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DSP applications using C and the TMS320C6X DSK (P4)

Finite Impulse Response Filters • • • Introduction to the z-transform Design and implementation of finite impulse response (FIR) filters Programming examples using C and TMS320C6x 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. 4.1 INTRODUCTION TO THE Z-TRANSFORM The z-transform is utilized for the analysis of discrete-time signals, similar to the Laplace transform for. | DSP Applications Using C and the TMS320C6x DSK. Rulph Chassaing Copyright 2002 John Wiley Sons Inc. ISBNs 0-471-20754-3 Hardback 0-471-22112-0 Electronic 4 Finite Impulse Response Filters Introduction to the -transform Design and implementation of finite impulse response FIR filters Programming examples using C and TMS320C6x code The -transform is introduced in conjunction with discrete-time signals. Mapping from the v-plane associated with the Laplace transform to the -plane associated with the -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. 4.1 INTRODUCTION TO THE Z-TRANSFORM The -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 -transform to solve a difference equation that represents a digital filter. Consider an analog signal x t ideally sampled xs t x t S t - kT 4.1 k 0 where 8 t - kT is the impulse delta function delayed by kT and T HFS is the sampling period. The function xv t is zero everywhere except at t kT. The Laplace transform of xv t is 102 Introduction to the z-Transform 103 Xs s j xs t e stdt jo x t d t x t d t - T e-stdt 4.2 From the property of the impulse function f f t d t - kT dt f kT Xs s in 4.2 becomes Xs s x 0 x T e -sT x 2T e-2 sT x nT e-nsT 4.3 n 0 Let z e T in 4.3 which becomes X z x nT z-n 4.4 n 0 Let the sampling period T be implied then x nT can be written as x n and 4.4 becomes X z x n z-n ZT x n 4.5 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 4.1 ZT of Exponential Function x n enk The ZT of x n enk n 0 and k a constant is X z enkz-n ekz-r 4.6 n 0 n 0 Using the .