Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function
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Design of Nonrecursive Digital Filters Using the Ultraspherical Window Function Stuart W. A. Bergen Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6 Email: [email protected]
Andreas Antoniou Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6 Email: [email protected] Received 13 September 2004; Revised 8 February 2005; Recommended for Publication by Ulrich Heute An efficient method for the design of nonrecursive digital filters using the ultraspherical window function is proposed. Economies in computation are achieved in two ways. First, through an efficient formulation of the window coefficients, the amount of computation required is reduced to a small fraction of that required by standard methods. Second, the filter length and the independent window parameters that would be required to achieve prescribed specifications in lowpass, highpass, bandpass, and bandstop filters as well as in digital differentiators and Hilbert transformers are efficiently determined through empirical formulas. Experimental results demonstrate that in many cases the ultraspherical window yields a lower-order filter relative to designs obtained using windows like the Kaiser, Dolph-Chebyshev, and Saram¨aki windows. Alternatively, for a fixed filter length, the ultraspherical window yields reduced passband ripple and increased stopband attenuation relative to those produced when using the alternative windows. Keywords and phrases: nonrecursive digital filters, FIR filters, window functions, ultraspherical window, digital differentiators, Hilbert transformers.
1.
INTRODUCTION
Window functions (or windows for short) are time-domain weighting functions that have found widespread usage in signal processing applications such as power spectral estimation, beamforming, and digital filter design. Windows can be categorized as fixed or adjustable [1]. Fixed windows have only one independent parameter, namely, the window length which controls the window’s mainlobe width. Adjustable windows have two or more independent parameters, namely, the window length, as in fixed windows, and one or more additional parameters that can control other window characteristics. Each of the adjustable windows has been derived by exploiting certain characteristics of well-known polynomials to satisfy a particular criterion. For instance, the Kaiser and Saram¨aki windows [2, 3] have two parameters and yield close approximations to discrete prolate functions, which have maximum energy concentration in the mainlobe. The Dolph-Chebyshev window [4] has two parameters and produces the minimum mainlobe width for a specified maximum sidelobe amplitude. The Kaiser, Saram¨aki, and DolphChebyshev windows can control the amplitude of the sidelobes relative to that of the mainlobe. The ultraspherical
window [5, 6, 7, 8] has three parameters and through the proper choice of these parameters, the amplitude of the sidelobes relative to t
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