# Showing an Autocorrelation Formula is True

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.

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To add a detail, the thesis topic was channel propagation model in wireless communications. – jrand Jan 31 '13 at 5:43