On maximally flat low-pass filters with ripple in the stopband

Maximally flat (MF) low-pass filters of order
n
with exactly
n
, or
n - 1
finite zeros are investigated and are compared with the same order Butterworth filter, i.e., the MF filter with no finite zeros. It is seen that the MF filter with finite imaginary zeros exhibits sharper cutoff at the edge of the passband and ripple in the stopband. It is shown that the MF filter that maximizes the magnitude of the slope at cutoff is characterized by equal ripple in the stopband. This filter is shown to be the Inverse Chebyshev. Expressions for cutoff slope and for transition bandwidth are derived and are compared with those of the Butterworth filter. The step responses of the Inverse Chebyshev and of the Butterworth are also compared.