Frequency Response: Sketch the frequency responses of these fourth-order low-pass filters with 3-dB cutoff frequency of 1 kHz: Butterworth, Chebyshev, and Bessel.

Answer: A low-pass Butterworth filter has a maximally flat response in the pass band as shown below. A fourth-order Butterworth filter rolls off at a rate of 80 dB per decade, which is why the gain is -80 dB at 10 kHz.

A low-pass Chebyshev filter has ripples in the pass band as shown below. The number of ripples equals half of the filter order, in this case, half of 4. This is why two passband ripples appear in the fourth-order Chebyshev response shown below. The tradeoff for accepting passband ripples is a faster rolloff, apparent because the gain reaches -80 dB at less than 10 kHz.

A low-pass Bessel filter has no ripples in the pass band as shown below. Furthermore, it rolls off slower than a Butterworth filter. This raises the question as to what advantage does a Bessel filter have. Why would anybody use it? The answer is it has the best step response. (See Step Response.)

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