All‐pole low‐pass digital filters with (2m‐ 1)‐th flat and r‐th equiripple group delay
Kyoto Kato, Tsuyoshi Takebe · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1989
Abstract As is well known, there are two kinds of constant delay all‐pole low‐pass digital filter functions: maximally flat type and equiripple type. In some cases of waveform transmission, transfer functions with inter mediate characteristics between the two forementioned types are preferable. This paper presents such transfer functions called (M‐r)FD function. In the n‐th order function, the delay flatness is reduced from the maximal order nO = (2n ‐ 1)‐th to (2m ‐ 1)‐th, where m < n. Then equiripple functions with a total of r maxima and minima of delay are derived utilizing the resulting r( = n – rn) degrees of freedom. The derivation of the transfer function is based on the method by Fettweis, i.e., in other words, the function providing the max‐ finally flat delay is obtained by forming the continued‐fraction expansion of tanh of the constant delay and truncating its expression at the n‐th‐term. Among n coefficients α2 of the expression, r from the last are given small deviations. While maintaining the (2m‐1)‐th flatness at zero frequency, an approximation is made by the traditional Newton‐Raphson method so that the delay deviation from the constant exhibits an equiripple behavior with the width of ±Δτ the (M‐r)DF function was derived for the case where the delay at τ = 0 is specified and for the case where the constant delay band width is specified. Keeping constant, the bandwidth for constant delay in the derived characteristics is increased with the increase of the number of ripples. the derived functions are summarized in a table.