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 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,
where δ(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
From the property of the impulse function
Xs(s) in (4.2) becomes
Let in (4.3) , which becomes
Let the ...