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

TL;DR

This paper introduces a simple design method for recursive and non-recursive filters that improve instantaneous frequency estimation and phase tracking, especially under noisy conditions, with applications in Doppler shift and time-delay measurements.

Contribution

It presents a unified procedure for designing low-complexity IIR and FIR filters optimized for polynomial signal tracking and noise minimization, with novel application to frequency estimation.

Findings

Incorporation of a prediction filter reduces unwrapping errors.

IIR estimators achieve lower error variance at reduced computational cost.

Guidelines for tuning filters enhance practical applicability.

Abstract

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.