**4**

**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,

(4.1)

where δ(*t*−*kT*) is the impulse (delta) function delayed by *kT* and *T* = 1/*F _{s}* is the sampling period. The function

*x*(

_{s}*t*) is zero everywhere except at

*t*=

*kT*. The Laplace transform of

*x*(

_{s}*t*) is

From the property of the impulse function

*X _{s}*(

*s*) in (4.2) becomes

Let in (4.3) , which becomes

Let the ...