Homework 5

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 5
Due March 4, 2003

Reading

Wozencraft & Jacobs: Chapter 4 and Chapter 8 pages 598-603.

Problems

1. Let $x$ and $y$ be elements of a normed linear vector space.
   1. Determine whether the following are valid inner products for the indicated space.
      1. $(x,y) = \underline{x}^T {\boldmath A} \underline{y}$, where $\boldmath A$ is a nonsingular,$NxN$ matrix and $\underline{x}$, $\underline{y}$ are elements of the space of $N$-dimensional vectors.
      2. $(x,y) = \underline{x} \underline{y}^T$, where $\underline{x}$ and $\underline{y}$ are elements of the space of $N$-dimensional (column!) vectors.
      3. $(x,y) = \int_0^T x(t)y(T-t) dt$, where $x$ and $y$ are finite energy signals defined over $[0,T]$.
      4. $(x,y) = \int_0^T w(t)x(t)y(t) dt$, where $x$ and $y$ are finite energy signals defined over $[0,T]$ and $w(t)$ is a non-negative function.
      5. $E[XY]$, where $X$ and $Y$ are real-valued random variables having finite mean-square values.
      6. $\mbox{Cov}(X,Y)$, the covariance of the real-valued random variables $X$ and $Y$. Assume that $X$ and $Y$ have finite mean-square values.
   2. Under what conditions is
      
      $$\int_0^T \! \int_0^T Q(t,u)x(t)y(u) dt du$$
      
      
      a valid inner product for the space of finite-energy functions defined over $[0,T]$?

2. Consider the vectors
   
   $$\begin{array}{ccc} s_0 = \left[ \begin{array}{c} 1 1 1... ... \begin{array}{c} 1 4 9 \end{array} \right]. \end{array}$$
   
   
   1. Use the Gram-Schmidt procedure to find orthonormal basis vectors which span the space of these vectors.
   2. What is the dimension of the space spanned by these three vectors?
   3. Compute the representation of the $s_i$ in terms of the orthonormal basis vectors determined in part (a).
   4. Repeat parts (a)-(c) for the signals
      
      $$\begin{array}{cc} s_0(t) = \left\{ \begin{array}{cl} 2 & \mb... ... t < 3$}\\ 0 & \mbox{else.} \end{array} \right. \end{array}$$
      
      

3. The following signals are used to communicate one of two equally likely messages over a channel perturbed by a zero mean, white Gaussian random process, $n(t)$, with spectral height $\frac{N_0}{2}$,
   
   $$\begin{array}{cc} s_0(t) = \left\{ \begin{array}{cl} 1 & 0\l... ...ac{T}{2} \\ 0 & \mbox{else} \end{array} \right. \end{array}$$
   
   
   1. Use the Gram-Schmidt procedure to find orthonormal functions $\Psi_0(t)$ and $\Psi_1(t)$ to represent $s_0(t)$ and $s_1(t)$.
   2. Sketch the signals $s_i(t)$ and the basis functions $\Psi_i(t)$.
   3. Express $s_0(t)$ and $s_1(t)$ in terms of the basis functions $\Psi_i(t)$.
   4. Define the random variables
      
      $$X_{ij} = \int_0^T s_j(t) (s_i(t)+n(t)) dt,\; i,j=0,1.$$
      
      
      Find the joint density function of $X_{i0}$ and $X_{i1}$.
   5. Define the random variables
      
      $$Y_{ij} = \int_0^T \Psi_j(t) (s_i(t)+n(t)) dt,\; i,j=0,1.$$
      
      
      Find the joint density function of $Y_{i0}$ and $Y_{i1}$.