Non-Gaussian statistical signal processing is important when signals and/or noise deviate from the ideal Gaussian model. Stable distributions are among the most important non-Gaussian models. They share defining characteristics with the Gaussian distribution, such as the stability property and central limit theorems, and in fact include the Gaussian distribution as a limiting case. To help engineers better understand the stable models and develop methodologies for their applications in signal processing. A tutorial review of the basic characteristics of stable distributions and stable signal processing is presented. The emphasis is on the differences and similarities between stable signal processing methods based on fractional lower-order moments and Gaussian signal processing methods based on second-order moments. >

Signal processing with fractional lower order moments: stable processes and their applications

Stochastic gradient-based adaptive algorithms are developed for the optimization of weighted myriad filters (WMyFs). WMyFs form a class of nonlinear filters, motivated by the properties of /spl alpha/-stable distributions, that have been proposed for robust non-Gaussian signal processing in impulsive noise environments. The weighted myriad for an N-long data window is described by a set of nonnegative weights {w/sub i/}/sub i=l//sup N/ and the so-called linearity parameter K>0. In the limit, as K/spl rarr//spl infin/, the filter reduces to the familiar weighted mean filter (which is a constrained linear FIR filter). Necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion. An implicit formulation of the filter output is used to find an expression for the gradient of the cost function. Using instantaneous gradient estimates, an adaptive steepest-descent algorithm is then derived to optimize the weights. This algorithm involves a very simple update term that is computationally comparable to the update in the classical LMS algorithm. The robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one-dimensional chirp-type signal in impulsive noise.

Adaptive weighted myriad filter algorithms for robust signal processing in /spl alpha/-stable noise environments

This paper develops fast algorithms to compute the output of the weighted myriad filter. Myriad filters form a large and important class of nonlinear filters for robust non-Gaussian signal processing and communications in impulsive noise environments. Just as the weighted mean and the weighted median are optimized for the Gaussian and Laplacian distributions, respectively, the weighted myriad is based on the class of /spl alpha/-stable distributions, which can accurately model impulsive processes. The weighted myriad is an M-estimator that is defined in an implicit manner; no closed-form expression exists for it, and its direct computation is a nontrivial and prohibitively expensive task. In this paper, the weighted myriad is formulated as one of the fixed points of a certain mapping. An iterative algorithm is proposed to compute these fixed points, and its convergence is proved rigorously. Using these fixed point iterations, fast algorithms are developed for the weighted myriad. Numerical simulations demonstrate that these algorithms compute the weighted myriad with a high degree of accuracy at a relatively low computational cost.

Fast algorithms for weighted myriad computation by fixed-point search

This paper proposes a new fast recursive total least squares (N-RTLS) algorithm to recursively compute the TLS solution for adaptive finite impulse response (FIR) filtering. The N-RTLS algorithm is based on the minimization of the constrained Rayleigh quotient (c-RQ) in which the last entry of the parameter vector is constrained to the negative one. As analysis results on the convergence of the proposed algorithm, we study the properties of the stationary points of the c-RQ. The high computational efficiency of the new algorithm depends on the efficient computation of the fast gain vector (FGV) and the adaptation of the c-RQ. Since the last entry of the parameter vector in the c-RQ has been fixed as the negative one, a minimum point of the c-RQ is searched only along the input data vector, and a more efficient N-RTLS algorithm is obtained by using the FGV. As compared with Davila's RTLS algorithms, the N-RTLS algorithm saves the 6M number of multiplies, divides, and square roots (MADs). The global convergence of the new algorithm is studied by LaSalle's invariance principle. The performances of the relevant algorithms are compared via simulations, and the long-term numerical stability of the N-RTLS algorithm is verified.

A fast recursive total least squares algorithm for adaptive FIR filtering

This paper develops a new fast recursive total least squares (N-RTLS) algorithm to recursively compute the total least squares (TLS) solution for adaptive infinite-impulse-response (IIR) filtering. The new algorithm is based on the minimization of the constraint Rayleigh quotient in which the first entry of the pa- rameter vector is fixed to the negative one. The highly computa- tional efficiency of the proposed algorithm depends on the efficient computation of the gain vector and the adaptation of the Reyleigh quotient. Using the shift structure of the input data vectors, a fast algorithm for computing the gain vector is established, which is referred to as the fast gain vector (FGV) algorithm. The computa- tional load of the FGV algorithm is smaller than that of the fast Kalman algorithm. Moreover, the new algorithm is numerically stable since it does not use the well-known matrix inversion lemma. The computational complexity of the new algorithm per iteration is also . The global convergence of the new algorithm is studied. The performances of the relevant algorithms are compared via simulations.

A Fast Recursive Total Least Squares Algorithm for

A new model of HF atmospheric radio noise (ARN) that leads to observed probability distributions of noise is briefly described. The new ARN model is then applied to the detection of known signals in the presence of noise to determine optimal receiver structure for ARN. Probabilities of errors for a linear matched filter and nonlinear (-1/log) and inverted (log)^{2} correlator receivers and a phase coherent binary detector are determined and compared. It is shown that the inverted (log)^{2} correlator combined with the linear matched filter is an optimal receiver in the presence of impulsive noise in tropics. The time statistics of burst noise is applied to design of signals for improved reliability.

S. Gupta

Papers

Signal Detection in the Presence of Atmospheric Noise in Tropics