Aufgaben:Exercise 4.15Z: Statements of the Covariance Matrix

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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

1

Which of the following statements are true? Give reasons for your findings.

	$\mathbf{K}$  can be with a suitable choice of   $A_1$, ... , $B_3$   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 can be  $\rho_{ij} = 0$.
	With appropriate choice of parameters   $A_1$, ... , $B_3$   exactly two of the correlation coefficients can be  $\rho_{ij} = 0$.
	With appropriate choice of parameters   $A_1$, ... , $B_3$  all three correlation coefficients  $\rho_{ij} \ne 0$.

2

What are the matrix elements of  $\mathbf{K}$  with   $A_1 = A_2 = - A_3$   and   $B_1 = B_2 = - B_3$ ?

$\rho_{12} \ = \ $

$\rho_{13} \ = \ $

$\rho_{23} \ = \ $

3

Calculate the coefficients  $\rho_{ij}$  for the case shown in the graph:   $A_1 = 1$,  $B_1 = 0$,  $A_2 = 0$,  $B_2 = 1$,  $A_3 = 0.8$,  $B_3 = 0.6$.

$\rho_{12} \ = \ $

$\rho_{13} \ = \ $

$\rho_{23} \ = \ $

Solution

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.