7Infinite Impulse Response Filters

Digital filters with an infinite impulse response (IIR), or recursive filters, have properties similar to those of analog filters, and consequently, their coefficients can be determined by similar techniques [1–3].

Before discussing the method for calculating the coefficients, it is useful to give some general expressions for the properties of these filters.

7.1 General Expressions for the Properties of IIR Filters

The general IIR filter is a system which, from the set of data x(n), produces the set y(n) such that:

(7.1)y left-parenthesis n right-parenthesis equals sigma-summation Underscript l equals 0 Overscript upper L Endscripts a 1 x left-parenthesis n minus 1 right-parenthesis minus sigma-summation Underscript k equals 1 Overscript upper K Endscripts b Subscript k Baseline y left-parenthesis n minus k right-parenthesis

The Z-transfer function for this system is written:

(7.2)upper H left-parenthesis upper Z right-parenthesis equals StartFraction sigma-summation Underscript l equals 0 Overscript upper L Endscripts a 1 upper Z Superscript negative 1 Baseline Over 1 plus sigma-summation Underscript k equals 1 Overscript upper K Endscripts b Subscript k Baseline upper Z Superscript negative k Baseline EndFraction

This is the quotient of two polynomials in Z, which are often of the same degree.

As the coefficients al and bk are real numbers, H(Z) is a complex number such that:

ModifyingAbove upper H left-parenthesis upper Z right-parenthesis With bar equals upper H left-parenthesis upper Z overbar right-parenthesis

and the frequency response of the filter can be written with the same conventions as in the earlier chapters:

upper H left-parenthesis omega right-parenthesis equals vertical-bar upper H left-parenthesis omega right-parenthesis vertical-bar e Superscript minus italic j phi left-parenthesis omega right-parenthesis

The modulus and the phase are expressed in terms of H(Z) by the following equations:

(7.3)StartAbsoluteValue upper H left-parenthesis omega right-parenthesis EndAbsoluteValue 2 equals left-bracket upper H left-parenthesis upper Z right-parenthesis upper H left-parenthesis upper Z Superscript negative 1 Baseline right-parenthesis right-bracket Subscript upper Z Baseline Subscript equals e Sub Superscript j omega

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