Help with covariance calculation please

[crimolvic]

Hello,

I have trying to understand this exercise for quite a time, but still with no results. So I thought maybe you can help me ;P

Here is the problem. I have a function given by:
$$x(t)=\sum_i A_i s(t-iT-\tau_i)+n(t)$$
where:
- $$\left\{A_i\right\}_{i\in Z}$$ is a random point process, which is assumed to be periodically correlated with a period $$Q$$ longer than $$T$$
- $$\left\{\tau_i\right\}_{i\in Z}$$ is a zero-mean delta-correlated point process with probability density function $$\phi_{\tau}(\tau_i)$$,
- $$\left\{n(t)\right\}_{t\in R}$$ is a zero-mean stationary process

The covariance function os a function$$x(t)$$ defined as follows:
$$C_{xx}(t,\tau)=E\left\{[x(t+\tau/2)-m_x(t+\tau/2)][x(t-\tau/2)-m_x(t-\tau/2)]\right\}$$
where $$m_x(t)$$ is the mathematical expectation (ensemble average) of $$x(t)$$. For our function one obtains $$m_x(t)=\sum_i\bar{A}_i s(t-iT)\ast\phi_{\tau}(t-iT)$$. So after substitution of $$m_x(t)$$ on the expression defining $$C_{xx}(t,\tau)$$ one could obtain the covariance function. By assuming that the processes $$\left\{A_i\right\}_{i\in Z}$$, $$\left\{\tau_i\right\}_{i\in Z}$$ and $$\left\{n(t)\right\}_{t\in R}$$ are mutually uncorrelated, one obtains somehow (and here is where I need your help, please) the result:

$$C_{xx}(t,\tau)=\sum_i \bar{A_i^2}[s(t+\tau/2-iT)s^{\ast}(t-\tau/2-iT)]\ast \phi_{\tau}(t)-\sum_i \bar{A}_i^2 \widetilde{s}(t+\tau/2-iT)\widetilde{s}^{\ast}(t-\tau/2-iT) + C_{nn}(\tau)$$

where $$\widetilde{s}(t)=s(t)\ast \phi_{\tau}(t)$$ (convolution) and the upperscript $$\ast$$ on for example $$s^{\ast}$$ stands for the complex conjugate.

I took the problem from the article "the relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals" from Randall, et. al published on Mechanical Systems and Signal Processing (2001) 15(5), 945-962, under the point 2.

thanking in advance,
crimolvic

 

[crimolvic]

Ups, it seems I did something wrong on using the Latex cappability of the forum. Does someone know how can I fix this?

crimolvic

 

[HallsofIvy]

Don't put spaces inside the brackets: $$e^{x^2}$$
not [ tex ]e^{x^2}[ /tex ]. Click on the LaTex above to see the difference.

 

[crimolvic]

thanks HallsofIvy, now is looking much better.
I hope now someone can help understand this problem

 

[HallsofIvy]

It's going to have to be someone better at probability than I am!