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Linear analog electronic filters |
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An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not).[citation needed] Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.
As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.
The gain of a lowpass elliptic filter as a function of angular frequency ω is given by:
where Rn is the nth-order elliptic rational function (sometimes known as a Chebyshev rational function) and
- is the cutoff frequency
- is the ripple factor
- is the selectivity factor
The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.
Properties
- In the passband, the elliptic rational function varies between zero and unity. The gain of the passband therefore will vary between 1 and .
- In the stopband, the elliptic rational function varies between infinity and the discrimination factor which is defined as:
- The gain of the stopband therefore will vary between 0 and .
- In the limit of the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor ε
- Since the Butterworth filter is a limiting form of the Chebyshev filter, it follows that in the limit of , and such that the filter becomes a Butterworth filter
- In the limit of , and such that and , the filter becomes a Chebyshev type II filter with gain
Poles and zeroes
The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on elliptic rational functions.
The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I Chebyshev filter. For simplicity, assume that the cutoff frequency is equal to unity. The poles of the gain of the elliptic filter will be the zeroes of the denominator of the gain. Using the complex frequency this means that:
Defining where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:
where and . Solving for w
where the multiple values of the inverse cd() function are made explicit using the integer index m.
The poles of the elliptic gain function are then:
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