What is a Tracker?

A Prism-based tracker accepts a sinusoidal input signal and generates sample by samples estimates for frequency, amplitude and/or phase of the signal.

Different Prism arrangements result in different trackers. The simplest, and perhaps the most efficient in terms of computational effort and tracking accuracy, is the Recursive Signal Tracker (RST).

In the figure above the triangle symbol represents a Prism, with characteristic frequency m and where h = 1. Its outputs over recent history are stored. Simple calculations provide a revised estimate of r, and from this value estimates of fA, and φ are derived from the latest outputs of the Prism.

Despite the simplicity of the calculation, for an input signal consisting of a sinusoid with added white noise, the RST performs close to the theoretical limit for parameter accuracy, as given by the Cramer-Rao Lower Bound (CRLB). The plot below shows the ratio of RST accuracy performance as a ratio of the CRLB and with varying r. Good tracking is achieved for r in the range [0.4, 0.6].

How might this work in practice? Suppose we are sampling a signal at 10 kHz, and wish to track a sinusoid with a frequency varying between 80 Hz and 120 Hz. An RST, with m for the Prism set at 200 Hz, will be able to track the sinusoid well, as the values of r will fall between 0.4 and 0.6 in this case.

The following plot shows the performance of an RST tracking a sinusoid in white noise where the SNR is 0 db. The tracked signal is constructed from the calculated amplitude and phase values. Despite the high level of white noise, good tracking is achieved.