Joel M. Morris

The Kalman filter: A robust estimator for some classes of linear quadratic problems

In this paper, theoretical justification is established for the common practice of applying the Kalman filter estimator to three classes of linear quadratic problems where the model statistics are not completely known, and hence specification of the filter gains is not optimum. The Kalman filter is shown to be a minimax estimator for one class of problems and to satisfy a saddlepoint condition in the other two classes of problems. Equations for the worst case covariance matrices are given which allow the specifications of the minimax Kalman filter gains and the worst case distributions for the respective classes of problems. Both time-varying and time-invariant systems are treated.

Wigner distribution decomposition and cross-term deleted representation

Abstract In this paper, we represent the Wigner Distribution (WD) of an arbitrary signal, via the Gabor expansion, in terms of a linear combination of elementary WDs, which can be easily partitioned into two subsets: auto WDs and cross WDs. The Gabor coefficients, Cm,n, for this decomposition are obtained with a Gaussian-shaped synthesis function. The optimally concentrated auto WDs are non-negative and entirely free of cross-terms; the sum of these auto WDs we call the cross-terms deleted representation (CDR). The sum of the cross WDs is an oscillating function with non-zero energy in general; it can be removed and returned depending on the users needs. Such a decomposition illustrates and isolates the mechanism of WD negative values and cross-term interference. Moreover, new information is provided to facilitate the design of valid joint time-frequency signal representations and time-varying filters. Also in this paper, analogous, yet more practical, results are shown for the Discrete Wigner Distribution (DWD) for finite or periodic discrete-time signals. Examples are presented to demonstrate the CDR technique and its performance in comparison with other joint time-frequency distributions. It is shown that the CDR has the high energy concentration of the WD without the interference problems that occur in many other approaches. Moreover, because only the Gabor coefficients, Cm,n, need be computed on-line, the CDR is suitable for on-line implementation.

Evaluation of the very low BER of FEC codes using dual adaptive importance sampling

We evaluate the error-correcting performance of a low-density parity-check (LDPC) code in an AWGN channel using a novel dual adaptive importance sampling (DAIS) technique based on multicanonical Monte Carlo (MMC) simulations, that allows us to calculate bit error rates as low as 10/sup -19/ for a (96,50) LDPC code without a priori knowledge of how to bias. Our results agree very well with standard MC simulations, as well as the union bound for the code.

Discrete Gabor expansion of discrete-time signals in l 2 Z via frame theory

Abstract This paper presents the Gabor representation for discrete-time signals in l 2 ( Z ) . The frame theory for discrete-time signals is introduced first. In this discrete Gabor expansion, the time-shift parameter N and the frequency-shift parameter M are taken to be integers with M ⩾ N for stable reconstruction. With the assumption that the Gabor analysis function has finite length, many useful formulae can be derived from frame theory. In particular, we discuss discrete-time signal decomposition and reconstruction and an algorithm for computing the dual frame by means of frame theory and time-frequency analysis. Important features of these results is that the decomposition and reconstruction can be easily computed, and perfect reconstruction is achieved. In particular: (1) we develop a frame theory for the discrete-time Gabor decomposition and reconstruction problem in l 2 ( Z ) and derive formulae that can be directly implemented via DSP methods (these results cannot be obtained simply from digitizing the equivalent continuous-time formulation of Daubechies);(2) there are no constraints on the time-shift parameter N and frequency-shift parameter M except M ⩾ N for stable reconstruction, i.e., the oversampling ratio R = M/N is a rational number, while in previous work that also dealt with discrete-time functions, the oversampling ratio can only be chosen from a few numbers dependent on the signal length; (3) the computation of the dual sequence is easy and fast in terms of Eqs. (1.9) and (1.12) (moreover, the matrix equation (1.12) for solving the dual can be separated into submatrix equations (1.15) thereby significantly reducing the computing time by a factor of M); and (4) examples are provided that illustrate the computation of dual frames (analysis sequences), the relationship between the dual frame and parameters M, N, and Q, comparisons with existing results, and the achievement of perfect reconstruction.

On turbo code decoder performance in optical-fiber communication systems with dominating ASE noise

In this paper, we study the effects of different ASE noise models on the performance of turbo code (TC) decoders. A soft-decoding algorithm, the Bahl, Cocke, Jelinek, and Raviv (BCJR) decoding algorithm, is generally used in TC decoders. The BCJR algorithm is a maximum a posteriori probability (MAP) algorithm, and is very sensitive to noise statistics. The Gaussian approximation of ASE noise is widely used in the study of optical-fiber communication systems, and there exist standard TCs for additive white Gaussian noise (AWGN) channels. We show that using a MAP decoding algorithm based on the Gaussian noise assumptions, however, may significantly degrade the TC decoder performance in an optical-fiber channel with non-Gaussian ASE noise. To take full advantage of TC, accurate noise statistics in optical-fiber transmissions should be used in the MAP decoding algorithm.

