Homework 4

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 4
Due Feb. 18, 2003

Reading

Wozencraft & Jacobs: Chapter 4 pages 210-273.

Problems

1. The stationary random process $X_t$ is passed through a linear filter with transfer function $H(f)$,
   
   $$H(f) = \frac{j2\pi f + a}{j2\pi f +2a}.$$
   
   
   The output process is labeled $Y_t$. The mean of $Y_t$ is measured to be $\frac{1}{2}$ and the covariance function of $Y_t$ is found to be
   
   $$K_Y(\tau) = a^2 e^{-2a \vert\tau \vert}.$$
   
   
   1. Compute the power spectral density of $Y_t$.
   2. Find the second order description of $X_t$.

2. In practice one often wants to measure the power spectral density of a stochastic process. For the purposes of this problem, assume the process $X_t$ is wide-sense stationary, zero mean, and Gaussian. The following measurement system is proposed.
   
   
   

   Here $H_1(f)$ is the transfer function of an ideal bandpass filter and $H_2(f)$ is an ideal lowpass,
   

   $$H_1(f) = \left\{ \begin{array}{cl} 1 & \mbox{for $f_0 - \... ... \frac{\Delta f}{2}$} 0 & \mbox{else} \end{array} \right.$$
   
   
   
   
   $$H_2(f) = \left\{ \begin{array}{cl} \frac{1}{2 \Delta f} &... ...vert \leq \Delta f$} 0 & \mbox{else.} \end{array} \right.$$
   
   
   Assume that $\Delta f$ is small compared to the range of frequencies over which $S_X(f)$ varies, i.e., you may assume that $S_X(f)$ is constant over intervals of width $\Delta f$.
   1. Find the mean and correlation function of $Y_t^2$ in terms of the second order description of $X_t$.
   2. Compute the the power spectral density of the process $Z_t$.
   3. Compute the expected value of $Z_t$.
   4. By considering the variance of $Z_t$, comment on the accuracy of this measurement of the power density of the process $X_t$.

3. The price of a certain stock can fluctuate during the day while the ``true'' value is rising or falling. To facilitate financial decisions, a Wall street broker decides to use stochastic process theory. The price $P_t$ of a stock is described by
   
   $$P_t = Kt +N_t\;\;\;0 \leq t \leq 1,$$
   
   
   where $K$ is the constant our knowledgeable broker is seeking and $N_t$ is a stochastic process describing the random fluctuations. $N_t$ is a white, Gaussian process having spectral height $\frac{N_0}{2}$. The broker decides to estimate $K$ according to:
   
   $$\hat{K} = \int_0^1 P_t g(t) dt,$$
   
   
   where the ``best'' function $g(t)$ is to be found.
   1. Find the probability density function of the estimate $\hat{K}$ for any $g(t)$ the broker might choose.
   2. A simple-minded estimate of $\hat{K}$ is to use simple averaging (i.e., set $g(t) =$ constant). Find the value of the constant which results in $E[\hat{K}] = K$. What is the resulting percentage error as expressed by $\sqrt{var[\hat{K}]}/\vert E[\hat{K}]\vert$?
   3. Use $g(t)=at$ and choose $a$ to yield $E[\hat{K}] = K$. How much better is this choice than simple averaging?
   4. DO you think $g(t)=at$ is the best possible choice? Why or why not?