Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

Abstract

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb’s phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using  Weighted Type Fractional Fourier Transform (WFrFt),  differentiator and integrator using weighted FrFt is also present.