1. Jul 21, 2008

crimolvic

Hello,

I have trying to understand this excercise 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.

crimolvic

2. Jul 21, 2008

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

3. Jul 21, 2008

HallsofIvy

Staff Emeritus
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.

4. Jul 21, 2008

crimolvic

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

5. Jul 21, 2008

HallsofIvy

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