Talk:Autocorrelation matrix

This page does not describe what a autocorrelation matrix is, let alone what it exactly does. Its just a bunch of numbers and for a layman, not very useful. Please include an explanation. — Preceding unsigned comment added by 129.132.45.243 (talk) 16:20, 17 October 2013 (UTC)

I created this page because there were links to "autocorrelation matrix" from other pages, and they really needed a definition of said matrix to make any sense. It is pretty short right now. If anyone wants to fill it out with more information about it's uses or special properties that would be good. Alternately, I would not be opposed to merging it with the autocorrelation page. Finally, I'm sure that older, more authoritative references exist, but this was the first one I found.

A real x does not necessarily imply that the ACM is circulant. For example: To be circulant a 3X3 ACM would have r(-1) = r(2). A real x does not imply E(x(0)x(-1)) = E(x(0)x(2)). Carbone1853 (talk) —Preceding undated comment added 10:54, 25 May 2011 (UTC).

Error: page doesn't make sense
Reason: Formula doesn't make sense. Reason: Meaning of notation used in the formula is not defined. Notation: something with a capital H in superscript. — Preceding unsigned comment added by 80.218.32.215 (talk) 03:14, 21 January 2014 (UTC)

Page should be deleted
This page is not right for a general stochastic process and relies on a very particular arrangement of data and assumptions placed on it - none of which is defined in the article.

If we have a general stochastic process $$\left\{\mathbf{X}_n : n \in \mathbb{Z}_0^\infty\right\}$$ with $$\mathbf{X}_n\in \mathfrak{C}^k\;\forall\;n$$, we can define the autocorrelation function as

$$\text{R}\left[n_1,n_2\right] = \mathbb{E}\mathbf{X}_{n_1}\mathbf{X}_{n_2}^\text{H} $$

where H denotes the complex-conjugate transpose. And in this way, we have a matrix-valued function whose domain is two nonnegative integers. Each entry is conceptually either a cross-correlation of two scalar-valued stochastic processes (the components in the vector) or an autocorrelation of that breed. For the scalar-minded:

$$\text{R}_{ij}\left[n_1,n_2\right] = \mathbb{E}\mathbf{X}_{n_1}(i)\mathbf{X}_{n_2}(j)^* $$

where the star denotes complex-conjugate and parenthetical indexing into the vector retrieves the $$i$$th component in that vector.

A common assumption to place on a stochastic process is for it to be wide-sense stationary. If this is the case,

$$\text{R}\left[n+\tau,n\right] = \text{R}\left[\tau,0\right]=: \text{R}\left[\tau\right] $$

The article uses a similar definition free of time-indexing, which makes the connection with stochastic processes dubious without further discourse, considering stochastic processes are temporal by definition. But the definition is wrong even we we say that it was meant to be the definition proposed with $$n_1 = n_2$$, because you cannot write the various cross-correlations in terms of a single function $$R_{XX}$$ as the author did without further assumption. In general, there would be $$\frac{n^2+n}{2}$$ different values, and the function will be symmetric. We can only write the autocorrelation matrix with $$n_1 = n_2$$ in the way he did if we assume our stochastic process is constructed by a single WSS scalar stochastic process, call it $$\left\{Y_n : n \in \mathbb{Z}_0^\infty\right\}$$ with $$Y_n\in \mathfrak{C}\;\forall\;n$$. Then, define our vector-valued stochastic process with $$\mathbf{X}_M(i) := Y_{i+kM}$$ for any M and all i from 1 to k.