On the behavior of minimax fir digital Hilbert transformers

Optimum (in a minimax sense) linear phase FIR Hilbert transformers can be designed efficiently using a Remez optimization procedure. This paper presents useful design data on wideband Hilbert transformers with even and odd values of N, the impulse response duration (in samples) of the filter. Based on these data, the following observations can be made: (i) In the case of equal lower and upper transition regions, Hilbert transformers with odd values of N can be realized more efficiently than those with even values of N, assuming the same peak errors of approximation for both cases. This is because every other impulse response sample is exactly zero for odd values of N. (ii) The peak approximation error for Hilbert transformers with odd values of N is determined primarily by the minimum of the values of the lower and upper transition widths. (iii) The peak approximation error for Hilbert transformers with even values of N is determined primarily by the lower transition width of the filter. (iv) The smaller the bandwidth of the Hilbert transformer, the faster the decrease of peak error of approximation with decreasing bandwidth of the Hilbert transformer. (v) The larger the value of N, the faster the decrease of peak approximation error with decreasing bandwidth of the Hilbert transformer. These observations lead to the conclusion that the bandwidth of the Hilbert transformer should be made as small as possible, and that odd values of N should be used, whenever possible, for efficient direct form realizations. A set of tables of values of the impulse response coefficients is included for several different bandwidth Hilbert transformers and for both even and odd values of N.