Recursive implementation of FIR differentiators with optimum noise attenuation

We introduce a computationally efficient recursive implementation of digital finite impulse response (FIR) filters for estimating the rate of change or slope of digitized signals. The proposed FIR differentiator is characterized by the optimal attenuation of white noise and an efficient suppression of upper-band frequencies. The basic structure needs only one multiplier, which becomes a power of two with an appropriate selection of the length of the impulse response. The structure does not need resetting and recovers from any bit errors. For long filters, sampling rate reduction by decimation gives further computational savings.