  A simple procedure for the design of recursive digital filters with an
infinite impulse response (IIR) and non-recursive digital filters with a finite
impulse response (FIR) is described. The fixed-lag smoothing filters are
designed to track an approximately polynomial signal of specified degree
without 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 specified
bandwidth, and excellent passband phase linearity. The filters are applied to
the problem of instantaneous frequency estimation, e.g. for Doppler-shift
measurement, for a complex exponential with polynomial phase progression in
additive white noise. For this classical problem, simulations show that the
incorporation of a prediction filter (with a one-sample lead) reduces the
incidence of (phase or frequency) angle unwrapping errors, particularly for
signals with high rates of angle change, which are known to limit the
performance of standard FIR estimators at low SNR. This improvement allows the
instantaneous phase of low-frequency signals to be estimated, e.g. for
time-delay measurement, and/or the instantaneous frequency of
frequency-modulated signals, down to a lower SNR. In the absence of unwrapping
errors, 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.
