Almost Equiripple Low-Pass FIR Filters
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Almost Equiripple Low-Pass FIR Filters Miroslav Vlˇcek · Pavel Zahradnik
Received: 9 February 2012 / Revised: 17 August 2012 © Springer Science+Business Media, LLC 2012
Abstract An approximation of the linear phase almost equiripple low-pass finite impulse response filter is introduced. The frequency response of an almost equiripple low-pass finite impulse response filter closely approaches the frequency response of an optimal equiripple low-pass finite impulse response filter in the Chebyshev sense. The presented approximation is based on the generating polynomial. Despite that the generating polynomial has no iso-extremal behavior, it is related to the class of iso-extremal polynomials. The zero phase transfer function of an almost equiripple low-pass finite impulse response filter follows from the generating polynomial. The closed form solution for the direct algebraic computation of the impulse response of the filter has been developed on the basis of generalization of the differential equation suitable for the half-band specifications. No numerical procedures are involved. The practical design procedure based on the developed approximation is presented. For illustration of the design procedure one example of the design is included here. Keywords FIR filter · Low-pass filter · Almost equiripple approximation · Zolotarev polynomials 1 Introduction The low-pass (LP) FIR filter is a fundamental building block for various applications in digital signal processing [4]. The most efficient LP FIR filters are equiripple (ER) filters. These are usually designed by the McClellan–Parks [3] numerical procedure, M. Vlˇcek Department of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical University in Prague, Na Florenci 25, 110 00 Praha 1, Czech Republic e-mail: [email protected] P. Zahradnik () Department of Telecommunication Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 16627 Praha 6, Czech Republic e-mail: [email protected]
Circuits Syst Signal Process
which provides the coefficients of the impulse response of the filter by a numerical optimization technique. Despite its practicality it does not solve the problem of approximation theory. After decades of research, the approximation problem of an ER LP FIR filter remains unresolved. This paper can be seen as an extension of our previous paper focused on almost equiripple (AER) half-band FIR filters [6]. We are primarily concerned with the approximation of a linear phase AER LP FIR filter. The frequency response of an AER LP FIR filter closely approaches the frequency response of an optimal filter in the Chebyshev sense—an ER LP FIR filter. Here, we present the differential equation for the generating polynomial of the AER LP FIR filter. It provides a class of novel polynomials related to the Zolotarev polynomials. Based on the differential equation we have developed an efficient recursive algorithm for evaluating the impulse response of the filter. 2 Basic Terminology We assume an impulse re
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