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).$

$P\left(\left\{ \mathbf{Y}\left(t\right)\leq r\right\} \right)=\Phi\left(\frac{r-\left(c_{1}-c_{2}\right)\mu_{\mathbf{X}}}{\sqrt{\left(c_{1}^{2}+c_{2}^{2}\right)\left(R_{\mathbf{X}}\left(0\right)-\mu_{\mathbf{X}}^{2}\right)+2c_{1}c_{2}\left(\mu_{\mathbf{X}}^{2}-R_{\mathbf{X}}\left(-\tau\right)\right)}}\right).$

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva