Are two jointly stationary white noise processes independent?

I am currently dealing with a problem concerning beamforming, where two "jointly stationary zero-mean white noise processes" form the input of an adaptive system. One of those processes resembles the actual signal $s[n]$, the other is an interference signal $v[n]$. Is it clear from the stated definition above that they are independent from each other, i.e. $E(XY)=E(X)E(Y)$? Why?

• No, it is not at all clear from the definition above that the processes are independent, and in any case, independence is not defined as $E[XY]=E[X]E[Y]$ holds; not even for random variables, let alone random processes. – Dilip Sarwate Sep 18 '18 at 2:08

Stationarity (WSS in particular) does not imply independence, even if they are white noises.

That's to say, there exist (at least one) jointly stationary zero mean white random processes which are also dependent. A simple example is this.

Consider a zero mean, wide sense stationary (WSS), white, real random process $x[n]$ with auto-correlation sequence (ACS), $r_{xx}[k] = \sigma_x^2 \delta[k]$, which is passed through a linear time invariant (LTI) filter with real impulse response $h[n] = M \delta[n]$, to produce the output $y[n] = h[n] \star x[n]$ :

$$y[n] = \sum_k h[k] x[n-k] = M x[n]$$

Now it can be shown that $y[n]$ will also be zero-mean, WSS white noise with its ACS given by $$r_{yy}[k] = h[k] \star h[-k] \star r_{xx}[k] = M^2 \sigma_x^2 \delta[k]$$,

and further, they are jointly WSS and white, as their cross-correlation depends on the lag $k=m-n$; $$\mathcal{E}\{x[n] y[m] \} = r_{yx}[k] = h[k] \star r_{xx}[k] = M \sigma_x^2 \delta[k]$$

However it can be shown that they are not independent, since the necessary condition for their independence does not hold: $$\mathcal{E}\{x[n] y[m] \} \neq \mathcal{E}\{x[n] \} \mathcal{E}\{y[m] \}$$

Where the left hand side is the cross-correlation sequence $r_{yx}[k] = M \sigma_x^2 \delta[k]$, as defined above, which is not zero for all $k$, whereas the right hand side is identical to zero for all $n,m$ as both processes are zero-mean WSS.

Hence these two jointly WSS and white random processes $x[n]$ and $y[n]$ cannot be independent and are in fact dependent, in line with intuition as the filtering creates a dependency between them.