3.8 Direct frequency domain iir filter design methods  (Page 4/6)

An interesting iterative design algorithm that can design to approximate complex or magnitude frequency responses has be recently proposed byJackson [link] . A different approach to the same problem was posed by Soewito [link] , [link] .

To illustrate this design method a sixth-order lowpass filter was designed with 41 frequency samples to approximate. The magnitude ofthose less than 0.2 Hz is one and of those greater than 0.2 is zero. The phase was experimentally adjusted to result in a good magnituderesponse. The design was performed with Program 9 in the appendix of [link] and the frequency response is shown in Figure 7-33 of [link] . Matlab programs have recently been written which are smaller and easier to understand than those in FORTRAN.

Summary

In this section an LS-error approximation method was posed to design IIR filters. By using an equation-error rather than asolution-error criterion, a problem resulted that required only the solution of simultaneous linear equations.

Like the FIR filter version, the IIR frequency sampling design method and the LS equation-error extension can be used for complexapproximation and, therefore, can design with both magnitude and phase specifications.

If the desired frequency-response samples are close to what an IIR filter of the specified order can achieve, this method will producea filter very close to what a true least-squared error method would. However, when the specifications are not consistent with what can beachieved and the approximating error is large, the results can be very poor and in some cases, unstable. It is particularly difficultto set realistic phase response specifications. With this method, it is even more important to have a design environment that will alloweasy trial-and-error procedure.

Newly published works which will be discussed here are [link] , [link] , [link] , [link] , [link] , [link] , [link] . Other references can be found in [link] , [link] , [link] , [link] , [link] , [link] . The Matlab command invfreqz() which is an inverse to the freqz() command gives a similar or, perhaps, the same result as the method described in thisnote but uses a different formulation [link] , [link] .

More

Practical problems occur in the design of a filter to separate signals according to their energy. Because the energycontent of a signal is the integral or sum of the square of the signal, a mean-squared-error measure is natural. Unfortunately,for the IIR filter design problem, the optimization procedure is nonlinear. This was pointed out in the last section where theequation error was used in order to have a linear problem.

Because of the nonlinear nature of the least-squared-error minimization, the method of solution becomes dependent on thedesired frequency response, and therefore, there is no single method for design. The mean-squared error for magnitudeapproximation is defined as

where $x$ is a vector of filter parameters chosen to minimize q, and the error is sampled at $L+1$ frequencies ${\omega }_i$ . Steiglitz [link] chose the parameter vector x to be the coefficients of a cascade structure in order to best fit an iterative optimizationscheme. He applied a standard optimization algorithm, the Fletcher-Powell method, to the minimization of [link] . Other methods which are more directly related to a squared-error measurecan also be used.