Basis expansion time-frequency analysis


Linear expansions in a single basis, whether it is a Fourier series, wavelet, or any other basis, are not suitable enough. A Fourier basis provided a poor representation of functions well localized in time, and wavelet bases are not well adapted to represent functions whose Fourier transforms have a narrow high frequency support. In both cases, it is difficult to detect and identify the signal patterns from their expansion coefficients, because the information is diluted across the whole basis. Therefore, we must large amounts of Fourier basis or Wavelets to represent whole signal with small approximation error. Some matching pursuit algorithms are proposed in reference papers to minimize approximation error when given the amount of basis.

Properties

For Fourier series
Some time-frequency analysis are also attempt to represent signal as the form below
when given the amount of basis M, minimize approximation error in mean-square sense

Examples

Three-parameter atoms

Since are not orthogonal, should be determined by a matching pursuit process.
Three parameters:

Four-parameter atoms (chirplet)

Four parameters:

[Short-time Fourier transform] of different basis