Random Process

Posted on 2023-11-10 Edited on 2024-09-22 In noise

Ergodicity & autocorrelation

The most important statistical properties of a random process \(x(t)\) are obtained by applying the expectation operator to some functions of the process itself

The expectation returns the probability-weighted average of the specific function at that specific time over all possible realizations of the process

In many real practical cases, though, data from many realizations of the process are not available. On the contrary, often only data from one of them are known

ensemble autocorrelation and temporal autocorrelation (time autocorrelation)

LTI Systems on WSS Processes

mean

autocorrelation

deterministic autocorrelation function

why \(\overline{R}_{hh}(\tau) \overset{\Delta}{=} h(\tau)*h(-\tau)\) is autocorrelation ? the proof is as follows:

\[\begin{align} \overline{R}_{hh}(\tau) &= h(\tau)*h(-\tau) \\ &= \int_{-\infty}^{\infty}h(x)h(-(\tau - x))dx \\ &= \int_{-\infty}^{\infty}h(x)h(-\tau + x))dx \\ &=\int_{-\infty}^{\infty}h(x+\tau)h(x))dx \end{align}\]

PSD

Topic 6 Random Processes and Signals [https://www.robots.ox.ac.uk/~dwm/Courses/2TF_2021/N6.pdf]

Alan V. Oppenheim, Introduction To Communication, Control, And Signal Processing [https://ocw.mit.edu/courses/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/a6bddaee5966f6e73450e6fe79ab0566_MIT6_011S10_notes.pdf]

Time Reversal \[ x(-t) \overset{FT}{\longrightarrow} X(-j\omega) \]

if \(x(t)\) is real, then \(X(j\omega)\) has conjugate symmetry \[ X(-j\omega) = X^*(j\omega) \]

Random Signals Sampling

sampling autocorrelation sequence

Alan V Oppenheim, Ronald W. Schafer. Discrete-Time Signal Processing, 3rd edition

[https://dsp.stackexchange.com/a/17348/59253]

Noise Aliasing

apply foregoing observation

The Periodogram

The periodogram is in fact the Fourier transform of the aperiodic correlation of the windowed data sequence

estimating continuous-time stationary random signal

periodogram.drawio

The sequence \(x[n]\) is typically multiplied by a finite-duration window \(w[n]\), since the input to the DFT must be of finite duration. This produces the finite-length sequence \(v[n] = w[n]x[n]\)

\[\begin{align} \hat{P}_{ss}(\Omega) &= \frac{|V(e^{j\omega})|^2}{LU} \\ &= \frac{|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2} \tag{1}\\ &= \frac{L|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} \tag{2} \end{align}\]

That is, by \((1)\) \[ \hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} = \frac{T_s|V(e^{j\omega})|^2}{\sum_{n=0}^{L-1}(w[n])^2}=\frac{|V(e^{j\omega})|^2}{f_{res}L\sum_{n=0}^{L-1}(w[n])^2} \]

That is, by \((2)\) \[ \hat{P}_{ss}(\Omega) = T_s\hat{P}_{xx(\omega)} = \frac{T_sL|V(e^{j\omega})|^2}{\sum_{k=0}^{L-1}(W[k])^2} = \frac{|V(e^{j\omega})|^2}{f_{res}\sum_{k=0}^{L-1}(W[k])^2} \]

!! ENBW

Wiener-Khinchin theorem

Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914

\(x(t)\), Fourier transform over a limited period of time \([-T/2, +T/2]\) , \(X_T(f) = \int_{-T/2}^{T/2}x(t)e^{-j2\pi ft}dt\)

With Parseval's theorem \[ \int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}|X_T(f)|^2df \] So that \[ \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt = \int_{-\infty}^{\infty}\frac{1}{T}|X_T(f)|^2df \]

where the quantity, \(\frac{1}{T}|X_T(f)|^2\) can be interpreted as distribution of power in the frequency domain

For each \(f\) this quantity is a random variable, since it is a function of the random process \(x(t)\)

The power spectral density (PSD) \(S_x(f )\) is defined as the limit of the expectation of the expression above, for large \(T\): \[ S_x(f) = \lim _{T\to \infty}\mathrm{E}\left[ \frac{1}{T}|X_T(f)|^2 \right] \]

The Wiener-Khinchin theorem ensures that for well-behaved wide-sense stationary processes the limit exists and is equal to the Fourier transform of the autocorrelation \[\begin{align} S_x(f) &= \int_{-\infty}^{+\infty}R_x(\tau)e^{-j2\pi f \tau}d\tau \\ R_x(\tau) &= \int_{-\infty}^{+\infty}S_x(f)e^{j2\pi f \tau}df \end{align}\]