Editing Time–frequency analysis (section)

== TF analysis and random processes<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>
:<math>R_x(t,\tau) = E[x(t+\tau/2)x^*(t-\tau/2)]</math> 
:In usual, we suppose that <math>E[x(t)] = 0 </math> for any t,

:<math>E[x(t+\tau/2)x^*(t-\tau/2)]</math>
:<math>=\iint x(t+\tau/2,\xi_1)x^*(t-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2</math>
:(alternative definition of the auto-covariance function)
:<math>\overset{\land}{R_x}(t,\tau)=E[x(t)x(t+\tau)]</math>
* Power spectral density (PSD) <math>S_x(t,f)</math>
:<math>S_x(t,f) = \int_{-\infty}^{\infty} R_x(t,\tau)e^{-j2\pi f\tau}d\tau</math>
* Relation between the [[Wigner distribution function|WDF (Wigner Distribution Function)]] and the PSD
:<math>E[W_x(t,f)] = \int_{-\infty}^{\infty} E[x(t+\tau/2)x^*(t-\tau/2)]\cdot e^{-j2\pi f\tau}\cdot d\tau</math>
:::::<math>= \int_{-\infty}^{\infty} R_x(t,\tau)\cdot e^{-j2\pi f\tau}\cdot d\tau</math><math>= S_x(t,f)</math>
* Relation between the [[ambiguity function]] and the ACF
:<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,
<math>R_x(\tau) = E[x(\tau/2)x^*(-\tau/2)]</math>
<math>=\iint x(\tau/2,\xi_1)x^*(-\tau/2,\xi_2)P(\xi_1,\xi_2)d\xi_1d\xi_2</math>PSD,
<math>S_x(f) = \int_{-\infty}^{\infty} R_x(\tau)e^{-j2\pi f\tau}d\tau</math> White noise:
<math>S_x(f) = \sigma</math>                                   , where <math>\sigma</math> is some constant.[[File:Stationary random process's WDF and AF.jpg|right|frameless|440x440px]]
* When x(t) is stationary,
<math>E[W_x(t,f)] = S_x(f)</math>                , (invariant with <math>t</math>)

<math>E[A_x(\eta,\tau)] = \int_{-\infty}^{\infty} R_x(\tau)\cdot e^{-j2\pi t\eta}\cdot dt</math>
<math>= R_x(\tau)\int_{-\infty}^{\infty} e^{-j2\pi t\eta}\cdot dt</math><math>= R_x(\tau)\delta(\eta)</math>        , (nonzero only when <math>\eta = 0</math>) 

=== Additive white noise ===
* For additive white noise (AWN),
:<math>E[W_g(t,f)] = \sigma</math>
:<math>E[A_x(\eta,\tau)] = \sigma\delta(\tau)\delta(\eta)</math>

* Filter Design for a signal in additive white noise
[[File:Filter design for white noise.jpg|left|thumb|440x440px]]





<math>E_x</math>: energy of the signal

<math>A</math>  : area of the time frequency distribution of the signal

The PSD of the white noise is <math>S_n(f) = \sigma</math>


<math>SNR \approx 10\log_{10}\frac{E_x}{\iint\limits_{(t,f)\in\text{signal part}} S_x(t,f)dtdf}</math>

<math>SNR \approx 10\log_{10}\frac{E_x}{\sigma\Alpha}</math>

=== Non-stationary random processes ===
* If <math>E[W_x(t,f)]</math> varies with <math>t</math> and <math>E[A_x(\eta,\tau)]</math> is nonzero when <math>\eta = 0</math>,  then <math>x(t)</math> is a non-stationary random process.
* If    
*# <math>h(t) = x_1(t)+x_2(t)+x_3(t)+......+x_k(t)</math>
*# <math>x_n(t)</math>'s have zero mean for all <math>t</math>'s
*# <math>x_n(t)</math>'s are mutually independent for all <math>t</math>'s and <math>\tau</math>'s
:then:
::<math>E[x_m(t+\tau/2)x_n^*(t-\tau/2)] = E[x_m(t+\tau/2)]E[x_n^*(t-\tau/2)] = 0</math> 

* if <math>m \neq n</math>, then

::<math>E[W_h(t,f)] = \sum_{n=1}^k E[W_{x_n}(t,f)]</math>
::<math>E[A_h(\eta,\tau)] = \sum_{n=1}^k E[A_{x_n}(\eta,\tau)]</math>

=== 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.