Noise reduction for NMR FID signals via Gabor expansion

The parameters in a nuclear magnetic resonance (NMR) free induction decay (FID) signal contain information that is useful in biological and biomedical applications and research. A real time-sampled FID signal is well modeled as a finite mixture of modulated exponential sequences plus noise. The authors propose to use the generalized Gabor expansion for noise reduction, where the generalized Gabor expansion represents a signal in terms of a collection of time-shifted and frequency-modulated versions of a single sequence (prototype sequence). For FID signal-fitting, the authors choose the exponential sequence as the prototype function. Using the generalized Gabor expansion and exponential prototype sequences for FID model-fitting, an NMR FID signal can be-well represented by the Gabor coefficients distributed in the joint time-frequency domain (JTFD). The Gabor coefficients reflect the weights of modulated exponential sequences in a signal. One of the important features is that the nonzero Gabor coefficients of a modulated exponential sequence will span a very small area in the JTFD, whereas the Gabor coefficients of the noise will not. If the exponent constant of the prototype sequence in the generalized Gabor expansion matches that of a modulated exponential sequence in the signal, then only one of the Gabor coefficients is nonzero in the JTFD. This is a very important property since it can be exploited to separate a signal from noise and to estimate modulated exponential sequence parameters.

Minimum-bandwidth discrete-time wavelets

In this paper we present a class of minimum-bandwidth, discrete-time orthonormal wavelets (MBDTWs). The wavelets were generated via the filter bank framework and were optimized using the global optimization technique, adaptive simulated annealing (ASA). The objective function is the average normalized bandwidth of the wavelets over all scales as obtained from the filter bank structure. We tabulate the wavelet-defining low-pass filter coefficients {g(n)} for filter lengths of N=4,8,10,12,14,16,18,24 and 32 and for L=2,3 and 4. We provide comparisons with Daubechies’ discrete wavelets and other classes of optimum wavelets. Finally, we present examples that demonstrate the advantage of our MBDTWs for certain narrowband applications: de-noising of an ECG signal, and compression of an ECG signal and a bird call signal. We compare the performance of our wavelets in these examples with that of Daubechies’ least-asymmetric wavelets which are closest to the MBDTWs with respect to our bandwidth measure.

Design of orthonormal wavelets with better time-frequency resolution

The signal decomposition techniques are an important tool for analyzing nonstationary signals. The time-frequency resolution of the decomposition basis functionals is essential to a variety of signal processing applications. The recently introduced wavelet transform is a very promising tool for signal analysis, but little attention has been paid to the time-frequency resolution property of wavelets. This paper describes a procedure to design wavelets with better time-frequency resolution. Some design examples and comparisons with traditional wavelets are also presented.

On alias-free formulations of discrete-time Cohen's class of distributions

The transition of the Cohens (1989) class of distributions from the continuous-time case to the discrete-time case is not straightforward because of aliasing problems. We classify the aliasing problems, which occur for joint time-frequency representations (TFRs), into two categories: type-I and type-II aliasings. Type-I aliasing can be avoided by properly defined discrete-time versions of some members of Cohens class (in particular, properly defined kernels), whereas type-II aliasing can be reduced and/or eliminated by increasing the sampling rate. A type-I alias-free formulation of the discrete-time Cohens class (AF-DTCC), which is equivalent to the AF-GDTFT of Joeng and Williams (see ibid., vol.40, no.2, p.1084, 1992) is then introduced based on the fact that the Cohens class can be expressed as the 2-D Fourier transform of the generalized ambiguity function (AF). Based on this definition, two discretization schemes for kernel functions are presented in both the AF domain and the time-lag domain, and are shown to be equivalent under certain conditions. We also do the following: (1) we show that a discrete-time Wigner-Ville distribution (DWVD) and discrete-time spectrogram (DSPG) are type-I alias-free and members of AF-DTCC; (2) we use all the available correlation information from a given data sequence by using the Woodward AF instead of the Sussman AF; (3) we give kernel constraints in the AF domain for various distribution properties; and (4) we provide a type-I and type-II alias-free formulation for those distributions whose kernel functions satisfy the finite frequency-support constraint.

Generalized Gabor expansions of discrete-time signals in l/sup 2/(Z) via biorthogonal-like sequences

In this paper, a biorthogonal-like sequences (BLS) theory and its application to the generalized Gabor expansions (equivalently, the generalized short-time Fourier transform/filterbank summation) are presented. A pair of BLSs are defined to be two sequences satisfying a biorthogonal-like condition (BLC), which is a moment equation and equivalent to a linear difference equation. We show that two collections in a Hilbert space generated by a pair of BLSs in the joint time-frequency domain are complete, either can be used as an analysis filter, and the other can be used as a synthesis filter for a generalized Gabor expansion of discrete-time signals. A sufficient and necessary condition on the existence of BLSs based on the moment equation is presented, which is simpler to use than frame theory. Given a filter generating a frame, its BLSs also generate frames. The dual frame is one of them. Given a FIR analysis/synthesis filter, there is a FIR synthesis/analysis filter if BLSs exist. The algorithm to compute FIR analysis and synthesis filters based on the linear difference equation is presented in this paper, which is simpler than frame operator.