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Dragon Notes

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  UNDER CONSTRUCTION

Randomness & Probability:
Multiple WSS Random Processes




Jointly WSS rp's

\(\up\)Two rp's, \(X[n]\) and \(Y[n]\), are jointly Wide Sense Stationary when the below are met:
\(\begin{align}\mu_X[n] &= E[X[n]]=\mu_X \\ r_X[k] &= E[X[n]X[n+k]] \\ \mu_Y[n] &= E[Y[n]]=\mu_Y \\ r_Y[k] &= E[Y[n]Y[n+k]] \\ r_{X,Y}[k] &= E[X[n]Y[n+k]] \end{align}\)

Cross-correlation Sequence

\(\up\)The Cross-correlation sequence of two rp's \(X[n]\) and \(Y[n]\) is defined as
\(\ds r_{X,Y}[k]=E[X[n]Y[n+k]]\quad k=..., -1,0,1, ...\)
and is a measure of the similarity of two rp's as a function of relative displacement.
Cross-correlation sequence
[\(n\)-independence] If \(E[...]\) depends on \(n\), a CCS cannot be defined
[Non-symmetry] \(\boxed{r_{X,Y}[-k]\neq r_{X,Y}[k]}\) (CCS is not necessarily symmetric)
[\(\t{max}\lrpar{r_{X,Y}}\) can occur at any \(k\)] The maximum of the CCS can occur for any value of \(k\)
[Bounded] \(\boxed{\abs{r_{X,Y}[k]}\leq\sqrt{r_X[0]r_Y[0]}}\) (The maximum value of the CCS is bounded)
[Interchange-symmetry] \(\boxed{r_{Y,X}[k]=r_{X,Y}[-k]}\) (Interchanging \(X[n]\) and \(Y[n]\) flips the CCS about \(k=0\))
[LSI relation] \(\boxed{r_{X,Y}[k]=\iisum{l}h[l]r_X[k-l]}\) \(\Lra\) \(\boxed{r_{X,Y}[k]=h[k]*r_X[k]}\ \)
asm
\(X[n]\ra\boxed{\t{LSI}}\ra Y[n]\)
Estimating CSS can be done with
\(\hat{r}_{X,Y}[k]=\left\{\nmtx{\frac{1}{N-k}\sum_{n=0}^{N-1-k}x[n]y[n+k]\ \ k=0,1,..., M\hspace{110px}} {\frac{1}{N-\abs{k}}\sum_{n=\abs{k}}^{N-1}x[n]y[n+k]\ \ \ k=-M,-(M-1),...,-1}\right.\)
where data records used for estimation were first shifted to place the maximum at \(k=0\) (time alignment). CCS is then estimated for \(\abs{k}\leq M\), where \(M\) is some quantity for which \(r_{X,Y}[k]\approx 0\) if \(\abs{k}>M\).

Cross-Spectral Density

\(\up\)The Cross-Power Spectral Density (CPSD), or Cross-spectral Density (CSD), of two rp's \(X[n]\) and \(Y[n]\) (or \(X(t)\) & \(Y(t)\)), is defined as
\(\ds \begin{align}P_{X,Y}(f)&=\ilim{M}\frac{1}{2M+1}E\lrbra{\lrpar{\sum_{n=-M}^{M}X[n]\t{exp}(-j2\pi fn)}^*\lrpar{\sum_{n=-M}^{M}Y[n]\t{exp}(-j2\pi fn)}} \\ P_{X,Y}(f) &= \ilim{T}\frac{1}{T}E\lrbra{\lrpar{\int_{-T/2}^{T/2}X(t)\t{exp}(-j2\pi Ft)dt}^*\lrpar{\int_{-T/2}^{T/2}Y(t)\t{exp}(-j2\pi Ft)dt}}\end{align}\)
and is a measure of power distribution over frequency of a joint rp.
[CSD-CCS relation] \(\boxed{P_{X,Y}(f)=\iisum{k}r_{X,Y}[k]\t{exp}(-j2\pi fk)}\) (CSD is the Fourier transform of the CCS)
[Hermitian] \(\boxed{P_{X,Y}(-f)=P^*_{X,Y}(f)}\) (\(g(f)\) is hermitian if \(\Re\lrpar{g(f)}=\) even and \(\Im\lrpar{g(f)}=\) odd about \(f=0\))
[Bounded] \(\boxed{\abs{P_{X,Y}(f)}\leq\sqrt{P_X(f)P_Y(f)}}\) (the maximum value of the CSD is bounded)
[0 for zero-mean & uncorr. rp's] \(\boxed{P_{X,Y}(f)=0}\ \)
asm
\(\mu_X=0, \mu_Y=0\), \(X[n]\) & \(Y[n]\) uncorr.
[Complex conjugate-symmetry] \(\boxed{P_{Y,X}(f)=P^*_{X,Y}(f)}\) (CSD of \((Y[n],X[n])\) is the complex conjugate of the CSD of \((X[n],Y[n])\))
[Coherence / Complex corr.] \(\boxed{\gamma_{X,Y}(f) = \frac{P_{X,Y}(f)}{\sqrt{P_X(f)P_Y(f)}}}\); \(\gamma_{X,Y} =\) coherence function, giving the complex corr. coefficient at \(f\); measures the correlation between the Fourier transforms of two jointly WSS rp's at \(f\)
[CSD-PSD relation] \(\boxed{P_{X,Y}(f)=H(f)P_X(f)}\ \)
asm
\(X[n]\ra\boxed{\t{LSI}}\ra Y[n]\)
[LSI coherence] \(\boxed{\abs{\gamma_{X,Y}(f)}=1}\) \(\boxed{\gamma_{X,Y}(f)=\t{exp}(j\phi (f))}\), \(\phi (f)=\angle H(f)\ \)
asm
\(X[n]\ra\boxed{\t{LSI}}\ra Y[n]\)
implies that \(Y[n]\) is perfectly predictable from \(X[n]\), and vice versa (since \(\rho_{X,Y}=\rho_{Y,X}\))
[LSI input-output] \(\boxed{Y[n]=\iisum{k}h[k]X[n-k]}\ \)
asm
\(X[n]\ra\boxed{\t{LSI}}\ra Y[n]\)





Dragon Notes,   Est. 2018     About

By OverLordGoldDragon