3.3 Chebyshev filter properties  (Page 3/5)

A partially factored form for the Butterworth filter and for the Chebyshev filter can be written forthe inverse-Chebyshev filter using the zero locations from [link] and the pole locations from the regular Chebyshev filter. For N even, this becomes

for $k=1,3,5,\cdots ,N-1$ . For N odd, F(s) has a single pole, and therefore, is of the form

for $k=2,4,6,\cdots ,N-1$

Because of the relationships between the locations of the poles of the Butterworth, Chebyshev, and inverse-Chebyshevfilters, it is easy to write a design program with many common calculations. That is illustrated in the program in the appendix.

Inverse-chebyshev filter design procedures

The natural form for the specifications of an inverse-Chebyshev filter is in terms of the flatness of the response at $\omega$ to determine the passband, and a maximum allowable response in the stopband. The filter order and the stopband ripple are theparameters to be determined by the specifications. The rate of dropoff near the transition from pass to stopband is similar tothe regular Chebyshev filter. Because practical specifications often allow more passband ripple than stopband ripple, theregular Chebyshev filter will usually have a sharper dropoff than the inverse-Chebyshev filter. Under those conditions, theinverse-Chebyshev filter will have a smoother phase response and less time-domain echo effects.

The stopband ripple d is simply defined as the maximum value that $\left|F\right(j\omega \left)\right|$ assumes in the stopband, which is the set of frequencies $1<\omega <\infty$ . An alternative specification is the minimum-allowed attenuation over stopband expressed in dB as b.The following formulas relate the stopband ripple $\delta$ , the stopband attenuation b in positive dB, and the transfer functionparameter $ϵ$ in [link]

In some cases passband performance is not given in terms of degree of flatness at $\omega =0$ , but in terms of a minimum-allowed magnitude $G$ in the passband up to a certain frequency ${\omega }_p$ , i.e., $1>\left|F\right|>G$ for $0<\omega <{\omega }_p<1$ . For a given $ϵ$ , this requirement will determine the order as the smallest positive integersatisfying

The design of an inverse-Chebyshev filter is summarized in the following steps:

1. The maximum-allowed stopband response must be given in the form of $\delta$ or b. From this, the parameter $ϵ$ is calculated using [link] .
2. The order N is determined from the desired flatness at $\omega =0$ , or from a minimum allowed response for frequencies up to ${\omega }_p$ using [link] .
3. ${\nu }_0$ and $sinh\left({\nu }_0\right)$ and $cosh\left({\nu }_0\right)$ are calculated just as for the regular Chebyshevfilter.
4. The pole locations for the prototype Chebyshev filter are calculated from [link] and [link] and then "inverted" according to [link] to give the inverse- Chebyshev filter pole locations.
5. The pole locations are combined in [link] to give the final filter transfer function denominator.
6. The zero locations are calculated from [link] and combined with the pole locations to give the totaltransfer function [link] or [link] .