Algorithms for processing periodic and non-periodic signals

In this work, we present algorithms for the processing of periodic and non-periodic signals or data, both, in the spatial and temporal domains. For periodic data, we propose a Newton–Raphson-based algorithm that identifies the amplitudes, frequencies and phases of the sinusoidal components in the input signal to a high degree of accuracy. The algorithm is based firstly on the systematic identification of the candidate periodic functions that pass through most of the sampled data, and then identifying the ‘correct solution’ from among these multiple candidate solutions, i.e., identifying the input signal that actually generated the data. Since the final solution passes through all the data points almost exactly, and since it is periodic, the ‘leakage loss’ is almost zero. For non-periodic temporal signals, we devise an approximation for the continuous time Fourier transform, and show by means of various examples that it yields accurate results. Regarding computational efficiency, the Newton–Raphson based algorithm for periodic signals is obviously computationally more expensive than standard techniques which treat parameter estimation as a linear problem, but the emphasis at this stage in on developing a robust algorithm. The algorithm for non-periodic temporal signals is, however, computationally efficient as well, since it just involves a summation of derived expressions over the number of time intervals used in the approximation.