‘Almost nonunique’ solutions to parameter estimation in periodic signals

Provided the Nyquist criterion is met, by the Nyquist–Shannon sampling theorem one can reconstruct the signal uniquely using the sampled data. In this work, by means of an algorithm that computes the amplitudes, frequencies and phases of a signal to machine precision, we show that one can get two solutions with very different amplitudes, frequencies and phases that pass through the same set of sampled data to within machine precision. Although this is not a counterexample to the Nyquist–Shannon sampling theorem (since, mathematically speaking, a very small error norm is still not perfectly zero), for all practical purposes this is an example of ‘nonuniqueness’ (hence the use of the word ‘almost’ in the title). The implication is that when working within the finite precision of a computer, the parameters that one obtains from a given set of data points, even using a high-precision algorithm, can be completely different than those in the actual signal. This is illustrated by means of several examples.