Chebyshev Functions-Based New Designs of Halfband Low/Highpass Quasi-Equiripple FIR Digital Filters
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Chebyshev Functions-Based New Designs of Halfband Low/Highpass Quasi-Equiripple FIR Digital Filters Ishtiaq Rasool Khan Department of Information and Media Sciences, The University of Kitakyushu, 1-1 Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan Collaboration Center, Kitakyushu Foundation for the Advancement of Industry, Science and Technology, 2-1 Hibikino, Wakamatsu-ku, Kitakyushu 808-0135, Japan Email: ir [email protected]
Ryoji Ohba Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan Email: [email protected] Received 12 April 2002 and in revised form 7 August 2002 Chebyshev functions, which are equiripple in a certain domain, are used to generate equiripple halfband lowpass frequency responses. Inverse Fourier transformation is then used to obtain explicit formulas for the corresponding impulse responses. The halfband lowpass FIR digital filters designed in this way are quasi-equiripple, having performances very close to those of true equiripple filters, and are comparatively much simpler to design. Keywords and phrases: digital filters, FIR, halfband, equiripple, Chebyshev functions.
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INTRODUCTION
The simplest way of designing finite impulse response (FIR) digital filters (DFs) is to truncate the infinite Fourier series of the desired frequency responses, using a window of finite length [1]. These windows-based designs provide very simple formulas for the impulse responses (tap coefficients); however, truncation of the Fourier series results in large ripples on the frequency responses, especially close to the transition edges. This builds up a need for development of new design procedures of FIR DFs having better frequency responses. One approach to a better frequency response leads to maximally flat (MAXFLAT) designs [2, 3], which have completely ripple-free frequency responses. However, a price is paid in terms of wider transition bands, which limits the applications of these otherwise excellent filters. Classical MAXFLAT designs have closed form expressions for the frequency responses, and inverse Fourier transformation is needed to find the corresponding impulse responses. Some recent developments [4, 5, 6, 7] have made MAXFLAT designs as simple as window-based designs by giving explicit formulas for the impulse responses. An entirely different approach to better frequency response is to spread the ripple uniformly over the entire frequency band. This ensures the minimum of the maximum size of ripple for a certain set of design specifications. The Remez exchange algorithm [8] offers a very flexible design pro-
cedure for such equiripple filters, and gives excellent tradeoff between the transition width and the ripple size. However, this procedure is relatively complex as it calculates the filter coefficients in an iterative manner and each iteration involves intensive search of extrema over the entire frequency band. Several other filter design techniques can be found in literature [9, 10, 11, 12, 13, 14, 15, 16] and some of them allow quasi-equ
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