3.8 Direct frequency domain iir filter design methods  (Page 3/6)

The number of frequency samples specified, $L+1$ , will be made larger than the number of filter coefficients, $M+N+1$ . This means that ${\mathbf{H}}_2$ is rectangular and, therefore, [link] cannot in general be satisfied. To formulate an approximationproblem, a length- $(L+1)$ error vector $\epsilon$ is introduced in [link] and [link] to give

Equation [link] becomes

where now ${\mathbf{H}}_2$ is rectangular with $(L-M)>N$ . Using the same methods as used to derive [link] , the error $\epsilon$ is minimized in a least-squared error sense by the solution of the normal equations [link]

If the equations are not singular, the solution is

If the normal equations are singular, the pseudo-inverse [link] can be used to obtain a minimum norm or reduced order solution.

The numerator coefficients are found by the same techniques as before in [link]

which results in the upper $M+1$ terms in $\epsilon$ being zero and the total squared equation error being minimum.

As is true for LS-error design of FIR filters, [link] is often numerically ill-conditioned and [link] should not be used to solve for ${\mathbf{a}}^{*}$ . Special algorithms such as those used by Matlab and LINPACK [link] , [link] should be employed.

The error $\epsilon$ defined in [link] can better be understood by considering the frequency-domain formulation. Takingthe DFT of [link] gives

where $\epsilon$ is the error in trying to satisfy [link] when the equations are over-specified. This can be reformulated in terms of $\mathcal{E}$ , the difference between the frequency response samples of the designed filter and the desired response samples, by dividing [link] by $A_k$ to give

$\mathcal{E}$ is the error in the solution of the approximation problem, and $\epsilon$ is the error in the equations defining the problem. The usual statement of a frequency-domain approximationproblem is in terms of minimizing some measure of $\mathcal{E}$ , but that results in solving nonlinear equations. The design proceduredeveloped in this section minimizes the squared error $\epsilon$ , thus only requiring the solution of linear equations. There is animportant relation between these problems. [link] shows that minimizing $\epsilon$ is the same as minimizing $\mathcal{E}$ weighted by $A$ . However, $A$ is unknown until after the problem is solved.

Although this is posed as a frequency-domain design method, the method of solution for both the interpolation problem and the LSequation-error problem is the same as the time-domain Prony's method, discussed in Complex and Minimum Phase Approximation of reference [link] .

Numerous modifications and extensions can be made to this method. If the desired frequency response is close to what can be achieved by an IIRfilter, this method will give a design approximately the same as that of a true least-squared solution-error method. It can be shown that $\epsilon =0\leftrightarrow \mathcal{E}=0$ . In some cases, improved results can be obtained by estimating $A_k$ and using that as a weight on $\epsilon$ to approximate minimizing $\mathcal{E}$ . There are iterative methods based on solving [link] and [link] to obtain values for $A_k$ . These values are used as weights on $\epsilon$ to solve for a new set of $A_k$ used as a new set of weights to solve again for $A_k$ [link] [link] . We found this approach to converge slowly, but a recent paper using the log-magnitude [link] was more successful. Other approaches are given in [link] , [link] , [link] . The solution of [link] and [link] is sometimes used to obtain starting values for other iterative optimization algorithms that need good startingvalues for convergence.