- #1
crimolvic
- 3
- 0
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 functionx(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
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 functionx(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
