Properties of iir filters

Digital filters with an Infinite-duration Impulse Response (IIR) have characteristics that make them useful in manyapplications. This section develops and discusses the properties and characteristics of these filters [link] .

Because of the feedback necessary in an implementation, the Infinite Impulse Response (IIR) filter is also called a recursivefilter or, sometimes, an autoregressive moving-average filter (ARMA). In contrast to the FIR filter with a polynomial transferfunction, the IIR filter has a rational transfer function. The transfer function being a ratio of polynomials means it has finitepoles as well as zeros, and the frequency-domain design problem becomes a rational-function approximation problem in contrast to thepolynomial approximation for the FIR filter [link] . This gives considerably more flexibility and power, but brings with it certain problems inboth design and implementation [link] , [link] , [link] .

The defining relationship between the input and output variables for the IIR filter is given by

The second summation in [link] is exactly the same moving average of the present plus past $M$ values of the input that occurs in the definition of the FIR filter. The difference arisesfrom the first summation, which is a weighted sum of the previous $N$ output values. This is the feedback or recursive part which causes the response to an impulse input theoretically to endureforever. The calculation of each output term y(n) from [link] requires $N+M+1$ multiplications and $N+M$ additions. There are other algorithms or structures for calculating $y\left(n\right)$ that may require more or less arithmetic.

In addition to the number of calculations required to calculate each output term being a measure of efficiency, the amount of storage forcoefficients and intermediate calculations is important. DSP chips are designed to efficiently implement calculations such as [link] by having a single cycle operation that multiplies a variable by a constant and accumulates it. In parallel with that operation, it issimultaneously calculating the address of the next variable.

Just as in the case of the FIR filter, the output of an IIR filter can also be calculated by convolution.

In this case, the duration of the impulse response $h\left(n\right)$ is infinite and, therefore, the number of terms in [link] is infinite. The $N+M+1$ operations required in [link] are clearly preferable to the infinite number required by [link] . This gives a hint as to why the IIR filter is very efficient. Thedetails will become clear as the characteristics of the IIR filter are developed in this section.

Frequency-domain formulation of iir filters

The transfer function of a filter is defined as the ratio $Y\left(z\right)/X\left(z\right)$ , where $Y\left(z\right)$ and $X\left(z\right)$ are the z-transforms of the output $y\left(n\right)$ and input $x\left(n\right)$ , respectively. It is also the z-transform of the impulse response. Using the definition of thez-transform in Equation 32 from Discrete-Time Signals , the transfer function of the IIR filter defined in [link] is

This transfer function is also the ratio of the z-transforms of the $a\left(n\right)$ and $b\left(n\right)$ terms.