On maximally flat sharp cutoff low-pass filters

Abstract

Maximally flat (MF) low-pass filters with multiple pairs of coincident or distinct imaginary axis zeros are investigated and compared with the all pole Butterworth filters. It is shown that for the same order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , finite zero filters provide much sharper cutoff than Butterworth filters, and that the cutoff slope increases with increasing number of zeros. Expressions for cutoff slope and minimum attenuation in the stopband are derived in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , the number of pairs of zeros ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> ) and their locations. In the case of coincident multiple zeros, the stopband performance is found to be an optimum for a particular value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> . The information required for design of finite zero filters is provided in the form of universal graphs and use of these graphs is illustrated by a design example.