Time–frequency analysis: Difference between revisions

Revision as of 14:59, 26 April 2024

Revision as of 15:02, 26 April 2024

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But time–frequency analysis can.

== TF analysis and Random ProcessProcesses<ref>{{Cite book |last=Ding |first=Jian-Jiun |title=Time frequency analysis and wavelet transform class notes |publisher=Graduate Institute of Communication Engineering, National Taiwan University (NTU) |year=2022 |___location=Taipei, Taiwan}}</ref> ==

For a random process x(t), we cannot find the explicit value of x(t).

The value of x(t) is expressed as a probability function.

=== General random processes ===

* Auto-covariance function (ACF) <math>R_x(t,\tau)</math>

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<math>E[A_X(\eta,\tau)] = \int_{-\infty}^{\infty} E[x(t+\tau/2)x^*(t-\tau/2)]e^{-j2\pi t\eta}dt</math>

<math>= \int_{-\infty}^{\infty} R_x(t,\tau)e^{-j2\pi t\eta}dt</math>

=== Stationary random processes ===

* [[Stationary process|Stationary random process]]: the statistical properties do not change with t. Its auto-covariance function:

<math>R_x(t_1,\tau) = R_x(t_2,\tau) = R_x(\tau)</math> for any <math>t</math>, Therefore,