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Problem

Let $\{X_n, n \in \mathbb{Z} \}$ be a discrete-time random process, defined as \begin{align*} X_n &=2 \cos \left(\frac{\pi n}{8}+ \Phi\right), \end{align*} where $\Phi \sim Uniform(0, 2 \pi)$.

1. Find the mean function, $\mu_X(n)$.
2. Find the correlation function $R_X(m,n)$.
3. Is $X_n$ a WSS process?

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Problem

Let $\{X(t), t \in \mathbb{R} \}$ be a continuous-time random process, defined as \begin{align*} X(t) =A \cos \left(2t+ \Phi\right), \end{align*} where $A \sim U(0,1)$ and $\Phi \sim U(0, 2\pi)$ are two independent random variables.

1. Find the mean function $\mu_X(t)$.
2. Find the correlation function $R_X(t_1,t_2)$.
3. Is $X(t)$ a WSS process?

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Problem

Let $\{X(n), n \in \mathbb{Z}\}$ be a WSS discrete-time random process with $\mu_X(n)=1$ and $R_X(m,n)=e^{-(m-n)^2}$. Define the random process $Z(n)$ as \begin{align*} Z(n)=X(n)+X(n-1), \quad \textrm{ for all }n \in \mathbb{Z}. \end{align*}

1. Find the mean function of $Z(n)$, $\mu_Z(n)$.
2. Find the autocorrelation function of $Z(n)$, $R_Z(m,n)$.
3. Is $Z(n)$ a WSS random process?

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Problem

Let $g:\mathbb{R}\mapsto \mathbb{R}$ be a periodic function with period $T$, i.e., \begin{equation*} g(t+T)=g(t), \quad \textrm{ for all }t \in \mathbb{R}. \end{equation*} Define the random process $\{X(t), t \in \mathbb{R} \}$ as \begin{equation*} X(t)=g(t+U), \quad \textrm{ for all }t \in \mathbb{R}, \end{equation*} where $U \sim Uniform(0,T)$. Show that $X(t)$ is a WSS random process.

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Problem

Let $\{X(t), t \in \mathbb{R} \}$ and $\{Y(t), t \in \mathbb{R}\}$ be two independent random processes. Let $Z(t)$ be defined as \begin{equation*} Z(t)=X(t)Y(t), \quad \textrm{ for all }t \in \mathbb{R}. \end{equation*} Prove the following statements:

1. $\mu_Z(t)=\mu_X(t) \mu_Y(t)$, for all $t \in \mathbb{R}$.
2. $R_Z(t_1,t_2)=R_X(t_1,t_2)R_Y(t_1,t_2)$, for all $t \in \mathbb{R}$.
3. If $X(t)$ and $Y(t)$ are WSS, then they are jointly WSS.
4. If $X(t)$ and $Y(t)$ are WSS, then $Z(t)$ is also WSS.
5. If $X(t)$ and $Y(t)$ are WSS, then $X(t)$ and $Z(t)$ are jointly WSS.

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Problem

Let $X(t)$ be a Gaussian process such that for all $t>s \geq 0$ we have \begin{align*} X(t)-X(s) \sim N\left(0, t-s\right). \end{align*} Show that $X(t)$ is mean-square continuous at any time $t \geq 0$.

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Problem

Let $X(t)$ be a WSS Gaussian random process with $\mu_X(t)=1$ and $R_X(\tau)=1+4 \textrm{sinc} (\tau)$.

1. Find $P(1 \lt X(1) \lt 2)$.
2. Find $P(1 \lt X(1) \lt 2, X(2) \lt 3)$.

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Problem

Let $X(t)$ be a Gaussian random process with $\mu_X(t)=0$ and $R_X(t_1,t_2)=\min(t_1,t_2)$. Find $P(X(4)<3 | X(1)=1)$.

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Problem

Let $\{X(t), t \in \mathbb{R} \}$ be a continuous-time random process, defined as \begin{align*} X(t) &=\sum_{k=0}^{n} A_k t^k, \end{align*} where $A_0$, $A_1$, $\cdots$, $A_n$ are i.i.d. $N(0,1)$ random variables and $n$ is a fixed positive integer.

1. Find the mean function $\mu_X(t)$.
2. Find the correlation function $R_X(t_1,t_2)$.
3. Is $X(t)$ a WSS process?
4. Find $P(X(1) \lt 1)$. Assume $n=10$.
5. Is $X(t)$ a Gaussian process?

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Problem

(Complex Random Processes) In some applications, we need to work with complex-valued random processes. More specifically, a complex random process $X(t)$ can be written as \begin{align*} X(t)=X_r(t)+jX_i(t), \end{align*} where $X_r(t)$ and $X_i(t)$ are two real-valued random processes and $j=\sqrt{-1}$. We define the mean function and the autocorrelation function as \begin{align*} \mu_X(t)&=E[X(t)]\\ &=E[X_r(t)]+jE[X_i(t)]\\ &=\mu_{X_r}(t)+j\mu_{X_i}(t); \end{align*} \begin{align*} R_X(t_1,t_2)&=E[X(t_1)X^{*}(t_2)]\\ &=E\left[\big(X_r(t_1)+jX_i(t_1)\big)\big(X_r(t_2)-jX_i(t_2)\big)\right]. \end{align*} Let $X(t)$ be a complex-valued random process defined as \begin{align*} X(t)=A e^{j(\omega t+\Phi)}, \end{align*} where $\Phi \sim Uniform(0, 2 \pi)$, and $A$ is a random variable independent of $\Phi$ with $EA=\mu$ and $\textrm{Var}(A)=\sigma^2$.

