Richard J. Kozick

Maximum-likelihood array processing in non-Gaussian noise with Gaussian mixtures

Many approaches have been studied for the array processing problem when the additive noise is modeled with a Gaussian distribution, but these schemes typically perform poorly when the noise is non-Gaussian and/or impulsive. This paper is concerned with maximum likelihood array processing in non-Gaussian noise. We present the Cramer-Rao bound on the variance of angle-of-arrival estimates for arbitrary additive, independent, identically distributed (iid), symmetric, non-Gaussian noise. Then, we focus on non-Gaussian noise modeling with a finite Gaussian mixture distribution, which is capable of representing a broad class of non-Gaussian distributions that include heavy tailed, impulsive cases arising in wireless communications and other applications. Based on the Gaussian mixture model, we develop an expectation-maximization (EM) algorithm for estimating the source locations, the signal waveforms, and the noise distribution parameters. The important problems of detecting the number of sources and obtaining initial parameter estimates for the iterative EM algorithm are discussed in detail. The initialization procedure by itself is an effective algorithm for array processing in impulsive noise. Novel features of the EM algorithm and the associated maximum likelihood formulation include a nonlinear beamformer that separates multiple source signals in non-Gaussian noise and a robust covariance matrix estimate that suppresses impulsive noise while also performing a model-based interpolation to restore the low-rank signal subspace. The EM approach yields improvement over initial robust estimates and is valid for a wide SNR range. The results are also robust to PDF model mismatch and work well with infinite variance cases such as the symmetric stable distributions. Simulations confirm the optimality of the EM estimation procedure in a variety of cases, including a multiuser communications scenario. We also compare with existing array processing algorithms for non-Gaussian noise.

Computation of discrete cosine transform using Clenshaw's recurrence formula

Clenshaws recurrence formula is used to derive recursive algorithms for the discrete cosine transform (DCT) and the inverse discrete cosine transform (IDCT). The recursive DCT algorithm presented requires one fewer delay element per coefficient and one fewer multiply operation per coefficient compared with two other proposed methods. Clenshaws recurrence formula provides a unified development for the recursive DCT and IDCT algorithms. The recursive algorithms apply to arbitrary length algorithms and are appropriate for VLSI implementation.<<ETX>>

Bounds on bearing and symbol estimation with side information

We develop Cramer-Rao bounds (CRBs) for bearing, symbol, and channel estimation of communications signals in flat-fading channels. We do this using the constrained CRB formulation of German and Hero (1990), and Stoica and Ng (see IEEE Signal Processing Lett., vol.5, p.177-79, 1998), with the unknown parameters treated as deterministic constants. The equality constraints may be combined arbitrarily, e.g., we may develop CRBs for bearing estimation of constant modulus (CM) signals where a subset of the symbols are known (semi-blind, CM case). The results establish the value of side information in a large variety of communications scenarios. We focus on the CM and semi-blind properties and develop closed-form CRBs for these cases. Examples are presented indicating the relative value of the training and CIM property. These show the significant amount of signal processing information provided under these two conditions. In addition, we consider the performance of the maximum-likelihood beamformer for the semi-blind case, assuming the bearings are known. This semi-blind beamformer achieves the appropriate (constrained) CRB with finite data at finite SNR. Analysis also reveals that in a semi-blind scenario with two closely spaced sources, ten or more training symbols are sufficient to achieve the asymptotic training regime. Together with previous results on angle estimation for known sources, these results indicate that relatively few training samples enable both angle estimation and closely spaced co-channel source separation that approaches the CRB with finite data and finite SNR.

An adaptive spatial diversity receiver for non-Gaussian interference and noise

Standard linear diversity combining techniques are not effective in combating fading in the presence of non-Gaussian noise. An adaptive spatial diversity receiver is developed for wireless communication channels with slow, flat fading and additive non-Gaussian noise. The noise is modeled as a mixture of Gaussian distributions, and the expectation-maximization (EM) algorithm is used to derive estimates for the model parameters. The parameter estimates are used in a generalized likelihood ratio test to reproduce the transmitted signals. The new receiver is shown to be relatively insensitive to errors in the parameter estimates as well as to errors in modeling the actual noise distribution.

