Spectrum continuation analysis

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

Remind that a Fourier series is in fact specific to periodic (or finite-domain) functions f(x) (with period 2π) and can be expressed as an infinite series of sinusoids:

$f(x)=\sum _{n=-\infty }^{\infty }F_{n}.\,e^{inx}$

where $F_{n}$ is the complex amplitude.

In SCA however, the frequency of each of the discrete functions that compose the sampled function fragment can no longer be considered to be multiples of the fundamental frequency (the period of the sampled function is considered to be infinite):

$f(x)=\sum _{n=0}^{\infty }F_{n}.\,e^{\omega _{n}.x}$

For real-valued functions, the SCA series should than be written as:

$f(x)=\sum _{n=1}^{\infty }\left[A_{n}.\cos(\omega _{n}.x)+B_{n}.\sin(\omega _{n}.x)\right]+C_{0}$

where A_n and B_n are the series amplitudes.