2.2 Least squared error design of fir filters  (Page 9/13)

The programs are written to be consistent with Matlab's convention of normalizing for two samples per second sampling rate. The equationsmost of this book, however, are normalized for one sample per second.

Design examples

To show the results of using this new design approach, two examples of multiband filter design are presented here. The first is a filter with astopband from $\omega =0$ to $\omega =0.2$ , a transition band from $\omega =0.2$ to $\omega =0.25$ , a passband with gain equalto 0.7 from $\omega =0.25$ to $\omega =0.5$ , a transition band from $\omega =0.5$ to $\omega =0.55$ , a passband with gain equalto 0.5 from $\omega =0.55$ to $\omega =0.7$ , a transition band from $\omega =0.7$ to $\omega =0.73$ , a stopband from $\omega =0.73$ to $\omega =0.85$ , a transition band from $\omega =0.85$ to $\omega =0.9$ , and a passband with gain equal one from $\omega =0.9$ to $\omega =1$ . This is called with the Matlab program by

and the amplitude response plot shown in [link] a. The response for length of $N=101$ is shown in Figures  [link] b and in [link] c the zero locations are given.

As an example of how versatile this approach can be, a length-101 linear phase multiband FIR filter wasdesigned with different types of filtering being done in different bands. The signal with frequencies in the band from $0<f<0.2$ is differentiated, in the band from $0.23<f<0.4$ is rejected, in $0.43<f<0.6$ is Hilbert transformed, in $0.63<f<0.8$ is rejected, and $0.83<f<1.0$ is highpass filtered. In the transition bands between each of these processing bands, there is an optimal spline transition function. The amplitude response is shown in Figure 7d. This is a truly versatile multibanddesign technique with the only major limitation being that weighting is not possible. However, that limitation is removed in the next secession.

Weighted least integral squares fir filter design

If the FIR filter design problem is posed as a weighted integral squared error approximation problem, a simple analytical design formula as in [link] or [link] is not possible (Recall that it is possible to easily introduce weights in the discrete approximation problem [link] ). In this section we consider a multiband generalization [link] of an approach which is a mixture of analytical formulas and numerical solution of Toeplitz plus Hankel matrices which have beenpresented in [link] , [link] , [link] .

The most general definition of the linear phase weighted least squares FIR filter design problem [link] , [link] , [link] defines the error measure as in Equation 10 from Constrained Approximation and Mixed Criteria by

where $\Omega$ is the set of frequencies that contribute to the error.

We set up the conditions for minimizing the error in [link] for odd $N$ by using the same approach used in [link] which substitutes

from Equation 34 from FIR Digital Filters into [link] , differentiates $q$ in respect to each $a\left(m\right)$ , and then sets it equal to zero to give

where we can obtain the $h\left(n\right)$ from the $a\left(n\right)$ by the scaling and shifting in Equation 35 from FIR Digital Filters . We denote this in matrix form by