Finite Impulse Response Filters Introduction to the z-transform Design and implementation of finite impulse response (FIR) filters Programming examples using C and TMS320C3x code The z-transform is introduced in conjunction with discrete-time signals. Mapping from the s-plane, associated with the Laplace transform, to the z-plane, associated with the z-transform, is illustrated. FIR filters are designed with the Fourier series method and implemented by programming a discrete convolution equation. Effects of window functions on the characteristics of FIR filters are covered. . | Digital Signal Processing Laboratory Experiments Using C and the TMS320C31 DSK Rulph Chassaing Copyright 1999 John Wiley Sons Inc. Print ISBN 0-471-29362-8 Electronic ISBN 0-471-20065-4 4 Finite Impulse Response Filters Introduction to the z-transform Design and implementation of finite impulse response FIR filters Programming examples using C and TMS320C3x code The z-transform is introduced in conjunction with discrete-time signals. Mapping from the s-plane associated with the Laplace transform to the z-plane associated with the z-transform is illustrated. FIR filters are designed with the Fourier series method and implemented by programming a discrete convolution equation. Effects of window functions on the characteristics of FIR filters are covered. INTRODUCTION TO THE z-TRANSFORM The z-transform is utilized for the analysis of discrete-time signals similar to the Laplace transform for continuous-time signals. We can use the Laplace transform to solve a differential equation that represents an analog filter or the z-transform to solve a difference equation that represents a digital filter. Consider an analog signal x t ideally sampled TO xs t x t 8 t - kT k 0 where 8 t - kT is the impulse delta function delayed by kT and T 1 Fs is the sampling period. The function xs t is zero everywhere except at t kT. The Laplace transform of xs t is 91 92 Finite Impulse Response Filters -o xs t e st dt x t 8 t x t 8 t -T . . . e st dt o From the property of the impulse function o f t 8 t - kT dt f kT Xs s in becomes Xs s x 0 x T e sT x 2T e-2sT . . . x nT e nsT n 0 Let z esT in which becomes X z x nT z-n n 0 Let the sampling period T be implied then x nT can be written as x n and becomes TO X z y x n z-n ZT x n n 0 which represents the z-transform ZT of x n . There is a one-to-one correspondence between x n and X z making the z-transform a unique transformation. Exercise ZT of Exponential Function x n enk The ZT of x n enk n 0 and k a .
