Optimal design of digital Hilbert transformers with a concavity constraint

A linear programming algorithm is described for designing FIR digital filters with the constraint that the magnitude response be concave over prescribed frequency bands. This is applied to odd-length Hilbert Transformers, and computational results are given. The concavity constraint avoids the ripple of the minimax design, and retains the advantage of maintaining half-band symmetry in the case of symmetric transition bands, so that alternate impulse response samples are zero. If N is the length of the impulse response, ΔF the (symmetric) transition width, and δ the maximum error, it is found that
N\DeltaF/\log_{10}^{/delta} \approx -1.1
, as opposed to the value of -0.61 in the minimax case (with ripple) reported by Rabiner and Schafer. Thus, the price paid for the absence of ripples is about twice the number of multiplications per sample.