# 2.3 Chebyshev or equal ripple error approximation filters  (Page 8/9)

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For problems where the signal and noise spectra are such that a specific frequency ${\omega }_{o}$ that separates the desired passband from the desired stopband can be specified but specific separate transition band edges, ${\omega }_{p}<{\omega }_{s}$ , cannot, we formulate [link] a design method where the pass and stop band ripple sizes, ${\delta }_{p}$ and ${\delta }_{s}$ are specified along with the separation frequency, ${\omega }_{o}$ . The algorithm described below will interpolate the specified ripple sizes exactly (asthe HOS algorithm does) but will allow exact control over the location of ${\omega }_{o}$ by not requiring maximum ripple. Although not set up to be an optimization procedure, it seems to minimize the transition band width.This formulation suits problems where there is no obvious transition band (“don't care band") having no signal or noise energy to be passed orrejected.

The optimal Chebyshev filter designed with this new algorithm is generally not extra ripple and, therefore, will have an extremalfrequency at $\omega =0$ or $\omega =\pi$ as the Parks-McClellan formulation does. Because we are trying to minimizing the transition bandwidth, we do not specify both the edges, ${\omega }_{p}$ and ${\omega }_{s}$ , but only one of them or, perhaps, the center of the transition band, ${\omega }_{o}$ . This results in $R$ equations which are used to find the $R$ coefficients $a\left(n\right)$ . The equations are formulated by adding the alternating peak pass and stop band ripples to the ${A}_{d}$ in [link] and not having the special last column of $C$ nor the unknown $\delta$ appended to $a$ as was done by Parks and McClellan in [link] . The resulting equation to be iterated in our new exchange algorithm has theform

$\left[\begin{array}{c}{A}_{d}\left({\omega }_{0}\right)\\ {A}_{d}\left({\omega }_{1}\right)\\ ⋮\\ {A}_{d}\left({\omega }_{o}\right)\\ {A}_{d}\left({\omega }_{s+1}\right)\\ ⋮\\ {A}_{d}\left({\omega }_{R-1}\right)\end{array}\right]+\left[\begin{array}{c}{\delta }_{p}\\ -{\delta }_{p}\\ ⋮\\ 0\\ {\delta }_{s}\\ ⋮\\ ±{\delta }_{s}\end{array}\right]=\left[\begin{array}{cccc}cos\left({\omega }_{0}0\right)& cos\left({\omega }_{0}1\right)& \cdots & cos\left({\omega }_{0}\left(R-1\right)\right)\\ cos\left({\omega }_{1}0\right)& cos\left({\omega }_{1}1\right)& \cdots & cos\left({\omega }_{1}\left(R-1\right)\right)\\ ⋮& & & ⋮\\ cos\left({\omega }_{o}\right)& cos\left({\omega }_{o}1\right)& \cdots & cos\left({\omega }_{o}\left(R-1\right)\right)\\ cos\left({\omega }_{s+1}0\right)& cos\left({\omega }_{s+1}1\right)& \cdots & cos\left({\omega }_{s+1}\left(R-1\right)\right)\\ ⋮& & & ⋮\\ cos\left({\omega }_{R-1}0\right)& cos\left({\omega }_{R-1}1\right)& \cdots & cos\left({\omega }_{R-1}\left(R-1\right)\right)\end{array}\right]\left[\begin{array}{c}a\left(0\right)\\ a\left(1\right)\\ a\left(2\right)\\ ⋮\\ a\left(\left(R-1\right)\right)\end{array}\right].$

The exchange algorithm is done as by Parks and McClellan finding new extremal frequencies at each iteration, but with fixed ripple sizes in bothpass and stop bands. This new algorithm reduces the transition band width as done by the Hofstetter, Oppenheim, and Siegel method but with thetransition band location controlled and without requiring the extra ripple solution. Note that any transition band frequency could be fixed. Itcould be ${A}_{d}\left({\omega }_{o}\right)=1/2$ to fix the half-power point. It could be ${A}_{d}\left({\omega }_{p}\right)=1-{\delta }_{p}$ to fix the pass band edge. Or it could be ${A}_{d}\left({\omega }_{s}\right)={\delta }_{s}$ to fix the stop band edge.

Extending this formulation and algorithm to the multiple transition band case complicates the problem as the solution may not be unique or may haveanomalous behavior in one of the transition bands. Details of the solution to this problem are given in [link] .

## Estimations of, the length of optimal chebyshev fir filters

All of the design methods discussed so far have assumed that $N$ ,the length of the filter, is given as part of the secifications. In many cases,perhaps even most, $N$ is a parameter that we would like to minimize. Often specifications are to meet certain pass and stopband ripplespecifications with given pass and stopband edges and with the shortest possible filter. None of our methods will do that. Indeed, it is notclear how to do that kind of optimization other than by some sort of search. In other words, design a set of filters of different lengthsand choose the one that meet the specifications with minimum length.

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I think
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