Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004

4. (30 pts.)

Assume that $\mathbf{X}\left(t\right)$ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\delta\left(t_{1}-t_{2}\right)$. Let $\mathbf{Y}\left(t\right)$ be a new random process defined by $\mathbf{Y}\left(t\right)=\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds$, where $T>0$ .

(a)

What is the mean of $\mathbf{Y}\left(t\right)$ ?

$E\left[\mathbf{Y}\left(t\right)\right]=E\left[\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds\right]=\frac{1}{T}\int_{t-T}^{t}E\left[\mathbf{X}\left(s\right)\right]ds=\frac{1}{T}\int_{t-T}^{t}0ds=0.$

(b)

What is the autocorrelation function of $\mathbf{Y}\left(t\right)$ ?

$R_{\mathbf{YY}}\left(t_{1},t_{2}\right)=E\left[\mathbf{Y}\left(t_{1}\right)\mathbf{Y}^{*}\left(t_{2}\right)\right]=E\left[\right]$

(c)

Write an expression for the second-order pdf $f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right)$ of $\mathbf{Y}\left(t\right)$ .

(d)

Under what conditions on $t_{1}$ and $t_{2}$ will $\mathbf{Y}\left(t_{1}\right)$ and $\mathbf{Y}\left(t_{2}\right)$ be statistically independent?

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Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010