On the behavior of minimax relative error fir digital differentiators

Optimum (in a minimax relative error sense) linear phase FIR digital differentiators can be designed in an efficient manner using a Remez optimization procedure. This paper presents data on wideband differentiators designed with even and odd values of N, the filter impulse response duration in samples. Based on these data, several interesting observations can be made, including: (i) Differentiators with even values of N have peak relative errors which are approximately one to two orders of magnitude smaller than identical bandwidth differentiators with odd values of N, and with the same number of multiplications per sample in a direct convolution realization. (ii) The smaller the bandwidth of the differentiator, the faster the decrease of the peak relative error with increasing N. (iii) The larger the value of N, the faster the decrease of the peak relative error with decreasing bandwidth. These observations lead to the conclusions that the bandwidth of a differentiator should be made as small as possible, and that even values of N should be used whenever possible. Complete tables of values of the impulse response coefficients are included for several wideband differentiators for even and odd values of N.