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Suddenly I am stuck at a question while studying Statistical signal processing: What are the differences between correlation sequence and matrix (can be auto/cross), and how are the two related? When do we use each of them?

For example, $E\{x[n+l] x[n]\} = r_{xx}[l]$ is the autocorrelation function for the discrete time signal $x[n]$, $l$ is the lag parameter. I am considering real-valued cases for simplicity.

$R_{XX} =$ \begin{bmatrix} E\{x[0]x[0]\} & E\{x[0]x[1]\} & \cdots & E\{x[0]x[N-1]\} \\ E\{x[1]x[0]\} & E\{x[1]x[1]\} & \cdots & E\{x[1]x[N-1]\} \\ \vdots & \vdots & \ddots & \vdots \\ E\{x[N-1]x[0]\} & E\{x[N-1]x[1]\} & \cdots & E\{x[N-1]x[N-1]\} \\ \end{bmatrix} is the auto-correlation matrix of a Random vector $\textbf{X}$.

Obviously, I can look at a discrete time signal $x[n]$ as a vector, where the value of signal at each instance $n = 0,1,\cdots N-1$ is a random variable.

Now, my question is how can I relate this matrix with the sequence for the same signal/vector $x$? Next, when is the sequence used and when is the matrix used?

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  • $\begingroup$ Your question is too open. For example, if the vector $X$ is drawn from a multivariate Gaussian distribution, then the covariance matrix is useful to determine the fundamental dependencies (and conditional independencies) among the variables $x(1), x(2), \ldots, x(N)$. More concretely, the support of the inverse of $R_{XX}$ (a.k.a. Precision matrix) represents the graph of direct interactions. It is often useful directly or indirectly within the scope of network identification. $\endgroup$ Commented Aug 1, 2024 at 14:21

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Calculating $E\{x[n+l]x[n]\} = r_{xx}[l]$ is a calculation of the statistical second moment of the random vector $x$ at lag $l$. The correlation matrix extends this to include the complete set of statistical second moments for $x$. Thus, it contains the set of second moments between all possible pairs of components of $x$. In many cases, it is actually advantageous to populate the correlation matrix with the values of the autocorrelation sequence under the assumption $E\{x[1]x[0]\}=E\{x[0]x[1]\}$.

As for when one is used vs. the other, there's all sorts of ways both can be used. For example, the matched filter is a correlation calculation, and the definition of the power spectral density is the Fourier transform of the autocorrelation sequence. The correlation matrix is used in adaptive filters and beamformers, and some other more advanced spectral estimation techniques. The autocorrelation sequence (combined with the partial correlation sequence) is useful in linear models, whereas the correlation matrix is useful in principal component analysis.

Bottom line, they are both tools to explore the second moments of random signals, but are in different forms so as to be used where necessary.

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