Properties of iir filters

Pole-zero locations for iir filters

The possible locations of the zeros of the transfer function of an FIR linear-phase filter were analyzed elsewhere. For the IIR filter,there are poles as well as zeros. For most applications, the coefficients $a\left(n\right)$ and $b\left(n\right)$ are real and, therefore, the poles and zeros occur in complex conjugate pairs or are real. A filter isstable if for any bounded input, the output is bounded. This implies the poles of the transfer function must be strictly inside the unitcircle of the complex $z$ plane. Indeed, the possibility of an unstable filter is a serious problem in IIR filter design, whichdoes not exist for FIR filters. An important characteristic of any design procedure is the guarantee of stable designs, and animportant ability in the analysis of a given filter is the determination of stability. For a linear filter analysis, thisinvolves the zeros of the denominator polynomial of [link] . The location of the zeros of the numerator, which are the zeros of $H\left(z\right)$ , are important to the performance of the filter, but have no effect on stability.

If both the poles and zeros of a transfer function are all inside or on the unit circle of the $z$ -plane, the filter is called minimum phase. The effects of a pole or zero at a radius of $r$ from the origin of the $z$ -plane on the magnitude of the transfer function are exactly the same as one at the same angle but at aradius of $1/r$ . However, the effect on the phase characteristics is different. Since only stable filters are generally used in practice,all the poles must be inside the unit circle. For a given magnitude response, there are two possible locations for each zero that is noton the unit circle. The location that is inside gives the least phase shift, hence the name “minimum- phase" filter.

The locations of the poles and zeros of the example in [link] are given in [link] b.

Since evaluating the frequency response of a transfer function is the same as evaluating $H\left(z\right)$ around the unit circle in the $z$ -plane, a comparison of the frequency-response plot in [link] a and the pole-zero locations in [link] b gives insight into the effects of pole and zero location on the frequency response. In thecase where it is desirable to reject certain bands of frequencies, zeros of the transfer function will be located on the unit circle atlocations corresponding to those frequencies.

By having both poles and zeros to describe an IIR filter, much more can be done than in the FIR filter case where only zerosexist. Indeed, an FIR filter is a special case of an IIR filter with a zero-order denominator. This generality and flexibility doesnot come without a price. The poles are more difficult to realize than the zeros, and the design is more complicated.

Summary

This section has given the basic definition of the IIR or recursive digital filter and shown it to a generalization of the FIR filterdescribed in the previous chapters. The feedback terms in the IIR filter cause the transfer function to be a rational function withpoles as well as zeros. This feedback and the resulting poles of the transfer function give a more versatile filter requiring fewercoefficients to be stored and less arithmetic. Unfortunately, it also destroys the possibility of linear phase and introduces thepossibility of instability and greater sensitivity to the effects of quantization. The design methods, which are more complicated thanfor the FIR filter, are discussed in another section, and the implementation, which also is more complicated, is discussed instill another section.