This correspondence introduces a new class of infinite impulse response (IIR) digital filters that unifies the classical digital Butterworth filter and the well-known maximally flat FIR filter. New closed-form expressions are provided, and a straightforward design technique is described. The new IIR digital filters have more zeros than poles (away from the origin), and their (monotonic) square magnitude frequency responses are maximally flat at /spl omega/=0 and at /spl omega/=/spl pi/. Another result of the correspondence is that for a specified cutoff frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z=-1 and the passband. This technique also permits continuous variation of the cutoff frequency. IIR filters having more zeros than poles are of interest because often, to obtain a good tradeoff between performance and implementation complexity, just a few poles are best.

Generalized digital Butterworth filter design

The paper provides, through the use of Bernstein polynomials, a new insight into the design of maximally-flat (MAXFLAT) FIR filters. In particular, it demonstrates a link between the various existing methods, and provides an analytical support to the empirical relation given by Herrmann. It is also shown to provide a new computationally efficient algorithm for the design of such filters.

Design of maximally-flat FIR filters using the Bernstein polynomial

This paper presents a formula-based method for the design of IIR filters having more zeros than (nontrivial) poles. The filters are designed so that their square magnitude frequency responses are maximally-flat at /spl omega/=0 and at /spl omega/=/spl pi/ and are thereby generalizations of classical digital Butterworth filters. A main result of the paper is that, for a specified half-magnitude frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z=-1 and the passband. IIR filters having more zeros than poles are of interest, because often, to obtain a good trade-off between performance and the expense of implementation, just a few poles are best.

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New techniques to approximate the magnitude response of infinite-impulse response digital filters with maximally flat or equiripple delay characteristics are presented. A polynomial with maximally flat or equiripple group delay characteristics is chosen to be the denominator of the filter transfer function. The coefficients of the numerator polynomial are found such that the group delay characteristic is unaffected, and the magnitude response either has desired degrees of flatness at /spl omega/=0 and/or /spl omega/=/spl pi/ or is flat in the passband and equiripple in the stopband. The magnitude response can also be matched to prescribed values at a number of frequencies between /spl omega/=0 and /spl omega/=/spl pi/. In the first case, the solution for the coefficients of the numerator polynomial is shown to be purely analytical. In the second case, the well-known Remez Exchange Algorithm is used to obtain the desired passband and stopband magnitude characteristics simultaneously. It is also shown that the proposed design methods can be used to obtain linear phase finite-impulse response filters with a magnitude response that is flat in the passband and flat or equiripple in the stopband.

Magnitude approximation of digital filters with specified degrees of flatness and constant group delay characteristics

An optimal design procedure for maximally flat FIR (finite impulse response) filters, based on the Bernstein polynomial, is proposed. Using a set of recurrence relations, this method simultaneously searches for the optimum order of filter and the orders of tangency at w=0 and w= pi , in a staircase-like manner. A detailed comparison to the earlier methods establishes the superiority of this approach. Being an optimal method, it results in the minimum order of the filter required to meet the given specifications. The method is shown to be suitable for the design of this class of filters with arbitrary magnitude specifications for which no method presently exists. >

Optimal design of maximally flat FIR filters with arbitrary magnitude specifications

Maximally flat f.i.r. digital filter design provides the advantage of giving a closed-form solution, but there still remains a problem of designing such a filter whose magnitude response passes through a prescribed cutoff frequency point. It is described here how to generate a class of transitional maximally flat f.i.r. digital filters to overcome such a difficulty.

Maximally flat f.i.r. filter with prescribed cutoff frequency

For a given order of a Chebyshev IIR digital filter, existing design techniques using the squared magnitude function or the bilinear transform do not allow the freedom of adjusting the cut-off rate. An available transform method is used to overcome this difficulty.

S. Banjongjit

Papers

Maximally flat f.i.r. filter with prescribed cutoff frequency

A Chebyshev IIR lowpass digital filter with adjustable cut-off rate