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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 . For N odd, F(s) has a single pole, and therefore, is of the form
for
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.
The natural form for the specifications of an inverse-Chebyshev filter is in terms of the flatness of the response at 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 assumes in the stopband, which is the set of frequencies . An alternative specification is the minimum-allowed attenuation over stopband expressed in dB as b.The following formulas relate the stopband ripple , the stopband attenuation b in positive dB, and the transfer functionparameter in [link]
In some cases passband performance is not given in terms of degree of flatness at , but in terms of a minimum-allowed magnitude in the passband up to a certain frequency , i.e., for . For a given , this requirement will determine the order as the smallest positive integersatisfying
The design of an inverse-Chebyshev filter is summarized in the following steps:
A third-order inverse-Chebyshev lowpass filter is desired with a maximum-allowed stopband ripple of or dB. This corresponds to an of 0.100504 and, together with , results in a . The scale factors are and . The prototype Chebyshev filter transfer function is
The zeros are calculated from [link] , and the poles of the prototype are inverted to give, from [link] , the desired inverse- Chebyshev filter transfer function of
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