×

Time-local Fourier analysis with a scalable, phase-modulated analyzing function: The \(S\)-transform with a complex window. (English) Zbl 1151.94329

Summary: The \(S\)-transform was originally defined as a method of determining the local spectrum of a time series, through the use of a translating, real Gaussian window that dilates to accomodate the different cycle durations of different frequencies. The \(S\)-transform “wavelet” is obtained by multiplying this real window with the complex Fourier sinusoid. Since the Fourier sinusoid has time-invariant frequency, the \(S\)-transform is consequently unsuitable for resolving waveforms whose frequency changes with time. This problem can be addressed by introducing a complex Gaussian window, with a user designed, complex phase function. The phase function modulates the frequency of the Fourier sinusoid to give a specific waveform, leading to better time-frequency localization of similar waveforms on the time series. The complex-window \(S\)-transform is similar to a wavelet transform, but has the fixed phase reference of the Fourier transform.

MSC:

94A11 Application of orthogonal and other special functions
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A14 Modulation and demodulation in information and communication theory
PDFBibTeX XMLCite
Full Text: DOI