Discussion
In filter design, what is the purpose of the Butterworth approximation ?
Answer: A
The Butterworth approximation aims to achieve a maximally flat response in the passband.
A Butterworth filter's magnitude response, \(|H(j\omega)|^{2} = 1/(1 + (\omega/\omega_{c})^{2n})\), has as many derivatives as possible equal to zero at \(\omega = 0\). The passband is therefore as flat as possible, with no ripple, and the response falls monotonically through the cut-off.
Equal ripple in both passband and stopband is the property of the elliptic (Cauer) filter; Chebyshev filters have equal ripple in one band. Minimum phase distortion is the aim of the Bessel filter, and that matters in the passband, not the stopband. Bandwidth is fixed by the design specification, not maximised by the approximation.
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