A new method of designing linear-phase FIR filters is proposed by minimizing a quadratic measure of the error in the passband and stopband. The method is based on the computation of an eigenvector of an appropriate real, symmetric, and positive-definite matrix. The proposed design procedure is general enough to incorporate both time- and frequency-domain constraints. For example, Nyquist filters can be easily designed using this approach. The design time for the new method is comparable to that of Remez exchange techniques. The passband and stopband errors in the frequency domain can be made equiripple by an iterative process, which involves feeding back the approximation error at each iteration. Several numerical design examples and comparisons to existing methods are presented, which demonstrate the usefulness of the present approach.

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Eigenfilters: A new approach to least-squares FIR filter design and applications including Nyquist filters

It has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response (FIR) filter, the weighted least squares solution is equiripple. The crux of the problem lies in the determination of the necessary least squares frequency response weighting function. A novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi-equiripple design is presented. The algorithm converges very rapidly. It typically produces a design which is only about 1 dB away from the minimax optimum solution in two iterations and converges to within 0.1 dB in six iterations. Convergence speed is independent of the order of the filter. It can be used to design filters with arbitrarily prescribed phase and amplitude response. >

A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design

We present a new approach to the least-squares design of stable infinite impulse response (IIR) digital filters. The design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function, and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function. When the iteration converges, a stable and truly least-squares solution is obtained. The method is then extended to address the least-squares design of stable IIR two-dimensional (2-D) filters. Examples are included to illustrate the proposed design techniques.

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A weighted least-squares method for the design of stable 1-D and 2-D IIR digital filters

The structure inherent to the centrosymmetric matrices is exploited to obtain factorization results leading to significant computational savings in many engineering applications. Several interesting properties of the matrices are discussed with a view toward algorithm computational complexity. It is shown that the multiplicative complexity involved in the process of principal component (eigen-value/eigenvector) extraction, and in the evaluation of the determinant and inverse of such matrices, can be reduced by nearly 75% by employing the results presented here. The theory presented hereunifies andgeneralizes previous computationally efficient results onseveral specialized generalized centrosymmetric matrices; for example, the class of symmetric centrosymmetric (or doubly symmetric) matrices is shown to be a special case of the class of centrosymmetric matrices, and since the results obtained here are applicable over the field of complex numbers, the factorization results available for centrosymmetric matrices are readilyextended to the complex field. The centrosymmetric matrices play an important role in a number of areas such as pattern recognition, antenna theory, mechanical and electrical systems, and quantum physics. Specific examples of pattern recognition feature selection, a uniform linear antenna array, vibration in structures, and the quantummechanical oscillator are discussed to demonstrate that the theory developed here has a wide range of applications. In addition, certain specialized cases of the centrosymmetric matrices have, in the past, proved their indisputable usefulness in the areas of communication theory, digital filters, linear systems, linear prediction, and speech analysis.

On the reducibility of centrosymmetric matrices—applications in engineering problems

The design of Finite Impulse Response (FIR) filters in one or several dimensions can be performed with good computational efficiency using a Weighted Least Square (WLS) design. Minimax design, which is often preferred, is computationally burdensome, principally in two dimensions. This paper draws attention to the design of minimax filters using iterative WLS techniques for one-dimensional filters and extends the approach to two-dimensional filters. For two dimensions the techniques apply to both rectangular and hexagonal sampling grids. Examples demonstrate flexibility and good computational efficiency. The paper also illustrates a promising new approach to filter design which couples the very general WLS methodology to the less manageable but often preferred minimax performance criterion.

Design of almost minimax FIR filters in one and two dimensions by WLS techniques

The problem of designing a finite duration impulse response (FIR) digital filter to approximate a desired spectral response is treated in this paper. The philosophy adopted is that for a given FIR filter structure, the filter coefficients can be designed to provide a minimum mean-squared error (MMSE) estimate of a random signal sequence (the design-signal) imbedded in a random noise sequence. By treating the signal and noise covariance functions as design parameters, one can design FIR filters with spectral responses that approximate the power spectral density of the design-signal. For signal processing applications that require some attention to signal fidelity, as well as noise rejection, the MMSE philosophy seems appropriate (as opposed to a maximum signal-to-noise ratio philosophy, for example). Several practical designs are presented that emphasize the simplicity of the design technique and illustrate the selection of design parameters. The designs show quite dramatically that the MMSE design technique can be competitive with existing low-pass and bandpass design techniques. Finally, considerable attention is given to an efficient Toeplitz matrix inversion algorithm that permits rapid inversion of the covariance matrices that arise in the MMSE design. The resulting computation times for the design of high-order filters (N = 128, e.g.) appear to be shorter than computation times for competing algorithms.

L. Scharf

Papers

Statistical design of nonrecursive digital filters