Instantaneous Spectral Analysis

Abstract

The standard Fourier Transform (FT) can be seen as a change of basis in which a “time domain” amplitude sequence is re-expressed as a sum of sinusoids with constant coefficients allowing the spectral analysis of the signal. The FT can be employed either as a tool for studying the sinusoidal properties of the amplitude sequence, or more actively as a prescription for transmitting the amplitude sequence using some range of frequencies. Introduced here, Instantaneous Spectral Analysis (ISA) is similar to the FT in usage, except that ISA expresses an amplitude sequence in terms of sinusoids with continuouslyvarying amplitudes. This makes ISA more suitable than the FT for studying situations in which the amplitude sequence is generated by a continuously time-varying (non-ergodic) source, corresponding to a non-stationary spectrum. Viewed prescriptively, ISA allows an amplitude sequence to be compressed into a much smaller range of frequencies than the FT, essentially because ISA is not restricted by an assumption in the proof of the sampling theorem, that the spectrum is stationary over the evaluation interval. Intuitively, the FT expresses increasing time-domain detail by using increasingly higher frequencies. ISA, instead, uses an increasingly dense set of sinusoids with timevarying amplitude, within a fixed frequency range.