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A simple procedure for the design of recursive digital filters with aninfinite impulse response (IIR) and non-recursive digital filters with a finiteimpulse response (FIR) is described. The fixed-lag smoothing filters aredesigned to track an approximately polynomial signal of specified degreewithout bias at steady state, while minimizing the gain of high-frequency(coloured) noise with a specified power spectral density. For the IIR variant,the procedure determines the optimal lag (i.e. the passband group delay)yielding a recursive low-complexity smoother of low order, with a specifiedbandwidth, and excellent passband phase linearity. The filters are applied tothe problem of instantaneous frequency estimation, e.g. for Doppler-shiftmeasurement, for a complex exponential with polynomial phase progression inadditive white noise. For this classical problem, simulations show that theincorporation of a prediction filter (with a one-sample lead) reduces theincidence of (phase or frequency) angle unwrapping errors, particularly forsignals with high rates of angle change, which are known to limit theperformance of standard FIR estimators at low SNR. This improvement allows theinstantaneous phase of low-frequency signals to be estimated, e.g. fortime-delay measurement, and/or the instantaneous frequency offrequency-modulated signals, down to a lower SNR. In the absence of unwrappingerrors, the error variance of the IIR estimators (with the optimal phase lag)reaches the FIR lower bound, at a significantly lower computational cost.Guidelines for configuring and tuning both FIR and IIR filters are provided.

Recursive and non-recursive filters for sequential smoothing and  prediction with instantaneous phase and frequency estimation applications  (extended version)