algorithm is not efficient in general. However, it approaches asymptotic efficiency when the SNRs (signal-to-noise ratios) of all sources tend to infinity. It is illustrated by several test cases that the relative efficiency of the MUSIC algorithm is nearly one under a wide range of parameter variations. The analytic performance evaluation thus confirms empirical evidence to the excellent performance of the MUSIC algorithm for narrowband signals. >

The root-MUSIC algorithm for direction finding with interpolated arrays

Abstract This paper presents a direction finding technique which uses the outputs of a virtual array, computed from the real array using a linear interpolation procedure over a given sector. The geometry of the virtual array is under the control of the designer. By using a linear virtual array, the root-MUSIC algorithm can be applied, even though the real array may have an arbitrary geometry. The root-MUSIC algorithm offers several advantages over the MUSIC algorithm, including significantly reduced computational requirements. Various issues related to the design of the interpolated array are discussed. The performance of the algorithm was tested by extensive simulations and was found to equal the performance of the MUSIC algorithm applied to the real array. Thus, the interpolation procedure does not introduce any performance degradation.

Detection of transient signals by the Gabor representation

Gabor representation is used for the detection of transient signals with unknown arrival times. A one-sided exponential window function is used which seems to be most appropriate for transient modelling. Explicit expressions for the Gabor coefficients are given for this window function. When the given signal is random, so are the coefficients. The second-order moments of the Gabor coefficients are computed for a white noise signal. These are then used to introduce a detection statistic based on the Gabor coefficients. The proposed detector is capable of separating transients having different arrival times, even in this case where their waveforms partially overlap. >