1. Find the mean function of $X(t)$, $\mu_X(t)$.
2. Find the autocorrelation function of $X(t)$, $R_X(t_1,t_2)$.

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Problem

(Time Averages) Let $\{X(t), t \in \mathbb{R} \}$ be a continuous-time random process. The time average mean of $X(t)$ is defined as (assuming that the limit exists in mean-square sense) \begin{align*} \left \lt X(t)\right> &=\lim_{T \rightarrow \infty} \left[\frac{1}{2T}\int_{-T}^{T} X(t) dt \right]. \end{align*} Consider the random process $\big\{X(t), t \in \mathbb{R}\big\}$ defined as \begin{align}%\label{} X(t)=\cos (t+U), \end{align} where $U \sim Uniform(0,2\pi)$. Find $\left \lt X(t)\right>$.

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Problem

(Ergodicity) Let $X(t)$ be a WSS process. We say that $X(t)$ is mean ergodic if $\left \lt X(t)\right>$ (defined above) is equal to $\mu_X$. Let $A_0$, $A_1$, $A_{-1}$, $A_2$, $A_{-2}$, $\cdots$ be a sequence of i.i.d. random variables with mean $EA_i=\mu \lt \infty$. Define the random process $\{X(t), t \in \mathbb{R}\}$ as \begin{equation*} X(t)=\sum_{k=-\infty}^{\infty} A_k g(t-k), \end{equation*} where, $g(t)$ is given by \begin{align} \nonumber g(t) = \left\{ \begin{array}{l l} 1& \quad 0 \leq t \lt 1 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{align} Show that $X(t)$ is mean ergodic.

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Problem

Let $\{X(t), t \in \mathbb{R}\}$ be a WSS random process. Show that for any $\alpha>0$, we have \begin{align*} P\big(|X(t+\tau)-X(t)|>\alpha \big) \leq \frac{2R_X(0)-2R_X(\tau)}{\alpha^2}. \end{align*}

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Problem

Let $\{X(t), t \in \mathbb{R}\}$ be a WSS random process. Suppose that $R_X(\tau)=R_X(0)$ for some $\tau>0$. Show that, for any $t$, we have \begin{align*} X(t+\tau)=X(t), \quad \textrm{with probability one.} \end{align*}

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Problem

Let $X(t)$ be a real-valued WSS random process with autocorrelation function $R_X(\tau)$. Show that the Power Spectral Density (PSD) of $X(t)$ is given by \begin{align*} S_X(f)= \int_{-\infty}^{\infty} R_X(\tau) \cos (2\pi f \tau) \; d\tau. \end{align*}

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Problem

Let $X(t)$ and $Y(t)$ be real-valued jointly WSS random processes. Show that \begin{align*} S_{YX}(f)=S^{*}_{XY}(f), \end{align*} where, $^{*}$ shows the complex conjugate.

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Problem

Let $X(t)$ be a WSS process with autocorrelation function \begin{align*} R_X(\tau)=\frac{1}{1+\pi^2\tau^2}. \end{align*} Assume that $X(t)$ is input to a low-pass filter with frequency response \begin{align*} H(f)=\left\{ \begin{array}{l l} 3 & \quad |f| \lt 2 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{align*} Let $Y(t)$ be the output.

1. Find $S_X(f)$.
2. Find $S_{XY}(f)$.
3. Find $S_Y(f)$.
4. Find $E[Y(t)^2]$.

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Problem

Let $X(t)$ be a WSS process with autocorrelation function \begin{align*} R_X(\tau)=1+\delta(\tau). \end{align*} Assume that $X(t)$ is input to an LTI system with impulse response \begin{align*} h(t)=e^{-t}u(t). \end{align*} Let $Y(t)$ be the output.

1. Find $S_X(f)$.
2. Find $S_{XY}(f)$.
3. Find $R_{XY}(\tau)$.
4. Find $S_Y(f)$.
5. Find $R_Y(\tau)$.
6. Find $E[Y(t)^2]$.

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Problem

Let $X(t)$ be a zero-mean WSS Gaussian random process with $R_X(\tau)=e^{-\pi \tau^2}$. Suppose that $X(t)$ is input to an LTI system with transfer function \begin{align*} |H(f)|=e^{-\frac{3}{2}\pi f^2}. \end{align*} Let $Y(t)$ be the output.

1. Find $\mu_Y$.
2. Find $R_Y(\tau)$ and $\textrm{Var}(Y(t))$.
3. Find $E[Y(3)|Y(1)=-1]$.
4. Find $\textrm{Var}(Y(3)|Y(1)=-1)$.
5. Find $P(Y(3) \lt 0|Y(1)=-1)$.

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Problem

Let $X(t)$ be a white Gaussian noise with $S_X(f)=\frac{N_0}{2}$. Assume that $X(t)$ is input to a bandpass filter with frequency response \begin{align*} H(f)=\left\{ \begin{array}{l l} 2 & \quad 1 \lt |f| \lt 3 \\ & \quad \\ 0 & \quad \text{otherwise} \end{array} \right. \end{align*} Let $Y(t)$ be the output.

1. Find $S_Y(f)$.
2. Find $R_Y(\tau)$.
3. Find $E[Y(t)^2]$.