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

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.

Digital Filters for Instantaneous Frequency Estimation

This technical note is on digital filters for the high-fidelity estimation of a sinusoidal signal's frequency in the presence of additive noise. The complex noise is assumed to be white (i.e. uncorrelated) however it need not be Gaussian. The complex signal is assumed to be of (approximately) constant magnitude and (approximately) polynomial phase such as the chirps emitted by bats, whale songs, pulse-compression radars, and frequency-modulated (FM) radios, over sufficiently short timescales. Such digital signals may be found at the end of a sequence of analogue heterodyning (i.e. mixing and low-pass filtering), down to a bandwidth that is matched to an analogue-to-digital converter (ADC), followed by digital heterodyning and sample rate reduction (optional) to match the clock frequency of the processor. The spacing of the discrete frequency bins (in cycles per sample) produced by the Fast Fourier Transform (FFT) is equal to the reciprocal of the window length (in samples). However, a long FFT (for fine frequency resolution) has a high complexity and a long latency, which may be prohibitive in embedded closed-loop systems, and unnecessary when the channel only contains a single sinusoid. In such cases, and for signals of constant frequency, the conventional approach involves the (weighted) average of instantaneous phase differences. General, naive, optimal, and pragmatic (recursive), filtering solutions are discussed and analysed here using Monte-Carlo (MC) simulations.