Exercise 4.15Z: Statements of the Covariance Matrix

Are the random signals correlated?

Let be given the two Gaussian random variables $u$ and $v$, each zero mean and with variance $\sigma^2 = 1$.

From these, three new random variables are formed by linear combination:

$$x_1 = A_1 \cdot u + B_1 \cdot v,$$

$$x_2 = A_2 \cdot u + B_2 \cdot v,$$

$$x_3 = A_3 \cdot u + B_3 \cdot v.$$

Assuming that in all cases considered $(i = 1, 2, 3)$ holds:

$$A_i^2 + B_i^2 =1.$$

The graph shows the signals $x_1(t)$, $x_2(t)$ and $x_3(t)$ for the case to be considered in the subtask (3) :

$A_1 = B_2 = 1$,
$A_2 = B_2 = 0$,
$A_3 = 0.8, \ B_3 = 0.6$,

The correlation coefficient $\rho_{ij}$ between the random variables $x_i$ and $x_j$ is given as follows:

$$\rho_{ij} = \frac{A_i \cdot A_j + B_i \cdot B_j}{\sqrt{(A_i^2 + B_i^2)(A_j^2 + B_j^2)}} = A_i \cdot A_j + B_i \cdot B_j.$$

Under the assumption implicit here $\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1$ the covariance matrix $\mathbf{K}$ is:

$${\mathbf{K}} =\left[ K_{ij} \right] = \left[ \begin{array}{ccc} 1 & \rho_{12} & \rho_{13} \\ \rho_{12} & 1 & \rho_{23} \\ \rho_{13} & \rho_{23} & 1 \end{array} \right] .$$

This is identical to the correlation matrix $\mathbf{R}$ for zero mean random variables.

Hints:

The exercise belongs to the chapter Generalization to N-Dimensional Random Variables.
Some basics on the application of vectors and matrices can be found on the pages Determinant of a Matrix and Inverse of a Matrix .

Questions

	$\mathbf{K}$ can be a diagonal matrix with a suitable choice of $A_1$, ... , $B_3$ be a diagonal matrix. Or in other words, $\rho_{12} = \rho_{13} = \rho_{23} = 0$ is possible.
	With appropriate choice of parameters $A_1$, ... , $B_3$ exactly one of the correlation coefficients $\rho_{ij} = 0$ can be.
	With appropriate choice of parameters $A_1$, ... , $B_3$ exactly two of the correlation coefficients $\rho_{ij} = 0$ can be.
	With appropriate choice of parameters $A_1$, ... , $B_3$ all three correlation coefficients $\rho_{ij} \ne be 0$ .

$\rho_{12} \ = \ $

$\rho_{13} \ = \ $

$\rho_{23} \ = \ $

$\rho_{12} \ = \ $

$\rho_{13} \ = \ $

$\rho_{23} \ = \ $

Solution

(1) Only the second and the last statement are true:

Statement 2 describes the case considered in the graph where two quantities $($here: $x_1$ and $x_2)$ are uncorrelated, while $x_3$ has statistical bindings with respect to $x_1$ $($about the quantity $u)$ and also with respect to $x_3$ $($due to the random variable $v)$ .
On the other hand, the combination $\rho_{12} = \rho_{13} = \rho_{23} = 0$ is not possible with the structure given here. For this, one would need a third statistically independent random variable $w$ and, for example, $x_1 = k_1 \cdot u$ , $x_2 = k_2 \cdot v$ and $x_3 = k_3 \cdot w$ would have to hold.
The third statement is also not true: If $x_1$ and $x_2$ are uncorrelated and at the same time also $x_1$ and $x_3$, then no statistical bindings can exist between $x_2$ and $x_3$ either.
In general, however, both $\rho_{12}$ and $\rho_{13}$ will be different from zero.
A very simple example of this is considered in the subtask (2) .

(2) In this case, the quantities $x_1 = x_2$ are completely $($to $100\%)$ correlated.

With $A_2 = A_1$ and $B_2 = B_1$ we obtain for the joint correlation coefficient:

$$\rho_{12} = A_1 \cdot A_2 + B_1 \cdot B_2 = A_1^2 + B_1^2 \hspace{0.15cm}\underline{=1}.$$

In the same way, with $A_3 = -A_1$ and $B_3 = -B_1$:

$$\rho_{13} = A_1 \cdot A_3 + B_1 \cdot B_3 = -(A_1^2 + B_1^2) \hspace{0.15cm}\underline{=-1 \hspace{0.1cm}(= \rho_{23})}.$$

(3) With this set of parameters, $x_1$ is identical to the random variable $u$, while $x_2 = v$ holds.

Since $u$ and $v$ are statistically independent of each other, we get $\rho_{12} \hspace{0.15cm}\underline{ = 0}.$
In contrast, for the other two correlation coefficients:

$$\rho_{13} = A_1 \cdot A_3 + B_1 \cdot B_3 = 1 \cdot 0.8 + 0 \cdot 0.6 \hspace{0.15cm}\underline{ = 0.8},$$

$$\rho_{23} = A_2 \cdot A_3 + B_2 \cdot B_3 = 0 \cdot 0.8 + 1 \cdot 0.6 \hspace{0.15cm}\underline{ = 0.6}.$$

For a (very well) trained eye, it can be seen from the graph on the information page that the signal $x_3(t)$ has more similarities with $x_1(t)$ than with $x_2(t)$.
This fact is also expressed by the calculated correlation coefficients.
Don't be frustrated, however, if you don't recognize the different correlation in the signal courses.