# Signal-to-Noise ratio of multivariate stochastic process from Correlation Matrix

I'm not in signal processing, I'm from an another discipline. I've derived a simple result which I presume must be well known in SP and I'd like to know whether there's a paper or textbook that has it that I can cite.

Suppose I have an n-dimensional stochastic process

$$\mathbf{x}_{t}\;=\;\left[\begin{array}{cccc} x_{t}^{\left(1\right)} & x_{t}^{\left(2\right)} & \cdots & x_{t}^{\left(n\right)}\end{array}\right]$$

where $$t$$ denotes an observation. Each of the components $$j$$ is the combination of a signal $$x_{t}^{*}$$ and some random idiosyncratic noise:

$$x_{t}^{\left(j\right)}\;=\;x_{t}^{*}b_{j}+\varepsilon_{t}^{\left(j\right)}\sqrt{1-b_{j}^{2}}$$

where $$\varepsilon^{(j)}_{t}$$ is iid distributed and uncorrelated across the $$n$$ components. For convenience, all these variables are already standardized (mean 0 variance 1).

$$b_{j}$$ and therefore the signal-to-noise of each $$j$$ component ( $$b_{j}/(1-b^2_{j})$$ ) are not observed.

The simple result is: you observe the correlation matrix $$\mathbf{R}$$ of the $$n$$ components of $$x_{t}$$. Then a simple quadratic system gives you the noise-to-signal ratios of each $$j$$ component of $$\mathbf{x}_{t}$$ .

Anybody can tell me whether you've seen this result in a paper or textbook?

Thanks!

• may I assume the $b_j$ are $\text{const.}$ and your ${}^*$ is just a "marker" for "this is the underlying value common to all $x_t^{(i)},i=1,\ldots, n$"? – Marcus Müller Feb 13 at 17:54
• Yes. Precisely. – bbecon Feb 13 at 18:16
• ok... then I don't see why you need a correlation matrix to get your result? If both $x_t^*$ and $\epsilon_t^{(j)}$ have variance 1 and mean 0, then the SNR of that component $x_t^{j}$ is $$\frac{b_j^2}{1-b_j^2}$$ per construction. – Marcus Müller Feb 13 at 18:49
• Thanks and sorry for not being clear enough. The whole point is that $b_{j}$ is unobserved. I have amended the question to clarify it. Also, I'm not looking for the answer. I have the answer. I want to confirm that this is a known result and where I can find it. – bbecon Feb 13 at 19:34
• yes, but $b_j^2$ is just the non-diagonal entries in your $\mathbf R$? – Marcus Müller Feb 13 at 20:40