FIR differentiators for quantized signals

The impact of quantization noise on a signal whose rate is to be estimated using a FIR differentiator is analyzed, concentrating on the important constant-rate case in order that the filter be optimized for systems with low-frequency rates of change. Formulae for the mean-squared error of the filter, the corresponding spectral characteristics, and general formulae governing the filler coefficients are derived. The characteristics of four specific differentiators, including a representative wideband differentiator, are examined and compared. It is shown that a differentiator that is optimum in terms of its attenuation of white noise can also be considered optimum with respect to quantization noise attenuation in certain circumstances. An elegant relationship is derived between worst-case rms error and the fractional value of the rate at which this error occurs. Minimization of this worst-case mean-squared error is shown to be achieved with a simple differentiator. However, the corresponding average error is poor, and a simple nonlinear filter that minimizes the worst-case error, while retaining a similar average mean-squared error to that of the "optimum" differentiator, is proposed. The equivalence between FIR differentiators and the decoders used in single-loop sigma-delta modulators is also highlighted.