Elliptic filter - Biblioteka.sk

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Elliptic filter
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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

The frequency response of a fourth-order elliptic low-pass filter with ε = 0.5 and ξ = 1.05. Also shown are the minimum gain in the passband and the maximum gain in the stopband, and the transition region between normalized frequency 1 and ξ
A closeup of the transition region of the above plot.
  • 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

Log of the absolute value of the gain of an 8th order elliptic filter in complex frequency space (s = σ + jω) with ε = 0.5, ξ = 1.05 and ω0 = 1. The white spots are poles and the black spots are zeroes. There are a total of 16 poles and 8 double zeroes. What appears to be a single pole and zero near the transition region is actually four poles and two double zeroes as shown in the expanded view below. In this image, black corresponds to a gain of 0.0001 or less and white corresponds to a gain of 10 or more.
An expanded view in the transition region of the above image, resolving the four poles and two double 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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