A new multiple-exchange ascent algorithm is presented for designing optimal Chebyshev digital FIR tilters with arbitrary magnitude and phase specifications. Compared to existing Chebyshev design techniques, the new design algorithm exhibits faster convergence while maintaining high accuracy, and is guaranteed to converge to the optimal solution. In addition, the proposed algorithm exactly reduces to the classic second Remez (Parks-McClellan) algorithm when real-only or imaginary-only filters are designed and is, therefore, a generalization of the classic Remez algorithm to the complex case. The described algorithm has been incorporated as part of the MATLAB signal processing toolbox. Design examples are presented to illustrate the performance of the proposed algorithm. (C) 1999 Elsevier Science B.V. All rights reserved.

A novel feature selection algorithm is presented which outperforms the well-known SFS (sequential forward selection) and SBS (sequential backward selection) algorithms for large-scale problems. The approach utilizes the solution to the similar problem of large-scale feature extraction by choosing a subset of the original measurements that are closest to the space spanned by the extracted (transformed) features. The authors develop a computationally efficient Frobenius subspace distance metric for the subspace comparisons, which reduces the complexity from order N taken k at a time to order N/sup 3/ operations. Finally, sufficient conditions for optimality of the algorithm are presented that demonstrate the relationship between the feature extraction and the feature selection solutions.<>

The purpose of this paper is to present a multidimensional MEM algorithm, valid for nonuniformly sampled arrays, which satisfies a "correlation-approximating" constraint. To this end, the correlation matching equality constraints of the usual MEM are replaced by a single inequality constraint whose form is based on a measure of the noise in the given autocovariance function (ACF). In this way, one can incorporate into the model knowledge of the noisy nature of the "given" ACF, since the "given" ACF is usually estimated from the samples of the wavefield. Specifically, the covariance matrix of the correlation estimates is used in a quadratic form that weights the difference between the "given" ACF and the one matched by the power spectrum. The maximization of entropy under this inequality constraint leads, ultimately, to a steepest-descent algorithm. The algorithm has been tested with 1-D synthetic data representing multiple sinusoids buried in additive white noise. The performance of this modified MEM algorithm is compared to a traditional MEM algorithm for extendible ACF's and for different SNR's. Examples of the MEM spectrum are given for the case of nonextendible ACF's.

The solution to the general multidimensional MEM spectral estimation problem is described. A detailed derivation of the dual optimisation problem, in which entropy is minimised, is given. A necessary and sufficient condition for the existence of a solution to the general problem is presented. The theory is also extended to a 'correlation-approximating' MEM spectral estimate. Algorithms applicable to the dual problem are discussed.

The array processing problem is briefly discussed and an abstract spectral estimation problem is formulated. This problem involves the estimation of a multidimensional frequency-wave vector power spectrum from measurements of the correlation function and knowledge of the spectral support. The investigation of correlation-matching spectral estimates leads to the extendibility question: does there exist any positive spectrum on the spectral support that exactly matches a given set of correlation samples? In answering this question, a mathematical framework is developed in which to analyze and design spectral estimation algorithms. Pisarenko's method of spectral estimation, which models the spectrum as a sum of impulses plus a noise component, is extended from the time series case to the more general array processing case. Pisarenko's estimate is obtained as the solution of a linear optimization problem, which can be solved using a linear programming algorithm such as the simplex method.

Phonocardiography, the analysis of heart sounds, is a noninvasive diagnostic method useful in studying heart valve function. Phonocardiograms (PCG's) of porcine prosthetic heart valves in the aortic position were analyzed by a parametric signal modeling method in order to derive frequency domain features suitable for the classification of the valve state.

A high-resolution method of spectral analysis, of the class generally called "maximum entropy method," was used in a study of aortic porcine valve closing sounds in 37 patients (ages 19 to 76). Spectra from 27 normal xenografts, implanted from 2 weeks to 61 months previously, were characterized by a dominant frequency peak, F1, at 89 +/- 15 Hz (mean +/- SD), with a lower amplitude peak, F2, at 154 +/- 25 Hz. Eight of nine patients with aortic porcine valve dysfunction were proved surgically to have leaflet degeneration or infection and had either F1 (139 +/- 54 Hz) and/or F2 (195 +/- 74 Hz) significantly higher than normal (p less than .001). In two patients with paravalvar leak but no leaflet abnormality, F1 and F2 were in the normal range. Estimation of F1 and F2 was highly reproducible and was unaffected by duration of implant up to 5 years. Spectral analysis of aortic porcine valve closing sounds by the maximum entropy method may be useful for detection of intrinsic xenograft dysfunction.

The problem of multidimensional maximum entropy method (MEM) spectral estimation from nonuniformly spaced correlation measurements is investigated. A necessary and sufficient condition is derived for the existence and uniqueness of the MEM spectral estimate in its usual form. It is shown that this condition is not satisfied in many multidimensional problems of interest, although it is satisfied in the important practical case of spectral supports composed of a finite number of points. When the existence condition is satisfied, calculation of the MEM estimate reduces to the solution of a finite-dimensional convex optimization problem. The application of standard optimization techniques to this problem results in iterative computational algorithms which are guaranteed to converge. The algorithms so obtained are compared to those previously proposed and a spectral estimation example is presented.

