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.
Highlights
F OURIER analysis has been the standard tool for analyzing signals in the frequency domain
This paper proposes a novel non-stationary frequency analysis tool, called instantaneous spectral analysis (ISA), which represents waveforms as sinusoids with continuously timevarying amplitude within the evaluation period
The signal was obtained by applying an inverse discrete Fourier transform (DFT) to a 16-sample frequency domain vector, where each sample corresponds to a point in the 16QAM constellation
Summary
F OURIER analysis has been the standard tool for analyzing signals in the frequency domain. This paper proposes a novel non-stationary frequency analysis tool, called instantaneous spectral analysis (ISA), which represents waveforms as sinusoids with continuously timevarying amplitude within the evaluation period. These sinusoids can be represented using complex spirals that are generalizations of the complex circles used by the FT. ISA conceptually differs from any other existing frequency analysis tool, including Fourier and time-frequency distributions, because it treats the time domain waveform as a polynomial or sequence of polynomials Another difference between ISA and the conventional frequency analysis tools lies in the fact that the ISA basis functions are complex spirals, i.e., complex exponentials with continuously increasing or decreasing amplitude, as opposed to Fourier’s (constant magnitude) complex circles.
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