Basis expansion time-frequency analysis

Introduction

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

$x(t)\approx \sum _{m=1}^{M}a_{m}\cdot e^{j{\tfrac {2\pi mt}{T}}}$

$a_{m}=\int _{0}^{T}x(t)e^{\frac {-j2\pi mt}{T}}\,\mathrm {d} t$

Some Time-Frequency Analysis are also attempt to represent signal as the form below

$x(t)\approx \sum _{m=1}^{M}a_{m}\cdot \phi _{m}(t)$

when given the amount of basis M, minimize approximation error in mean-square sense

$approximation\quad error=\int _{-\infty }^{\infty }|x(t)-\sum _{m=1}^{M}a_{m}\cdot \phi _{m}(t)|^{2}\,\mathrm {d} t$

Examples

Three Parameter Atoms

$x(t)\approx \sum a_{t_{0},f_{0},\sigma }\cdot \phi _{t_{0},f_{0},\sigma }(t)$

$\phi _{t_{0},f_{0},\sigma }(t)={\frac {2^{0.25}}{\sigma ^{0.5}}}e^{j2\pi f_{0}t-{\frac {\pi (t-t_{0})^{2}}{\sigma ^{2}}}}$

Since$\quad \phi _{t_{0},f_{0},\sigma }(t)\quad$ are not orthogonal,$\quad a_{t_{0},f_{0},\sigma }\quad$ should be determined by a Matching pursuit process.

3 parameters:

$t_{0}$:controls the central time

$f_{0}$:controls the central frequency

$\sigma$:controls the scaling factor

Four Parameter Atoms(Chirplet)

$x(t)\approx \sum a_{t_{0},f_{0},\sigma ,\eta }\cdot \phi _{t_{0},f_{0},\sigma ,\eta }(t)$

$\phi _{t_{0},f_{0},\sigma ,\eta }(t)={\frac {2^{0.25}}{\sigma ^{0.5}}}e^{j2\pi (f_{0}t+{\frac {\eta }{2}}t^{2})-{\frac {\pi (t-t_{0})^{2}}{\sigma ^{2}}}}$

4 parameters:

$t_{0}$:controls the central time

$f_{0}$:controls the central frequency

$\sigma$:controls the scaling factor

$\eta$:controls the chirp rate

• Short-time Fourier transform of different basis:

References

• S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Processing, vol. 41, no. 12, pp. 3397-3415, Dec. 1993.
• A. Bultan, “A four-parameter atomic decomposition of chirplets,” IEEE Trans. Signal Processing, vol. 47, no. 3, pp. 731–745, Mar. 1999.
• C. Capus, and K. Brown. "Short-time fractional Fourier methods for the time-frequency representation of chirp signals," J. Acoust. Soc. Am. vol. 113, issue 6, pp. 3253-3263, 2003.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2016