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where ${\mathbf{H}}_{\mathbf{1}}$ is $(M+1)$ by $(N+1)$ , ${\mathbf{h}}_{1}$ is length- $(L-M)$ , and ${\mathbf{H}}_{2}$ is $(L-M)$ by $N$ . The lower $(L-M)$ equations are written
or
which must be solved for ${\mathbf{a}}^{*}$ . The upper $M+1$ equations of [link] are written
which allows the calculation of $\mathbf{b}$ .
If $L=N+M$ , ${\mathbf{H}}_{2}$ is square. If ${\mathbf{H}}_{2}$ is nonsingular, [link] can be solved exactly for the denominator coefficients in ${\mathbf{a}}^{*}$ , which are augmented by the unity term to give $\mathbf{a}$ . From [link] , the numerator coefficients in $\mathbf{b}$ are found. If ${\mathbf{H}}_{2}$ is singular [link] and there are multiple solutions, a lower order problem can be posed. Ifthere are no solutions, the approximation methods must be used.
Note that any order numerator and denominator can be prescribed. If the filter is in fact an FIR filter, $\mathbf{a}$ is unity and ${\mathbf{a}}^{*}$ does not exist. Under these conditions, [link] states that ${b}_{n}={h}_{n}$ , which is one of the cases of FIR frequency sampling covered [link] . Also note that there is no control over the stability of the filter designed by this method.
In this section, an interpolation design method was developed and analyzed. Use of the DFT converted the frequency- domainspecifications to the time domain. A matrix partitioning allowed uncoupling the solution for the numerator from the solution of thedenominator coefficients. The use of the DFT prevents the possibility of unequally spaced frequency samples as was possiblefor FIR filter design. The solution of simultaneous equations would allow unequal spacing which is not as troublesome as with the FIRfilter because IIR filters are usually of lower order.
The frequency-sampling design of IIR filters is somewhat more complicated than for FIR filters because of the requirement that ${\mathbf{H}}_{2}$ be nonsingular. As for the FIR filter, the samples of the desired frequency response must satisfy the conditions to insurethat ${h}_{n}$ are real. The power of this method is its ability to interpolate arbitrary magnitude and phase specification. In contrastto most direct IIR design methods, this method does not require any iterative optimization with the accompanying convergence problems.
As with the FIR version, because this design approach is an interpolation method rather than an approximation method, theresults may be poor between the interpolation points. This usually happens when the desired frequency-response samples are notconsistent with what an IIR filter can achieve. One solution to this problem is the same as for the FIR case [link] , the use of more frequency samples than the number of filter coefficients and the definition of an approximation errorfunction that can be minimized. There is no simple restriction that will guarantee stable filters. If the frequency-response samplesare consistent with an unstable filter, that is what will be designed.
In order to obtain better practical filter designs, the interpolation scheme of the previous section is extended to give anapproximation design method [link] . It should be noted at the outset that the method developed in this section minimizes anequation-error measure and not the usual frequency-response error measure.
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