Polynomials in more than one variable arise frequently in multidimensional signal processing applications. Unlike polynomials in a single variable, multidimensional polynomials cannot, in general, be factored. In this note, it is shown that the set of factorable multi-dimensional polynomials is extremely small in the sense that almost all polynomials in two or more variables are irreducible.

Iterative algorithms for signal reconstruction from partial time- and frequency-domain knowledge have proven useful in a number of application areas. In this paper, a general convergence proof, applicable to a general class of such iterative reconstruction algorithms, is presented. The proof relies on the concept of a nonexpansive mapping in both the time and frequency domains. Two examples studied in detail are time-limited extrapolation (equivalently, band-limited extrapolation) and phase-only signal reconstruction. The proof of convergence for the phase-only iteration is a new result obtained by this method of proof. The generality of the approach allows the incorporation of nonlinear constraints such as time- (or space-) domain positivity or minimum and maximum value constraints. Finally, the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique.

The ability of a modified covariance method "maximum entropy" spectral estimator to estimate the frequencies of several sinusoids in additive white Gaussian noise is studied. Analytical expressions for the variance of the spectral estimate peak positions at high signal-to-noise ratios ate derived. The calculated variance is compared to the Cramer-Rao lower bound and to the results of similar variance calculations for the more familiar covariance method. It is shown that performance approaching the Cramer-Rao bound can be obtained. Simulations demonstrate substantial agreement with the analytical results over a wide range of signal-to-noise ratios.

Bounds on the minimum number of data transfers (i.e., loads, stores, and copies) required by WFTA and FFT programs are presented. The analysis is applicable to those general-purpose computers with M general processor registers (e.g., the IBM 370, PDP-11, etc.) where transform length. It is shown that the 1008- point WFTA requires about 21 percent more data transfers than the 1024-point radix-4 FFT; on the other hand, the 120-point WFTA has about the same number of data transfers as the mixed radix (4 × 4 × 4 × 2) version of the 128-point FFT and 22 percent fewer than the radix-2 version. Finally, comparisons of the "total" program execution times (multiplications, additions, and data transfers, but not indexing or permutations) are presented.

The quantization error introduced by the Winograd Fourier transform algorithm (WFTA) when implemented in fixed-point arithmetic is studied and compared with that of the fast Fourier transform (FFT). The effect of ordering the computational modules and the relative contributions of data quantization error and coefficient quantization error are determined. In addition, the quantization error introduced by the Good-Winogzad (GW) algorithm, which uses Good's prime-factor decomposition for the discrete Fourier transform (DFT) together with Winograd's short length DFT algorithms, is studied. Error introduced by the WFTA is, in all cases, worse than that of the FFT. In general, the WFTA requires one or two more bits for data representation to give an error similar to that of the FFT. Error introduced by the GW algorithm is approximately the same as that of the FFT.

The hardware design and implementation of a Fermat number transform (FNT) is described. The arithmetic logic design is treated in detail and a new data representation for integers modulo a Fermat number is derived. In addition, the FNT is compared with the fast Fourier transform (FFT) on the basis of hardware required for a pipeline convolver.

Several properties of finite-duration impulse-response (FIR) digital filters designed to have the maximum possible number of ripples are discussed and illustrated with examples. Such filters have been called extraripple filters. Among the properties of such filters are as follows. 1) Extraripple low-pass filters with fixed passband ripple δ1and stopband ripple δ2achieve the local minimum of transition width in the class of linear phase filters with fixed impulse-response duration of N samples. 2) For the case δ1= δ2the minimum transition width is roughly independent of Fp, the passband cutoff frequency. 3) For the case δ2< δ1, the minimum transition width decreases with increasing bandwidth. Several figures are included to show the relation between the transition width and bandwidth for low-pass filters.

A method for designing finite-duration impulse-response (FIR) linear-phase digital filters is presented in which the four possible cases for such filters are treated in a unified approach. It is shown how to reduce each case to the proper form so that the Remez exchange algorithm can be used to compute the best approximation to the desired frequency response. The result is that a very flexible and fast technique is available for FIR linear-phase filter design.

An efficient procedure for the approximation of two‐dimensional (2‐D) fan filters is presented. The Chebychev (or min‐max) error criterion is used to measure the optimality of the approximation as opposed to previous designs which use the least‐squares norm. Since the direct solution of a 2‐D Chebychev approximation problem is difficult, a change of variables is introduced to convert the 2‐D approximation problem into an equivalent 1‐D low‐pass filter approximation problem. The response is opptimized in the 1‐D domain, and then the change of variables is applied to obtain the 2‐D filter. Thus the techniques of 1‐D Chebychev approximation are applied to yield a 2‐D fan filter.