Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2007

3. (25 Points)

Let $\mathbf{X}\left(t\right)$ be a real Gaussian random process with mean function $\mu\left(t\right)$ and autocovariance function $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

(a)

Write the expression for the $n$ -th order characteristic function of $\mathbf{X}\left(t\right)$ in terms of $\mu\left(t\right)$ and $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

ref.

There are the note about the n-th order characteristic function of Gaussians random process . The only difference between the note and this problem is that this problem use the $\mu\left(t\right)$ rather than $\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]$ .

Solution

$\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\mu_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\}$ .

(b)

Show that the probabilistic description of $\mathbf{X}\left(t\right)$ is completely characterized by $\mu\left(t\right)$ and autocovariance function $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

Solution

From (a), the characteristic function of $\mathbf{X}\left(t\right)$ is specified completely in terms of $\mu_{\mathbf{X}}\left(t\right)$ and $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ . Thus, probabilistic description of $\mathbf{X}\left(t\right)$ is completely characterized by the characteristic function.

Note

$f_{\mathbf{X}}\left(x\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\omega x}\Phi_{\mathbf{X}}\left(\omega\right)d\omega.$

(c)

Show that if $\mathbf{X}\left(t\right)$ is wide-sense stationary then it is also strict-sense stationary.

Note

You can use the theorem and its proof for solving this problem.

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