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Dragon Notes

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  UNDER CONSTRUCTION

Randomness & Probability:
Gaussian Distribution




Multivariate Gaussian PDF

Gaussian random vector, \(\bn{X}\sim\mathcal{N}(\bn{\mu},\bn{C})\), is an \(N\times 1\) rv \([X_1\ X_2\ ...\ X_N]^T\) with a joint PDF given by the multivariate Gaussian PDF
\(p_{\bn{X}}(\bn{x})=\lfrac{1}{(2\pi)^{N/2}\t{det}^{1/2}(\bn{C})}\t{exp}\lrbra{-\frac{1}{2}\lrpar{\bn{x}-\bn{\mu}^T\bn{C}^{-1}(\bn{x}-\bn{\mu})}},\)

\(\ds\bn{C}=\left[\mtxxxx{\var{X_1}}{\cov{X_1,X_2}}{...}{\cov{X_1,X_N}}{\cov{X_2,X_1}}{\var{X_2}}{...}{\cov{X_2,X_N}}{\vdots}{\vdots} {\ddots}{\vdots}{\cov{X_N,X_1}}{\cov{X_N,X_2}}{...}{\cov{X_N,X_N}}\right],\) \(\ds\ \bn{\mu}=\lrbra{\nmtttx{\mu_1}{\mu_2}{\vdots}{\mu_N}}=\lrbra{\nmtttx{E_{X_1}[X_1]}{E_{X_2}[X_2]}{\vdots}{E_{X_N}[X_N]}}\)


[1]: Only the first two moments, \(\bn{\mu}\) and \(\bn{C}\), are required to specify the entire PDF
[2]: Uncorrelated\({}^*\) \(\ra\) independent (* - all rv's)
[3]: A linear transformation of \(\bn{X}\) produces another Gaussian vector.
If \(\bn{Y}=\bn{GX}\), then \(\bn{Y}\sim\mathcal{N}(\bn{G\mu}, \bn{GCG}^T)\) -- (\(\bn{G} = M\times N\) matrix, \(M\leq N\))
Multivariate Gaussian PDF

Gaussian Random Process

An rp is Gaussian if all finite sets of samples, \(\bn{X}=[X[n_1]\ X[n_2]\ ...\ X[n_K]]^T\), have a multivariate Gaussian PDF for all \(\{n_1,n_2,...,n_K\}\) and all \(K\). Its properties follow:

[1]: Uncorrelated \(\ra\) independent

(a Gaussian rp with uncorrelated samples has independent samples)


[2]: WSS \(\ra\) SSS

(a WSS Gaussian rp is also stationary)


[3]: \(\t{LSI}\{\)Gaussian rp\(\}\) \(\ra\) Gaussian rp

(any linear transformation of a Gaussian rp produces another Gaussian rp)

[4]: \(X[n]\ra\lrbra{LSI}\ra Y[n]\), \(X[n]=\) WSS Gaussian rp with mean \(\mu_X\), ACS \(r_X[k]\), and \(\mu_Y=\mu_X H(0)\), \(\ P_Y(f)=\abs{H(f)}^2P_X(f)\).

(LSI = linear, shift-invariant filter)

For \(N\) successive output samples \(\bn{Y}=[Y[0]\ Y[1]\ ...\ Y[N-1]]^T\),
\(\ds p_{\bn{Y}}(\bn{y})=\frac{1}{(2\pi)^{N/2}\t{det}^{1/2}(\bn{C}_Y)}\t{exp}\lrbra{-\sfrac{1}{2}(\bn{y}-\bn{\mu}_Y)^T\bn{C}_Y^{-1}(\bn{y}-\bn{\mu}_Y)},\)
\(\dsup\bn{\mu}_Y=[\mu_X H(0)\cdots\mu_X H(0)],\)
\(\ds[\bn{C}_Y]_{mn}=r_U[m-n]-(\mu_X H(0))^2 = \int_{-1/2}^{1/2}\abs{H(f)}^2 P_X(f)\t{exp}(j2\pi f(m-n))df - (\mu_X H(0))^2\)
[5]: \(\t{(Higher-order)} = \sum\t{(Second-order)}\);
(higher-order joint moments of a multivariate Gaussian PDF can be expressed in terms of first- and second-order moments)

 - For rv \(\dsup\bn{X}=[X_1X_2X_3X_4]^T\ \bn{\mu}_X=0\),
\(\ds E[X_1X_2X_3X_4]=E[X_1X_2]E[X_3X_4]+ E[X_1X_3]E[X_2X_4]+E[X_1X_4]E[X_2X_3]\)
 - For rp \(\dsup X[n]\), \(\mu_X = 0\),
\(\ds \begin{align} E[X[n_1]X[n_2]X[n_3]X[n_4]] &= E[X[n_1]X[n_2]]\ E[X[n_3]X[n_4]] \\ &+ \hspace{2px} E[X[n_1]X[n_3]]\ E[X[n_2]X[n_4]] \\ &+ \hspace{2px} E[X[n_1]X[n_4]]\ E[X[n_2]X[n_3]] \end{align}\)
 - and if furthermore \(X[n]\) is WSS,
\(\ds\begin{align} E[X[n_1]X[n_2]X[n_3]X[n_4]] &= r_X[n_2-n_1]r_X[n_4-n_3] + r_X[n_3-n_1]r_X[n_4-n_2] \\ &+ \hspace{2px}r_X[n_4-n_1]r_X[n_3-n_2]\end{align}\)


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon