2.3 Chebyshev or equal ripple error approximation filters  (Page 5/9)

This iterative procedure is called a multiple exchange algorithm because all of the extremal frequencies are up-dated each iteration. If only thefrequency of the largest error is up-dated each iteration, it is called a single exchange algorithm which also converges but much more slowly. Somemodification of the Parks-McClellan method or the Remez exchange algorithm will not converge as a multiple exchange, but will as a single exchange.

The Alternation theorem states that there will be a minimum of $R+1$ extremal frequencies, even for multiband designs with arbitrary $A_d\left(\omega \right)$ . If $A_d\left(\omega \right)$ is piece-wise constant with $T$ transition bands, one can derive the maximum possible number of extremal frequencies and it is $R+2T$ . This comes from the maximum number of maxima and minima that a function of the form [link] or [link] can have plus two at the edges of each transition band. For a simple lowpass filter with one passband, one transition band, and onestopband, there will be a minimum of $R+1$ extremal frequencies and a maximum of $R+2$ . For a bandpass filter, the maximum is $R+4$ . If a design has more than the minimum number of extremal frequencies, it is called an extra ripple design. If it has the maximum number, it is called a maximum ripple design.

It is interesting to note that at each iteration, the approximation is optimal over that set of extremal frequencies and $\delta$ increased over the previous iteration. At convergence, $\delta$ has increased to the maximum error over $\Omega$ and that is the minimum Chebyshev error.

At each iteration, the exchange of a proper set of extremal frequencies with alternating signs of the errors is always possible. One can showthere will never be too few and if there are too many, one uses those corresponding to the largest errors.

In step 4 it is suggested that the amplitude response $A\left(\omega \right)$ be calculated over a dense grid in the pass and stopbands and in step 5the local extremes are found by searching over this dense grid. There are more accurate methods that use bisection methods and/or Newton'smethod to find the extremal points.

In step 2 it is suggested that the simultaneous equation of [link] be solved. Parks and McClellan [link] use a more efficient and numerically robust method of evaluating $\delta$ using a form of Cramer's rule. With that $\delta$ , an interpolation method can be used to find $a\left(n\right)$ . This is faster and allows longer filters to be designed than with the linear algebra based approach described here.

For the low pass filter, this formulation always has an extremal frequency at both pass and stop band edges, ${\omega }_p$ and ${\omega }_s$ , and at $\omega =0$ and/or at $\omega =\pi$ . The extra ripple filter has $R+2$ extremal frequencies including both zero and pi. If this algorithm is started with an incorrect number of extremal frequencies in the stop orpass band, the iterations will correct this. It is interesting and informative to plot the frequency response of the filters designed at eachiteration of this algorithm and observe how the correction takes place.

The Parks-McClellan algorithm starts with fixed pass and stop band edges then minimizes a weighted form of the pass and stop band error ripple. Insome cases it may be more appropriate to fix one of the ripples and minimize the other or to fix both ripples and minimize the transition bandwidth. Indeed Schüssler, Hofstetter, Tufts, and others [link] , [link] , [link] , [link] formulated some of these ideas before Parks and McClellan developed their algorithm. The DSP group at Rice has developed some modifications tothese methods and they are presented below.