I am reading a thesis and I am trying to show an autocorrelation formula is true.
We have the process: $$ P(t) = |V|^2 + |v(t)|^2 + Vv^*(t) + V^*v(t) $$
, where $V$ is a constant complex number and $v(t)$ is a zero-mean complex Gaussian process with autocorrelation function defined as: $$ r(\tau) = E[v^*(t)v(t+\tau)]. $$
Also, we know that $E[|v(t)|^2] = \sigma^2$.
How do I show that the autocorrelation of $P(t)$ is: $$ E[P(t)P(t+\tau)] = |V|^4 + 2|V|^2\sigma^2 + E[|v(t)|^2|v(t+\tau)|^2] + |V|^2 r(\tau) + |V|^2 r^*(\tau) $$
I have reached an expression which is close to the given expression, and I am trying to see how to progress. My work:
The expectation is a product of two 4-terms, therefore after expansion, there must be 16 terms, and some of those terms could be zero. $$ \begin{align*} E[P(t)P(t+\tau)] & = E[[|V|^2 + |v(t)|^2 + Vv^*(t) + V^*v(t)][|V|^2 + |v(t+\tau)|^2 + Vv^*(t+\tau) + V^*v(t+\tau)]] \\ & = E[|V|^2|V|^2 + |V|^2|v(t+\tau)|^2 + |V|^2Vv^*(t+\tau)+|V|^2V^*v(t+\tau) \\ & +\ |v(t)|^2|V|^2 + |v(t)|^2|v(t+\tau)|^2 + |v(t)|^2Vv^*(t+\tau)+|v(t)|^2V^*v(t+\tau) \\ & +\ Vv^*(t)[|V|^2 + |v(t+\tau)|^2 + Vv^*(t+\tau) + V^*v(t+\tau)] \\ & +\ V^*v(t)[|V|^2 + |v(t+\tau)|^2 + Vv^*(t+\tau) + V^*v(t+\tau)]]. \\ & = |V|^4 + |V|^2\sigma^2 + 0 + 0 \\ & +\ \sigma^2|V|^2 + E[|v(t)|^2|v(t+\tau)|^2] + E[|v(t)|^2Vv^*(t+\tau)]+E[|v(t)|^2V^*v(t+\tau)] \\ & +\ 0 + E[Vv^*(t)|v(t+\tau)|^2] + V^2E[v^*(t)v^*(t+\tau)] + |V|^2r(\tau) \\ & +\ 0 + E[V^*v(t)|v(t+\tau)|^2] + |V|^2r^*(\tau) + (V^*)^2E[v(t)v(t+\tau)] \end{align*} $$
Then, comparing the given expression and the expression I have, the following must true:
$$ \begin{align*} & E[|v(t)|^2Vv^*(t+\tau)]+E[|v(t)|^2V^*v(t+\tau)] \\ & +\ E[Vv^*(t)|v(t+\tau)|^2] + V^2E[v^*(t)v^*(t+\tau)] \\ & +\ E[V^*v(t)|v(t+\tau)|^2] + (V^*)^2E[v(t)v(t+\tau)] = 0. \end{align*} $$
How do I show that it is true, given that $v(t)$ is a zero-mean Complex Gaussian random variable, with the given autocorrelation function? It may have to do with the third moment of a Gaussian random variable, with zero-mean, being zero, but I wanted to be completely sure of the result.
Thanks.
