3.8 Direct frequency domain iir filter design methods

The preceding design methods have been based on designing an analog prototype filter and then converting it to a digitalfilter. This approach is appropriate for the class of approximations where analytical solutions are possible, but notfor many others. In the remaining part of this chapter, methods will be developed that directly design the desired digitalfilter. Most approaches are extensions of methods used for FIR filters, but they are more complicated for the IIR case whererational approximation is being performed rather than polynomial approximation.

In this section a frequency-sampling design method is developed such that the frequency response of the IIR filter will passthrough the given samples of a desired response. Since an IIR filter cannot have linear phase, the sampled response must contain both magnitudeand phase. The extension of the frequency- sampling method to a LS-error approximation is not as simple as for the FIR filter. The method presentedin this section uses a criterion based on the equation error rather than the more common error between the actual and desired frequencyresponse [link] . Nevertheless, it is a useful noniterative design method. Finally, a general discussion of iterative design methods forLS-frequency response error is given.

Frequency-sampling design of iir filters

The method for calculating samples of the frequency response of an IIR filter presented in the section on Properties of IIR Filterscan be reversed to design a filter much the same way it was for the FIR filter using frequency sampling.The z-transform transfer function for an IIR filter is given by

The frequency response of the filter is given by setting $z=e^{-j\omega }$ . Using the notation

Equally-spaced samples of the frequency response are chosen so that the number of samples is equal to the number of unknown coefficients in [link] . These $L+1$ = $M+N+1$ samples of this frequency response are given by

and can be calculated from the length- $(L+1)$ ) DFTs of the numerator and denominator.

where the indicated division is term-by-term division for each value of $k$ . Multiplication of both sides of [link] by $A_k$ gives

If the length- $(L+1)$ inverse DFT of $H_k$ is denoted by the length- $(L+1)$ sequence $h_n$ , equation [link] becomes cyclic convolution which can be expressed in matrix form by

Note that the $h_n$ in [link] are not the impulse response values of the filter as used in the FIR case. A more compact matrix notation of is

where $\mathbf{H}$ is $(L+1)$ by $(L+1)$ , $\mathbf{b}$ is length- $(M+1)$ , and $\mathbf{a}$ is length- $(N+1)$ . Because the lower $L-N$ terms of the right-hand vector of [link] are zero, the $\mathbf{H}$ matrix can be reduced by deleting the right-most $L-N$ columns to give ${\mathbf{H}}_0$ which causes [link] to become

Because the first element of $\mathbf{a}$ is unity, it is partitioned to remove the unity term and the remaining length- $N$ vector is denoted ${\mathbf{a}}^{*}$ . The simultaneous equations represented by [link] are uncoupled by further partitioning of the $\mathbf{H}$ matrix as shown in