Spectrum Continuation Analysis

Spectrum Continuation Analysis

Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain.

Recall that a Fourier series is only suitable to the analysis of periodic (or finite-domain) functions f(x) with period 2π. It can be expressed as an infinite series of sinusoids:

where is the amplitude of the individual harmonics.

In SCA however, one decomposes the spectrum into optimized discrete frequencies. As a consequence, and as the period of the sampled function is supposed to be infinite or not yet known, each of the discrete periodic functions that compose the sampled function fragment can not be considered to be a multiple of the fundamental frequency:

As such, SCA does not necessarily deliver periodic functions, as would have been the case in Fourier analysis. For real-valued functions, the SCA series can be written as:

where A_n and B_n are the series amplitudes. The amplitudes can only be solved if the series of values is previously optimized for a desired objective function (usually least residuals). is not necessarily the average value over the sampled interval: one might prefer to include predominant information on the behavior of the offset value in the time domain.