# Relation and differences between correlation sequence and correlation matrix

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?

• 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. Commented Aug 1 at 14:21

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 $$E\{x[1]x[0]\}=E\{x[0]x[1]\}$$.