Time–frequency analysis: Difference between revisions

Revision as of 16:31, 26 April 2024

Revision as of 16:32, 26 April 2024

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=== Short-time Fourier transform ===

#* Random process for [[Short-time Fourier transform|STFT (Short Time Fourier Transform)]]

<math>E[x(t)]\neq 0</math> should be satisfied. Otherwise,

<math>E[X(t,f)] = E[\int_{t-B}^{t+B} x(\tau)w(t-\tau)e^{-j2\pi f\tau}d\tau]</math>

<math>=\int_{t-B}^{t+B} E[x(\tau)]w(t-\tau)e^{-j2\pi f\tau}d\tau</math>for zero-mean random process, <math>E[X(t,f)] = 0</math>

#* Decompose by the AF and the FRFT. Any non-stationary random process can be expressed as a summation of the fractional Fourier transform (or chirp multiplication) of stationary random process.

==Applications==