A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design

Abstract

The sinc function is often used as the basis for the design of discrete linear-phase FIR filters. However the Fourier transform of the truncated sinc function exhibits ripple in the pass band due to the Gibbs phenomenon. This paper introduces an alternative function based on Chebyshev polynomials whose Fourier transform decreases monotonically in the pass band. Furthermore this function features an intrinsic window function with an adjustable parameter influencing the Fourier transform in the transition and stop bands.

28 pages, 6 figures, 10 tables. This revision retracts the erroneous last two paragraphs of Section 8. A detailed discussion of the anomalous result illustrated in Figure 6.3 is presented in Appendix 3. The proof of the convergence of the sinc function and the alternative to the sinc function in Section 2 has been replaced with a simpler proof