Magnitude response optimization of delta operator filters

Digital filters that use the delta operator have been shown to be less sensitive to filter coefficient quantization than filters using the shift operator when the poles and zeros lie near the point z=±1. It is customary to simply round the prototype filter coefficients to the nearest fixed-point numbers for the implementation. We show that when very few bits are allocated for each coefficient in a delta filter, some other quantized point than the directly rounded point is likely to offer a magnitude response that more closely matches the required characteristics. Furthermore, a neighborhood search may even find a solution that fulfills the given specifications with fewer bits than by using direct rounding. This is of particular importance in application-specific integrated circuit implementations where expensive die area can be saved by using fewer bits to represent a coefficient. Our experiments also confirm the result that delta parameterization regularly requires at least two or three bits less than shift parameterization to fulfill the same narrowband filter specifications