  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.
