**5**

**Infinite Impulse Response Filters**

- Infinite impulse response filter structures: direct form I, direct form II, cascade and parallel
- Bilinear transformation for filter design
- Sinusoidal waveform generation using difference equation
- Filter design and utility packages
- Programming examples using TMS320C6X and C code

The FIR filter discussed in Chapter 4 has no analog counterpart. In this chapter we discuss the infinite impulse response (IIR) filter that makes use of the vast knowledge already acquired with analog filters. The design procedure involves the conversion of an analog filter to an equivalent discrete filter using the bilinear transformation (BLT) technique. As such, the BLT procedure converts a transfer function of an analog filter in the *s*-domain into an equivalent discrete-time transfer function in the *z*-domain.

**5.1 INTRODUCTION**

Consider a general input–output equation of the form

or equivalently,

This recursive type of equation represents an IIR filter. The output depends on the inputs as well as past outputs (with feedback). The output *y*(*n*), at time *n*, depends not only on the current input *x*(*n*), at time *n*, and on past inputs *x*(*n* − 1), *x*(*n* − 2), …, *x*(*n* − *M*), but also on past outputs *y*(*n* − 1), *y*(*n* − 2), …, *y*(*n* − *N*).

If we assume all initial conditions to be ...