Regularity and strict identifiability in MIMO systems

We study finite impulse response (FIR) multi-input multi-output (MIMO) systems with additive noise, treating the finite-length sources and channel coefficients as deterministic unknowns, considering both regularity and identifiability. In blind estimation, the ambiguity set is large, admitting linear combinations of the sources. We show that the Fisher information matrix (FIM) is always rank deficient by at least the number of sources squared and develop necessary and sufficient conditions for the FIM to achieve its minimum nullity. Tight bounds are given on the required source data lengths to achieve minimum nullity of the FIM. We consider combinations of constraints that lead to regularity (i.e., to a full-rank FIM and, thus, a meaningful Cramer-Rao bound). Exploiting the null space of the FIM, we show how parameters must be specified to obtain a full-rank FIM, with implications for training sequence design in multisource systems. Together with constrained Cramer-Rao bounds (CRBs), this approach provides practical techniques for obtaining appropriate MIMO CRBs for many cases. Necessary and sufficient conditions are also developed for strict identifiability (ID). The conditions for strict ID are shown to be nearly equivalent to those for the FIM nullity to be minimized.

Detecting interfaces on ultrasound images of the carotid artery by dynamic programming

A dynamic programming edge following procedure is applied to ultrasound images of the carotid artery. The objective is to automatically determine the far wall interfaces of the common carotid artery. The far wall interfaces are then used to estimate the far wall thickness, which is an important metric for disease diagnosis and treatment evaluation. A current system uses human readers to determine the carotid artery interfaces using digitized images on a computer display. This process is time consuming and difficult to control, since readers tend to vary over time in the way in which they identify interfaces. In addition, different readers tend to identify interfaces in slightly different ways. The edge following procedure is designed to apply a consistent and objective criteria to all images in order to reduce the variability in far wall thickness estimates. The edge following procedure works by joining local peaks in the image gradient. The gradient is estimated by a Sobel operator, and dynamic programming is used to join the peaks into a smooth edge. The dynamic programming is necessary to combat the effects of noise and speckle in the ultrasound images. The paper describes the dynamic programming cost function formulation and discuses the algorithm performance.

Methods for designing efficient parallel-recursive filter structures for computing discrete transforms

Analytical and numerical approaches are presented for the design of first‐order and second‐order recursive digital filter banks for computing linear, discrete transforms. This subject has been studied extensively for the case of trigonometric transforms. The focus of this paper is on discrete polynomial transforms, and Clenshaws recurrence formulae are used to design the second‐order filters. The efficiency of the implementation for a specific transform is dependent upon the characteristics of recurrence relations for the transform basis vectors. Efficient implementations are derived for the discrete cosine transform and the inverse discrete Legendre transform from analytical expressions for basis vector recurrence relations. A numerical procedure is presented to search for the existence and parameters of an efficient implementation when analytical expressions for the basis vector recurrence relations are unknown.

Source localization with distributed sensor arrays and partial spatial coherence

We present performance analysis for source localization when wideband aeroacoustic signals are measured at multiple distributed sensor arrays. The acoustic wavefronts are modeled with perfect spatial coherence over individual arrays and with frequency-selective coherence between distinct arrays, thus allowing for random fluctuations due to the propagation medium when the arrays are widely separated. The signals received by the sensors are modeled as wideband Gaussian random processes, and we study the Cramer-Rao bound on source localization accuracy for varying levels of signal coherence between the arrays and for processing schemes with different levels of complexity. When the wavefronts at distributed arrays exhibit partial coherence, we show that the source localization accuracy is significantly improved if the coherence is exploited in the source localization processing. Further, we show that a distributed processing scheme involving bearing estimation at the individual arrays and time-delay estimation between pairs of sensors performs nearly as well as the optimum scheme that jointly processes the data from all sensors. We discuss tradeoffs between source localization accuracy and the bandwidth required to communicate data from the individual arrays to a central fusion center, and results from measured aeroacoustic data are included to illustrate partial spatial coherence at distributed arrays.

An integrated environment for modeling, simulation, digital signal processing, and control

An integrated laboratory for system modeling, simulation, real-time digital signal processing, and control is being developed at Bucknell University for undergraduate electrical engineering education. The laboratory bridges the gap between software simulation and testing of actual systems through a common visual programming interface. Students can explore the iterative process of developing a model and then refining the model until computer simulation results agree with experimental measurements. A liquid level control system is presented to illustrate the features of the laboratory environment. The key components of the laboratory are networked digital signal processing (DSP) hardware units and a simulation software package. The simulation software runs on workstations, and all of the laboratory equipment (including the DSP hardware) is connected to the Internet.