Answer 4
The passband ripples can be spaced fairly evenly, and can be thought of as a sine or cosine wave imposed on a flat frequency response. Because multiplication in the frequency domain corresponds to convolution in the time domain, and the transform of a sine wave is an impulse, the effect of the ripple in the frequency domain is low-level echoes in the time domain.

For example, the following filter was designed using the Parks-McClellan algorithm to produce an anti-alias filter with about 1 dB of ripple in the passband. The overall response:

A close-up of the passband shows the ripple:

The impulse response clearly shows the pre- and post-echoes at its ends.

Design example provided by Robert Bristow-Johnson in the Usenet newsgroup comp.dsp.

Contributor: Dave Tweed

Published May 2003