Transitional Butterworth-Legendre filters

A new class of monotonic passband low-pass filters, referred to as transitional Butterworth-Legendre (TBL) filters, is introduced. The closed form expressions of the characteristic functions of these filters depending on one variable parameter q are derived by generalization of the results previously obtained for Legendre sharp cut-off monotonic filters. The parameter q, the values of which are restricted to non-negative integer values (q¿n) controls the number of flatness conditions in the magnitude response at the origin and enables a trade-off between the passband loss and the stopband attenuation of the resulting filters. The Legendre, Halpern and Butterworth filters are shown to be special cases of TBL filters. Only all-pole filter functions are discussed although the results can easily be extended so as to include low-pass filters with finite real frequency transmission zeros.