Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2002
3. (25 Points)
Let $ \mathbf{X}\left(t\right) $ be a wide-sense stationary Gaussian random process with mean $ \mu_{\mathbf{X}} $ and autocorrelation function $ R_{\mathbf{XX}}\left(\tau\right) $ . Let $ \mathbf{Y}\left(t\right)=c_{1}\mathbf{X}\left(t\right)-c_{2}\mathbf{X}\left(t-\tau\right), $ where $ c_{1} $ and $ c_{2} $ are real numbers. What is the probability that $ \mathbf{Y}\left(t\right) $ is less than or equal to a real number $ \gamma $ ? Express your answer in terms of “phi-function”$ \Phi\left(x\right)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-z^{2}/2}dz. $
Solution
Since $ \mathbf{X}\left(t\right) $ is a WSS Gaussian random process, $ \mathbf{Y}\left(t\right) $ is a Gaussian process.
$ E\left[\mathbf{Y}\left(t\right)\right]=c_{1}E\left[\mathbf{X}\left(t\right)\right]-c_{2}E\left[\mathbf{X}\left(t-\tau\right)\right]=\left(c_{1}-c_{2}\right)\mu_{\mathbf{X}}. $
$ E\left[\mathbf{Y}^{2}\left(t\right)\right]=E\left[\left(c_{1}\mathbf{X}\left(t\right)-c_{2}\mathbf{X}\left(t-\tau\right)\right)^{2}\right] $$ =c_{1}^{2}E\left[\mathbf{X}^{2}\left(t\right)\right]-2c_{1}c_{2}E\left[\mathbf{X}\left(t\right)\mathbf{X}\left(t-\tau\right)\right]+c_{2}^{2}E\left[\mathbf{X}^{2}\left(t-\tau\right)\right] $$ =\left(c_{1}^{2}+c_{2}^{2}\right)R_{\mathbf{X}}\left(0\right)-2c_{1}c_{2}R_{\mathbf{X}}\left(-\tau\right). $
$ Var\left[\mathbf{Y}\left(t\right)\right]=E\left[\mathbf{Y}^{2}\left(t\right)\right]-E\left[\mathbf{Y}\left(t\right)\right]^{2} $$ =\left(c_{1}^{2}+c_{2}^{2}\right)R_{\mathbf{X}}\left(0\right)-2c_{1}c_{2}R_{\mathbf{X}}\left(-\tau\right)-\left(\left(c_{1}-c_{2}\right)\mu_{\mathbf{X}}\right)^{2} $$ =\left(c_{1}^{2}+c_{2}^{2}\right)R_{\mathbf{X}}\left(0\right)-2c_{1}c_{2}R_{\mathbf{X}}\left(-\tau\right)-\left(c_{1}^{2}+c_{2}^{2}\right)\mu_{\mathbf{X}}^{2}+2c_{1}c_{2}\mu_{\mathbf{X}}^{2} $$ =\left(c_{1}^{2}+c_{2}^{2}\right)\left(R_{\mathbf{X}}\left(0\right)-\mu_{\mathbf{X}}^{2}\right)+2c_{1}c_{2}\left(\mu_{\mathbf{X}}^{2}-R_{\mathbf{X}}\left(-\tau\right)\right). $
$ P\left(\left\{ \mathbf{Y}\left(t\right)\leq r\right\} \right)=\Phi\left(\frac{r-\left(c_{1}-c_{2}\right)\mu_{\mathbf{X}}}{\sqrt{\left(c_{1}^{2}+c_{2}^{2}\right)\left(R_{\mathbf{X}}\left(0\right)-\mu_{\mathbf{X}}^{2}\right)+2c_{1}c_{2}\left(\mu_{\mathbf{X}}^{2}-R_{\mathbf{X}}\left(-\tau\right)\right)}}\right). $
