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Chebyshev Filters

Module by: Richard Baraniuk

Summary: Briefly explains Chebyshev filters.

This module will cover Chebyshev filters with an assumed understanding of Butterworth filters. This module will also examine the lowpass example of these filters, leaving conversion to other types of filters for the Lowpass to Highpass Transformation, Lowpass to Bandpass Transformation and Lowpass to Bandstop Transformation modules.

Like Butterworth filters, Chebyshev filters contain only poles. However, while the poles of the Butterworth filter lie on a circle in the s-plane, those of the Chebyshev filter lie on an ellipse.

Figure 1: A pole-zero plot of a lowpass Chebyshev filter The poles are equally spaced around an ellipse in the left half of the complex plane.
Figure 1 (chebyshev1.png)

The result of this repositioning of poles is a "rippling effect" in the passband of the magnitude of the transfer function. Since each local maximum in this rippling reaches the same value and each local minimum reaches the same value, this rippling is described as equal ripple. It is important to notice that there is no rippling in the stopband and to be aware that this transition band will be narrower than a comparable Butterworth filter.

Figure 2: A sketch of the magnitude of a lowpass Chebyshev filter. Notice that the "equal rippling" in the passband and not in the stopband.
Figure 2 (chebyshev2.png)

The magnitude of the transfer function of a Chebyshev filter takes the form

|Hⅈω|=11+ε2 C n 2ω H ω 1 1 ε 2 C n 2 ω (1)
where the polynomial, C n ω=cosnarccosω C n ω n ω is known as the Chebyshev polynomial.

The design of Chebyshev filters is generally done the same way Butterworth filters are (with tables and Matlab). The most relevant Matlab command is cheb1ap, but there are others.

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