Homework 10

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 10
Due April 30, 2003

Reading

Wozencraft & Jacobs: Chapter 7, pp. 509-550.

Problems

1. Random Amplitude
   Let $\{s_i(t)\}$, $i=1,2,\ldots,M$ denote the signals comprising a transmitter's signal set. This set is said to be an orthogonal signal set if the signals are pairwise orthogonal,
   
   $$(s_i(t),s_j(t)) = 0,\;i \neq j.$$
   
   
   Usually, the signals in an orthogonal signal set have equal energies,
   
   $$\vert\vert s_i(t)\vert\vert^2 = E,\; i=1,2,\ldots ,M.$$
   
   
   Under these conditions and assuming the members of the set are equally likely, solve the following problems.
   1. Sketch a block diagram of the minimum probability of error receiver when the channel adds white Gaussian noise to the transmitted signal.
   2. Now assume the channel introduces a random amplitude $A$ during transmission. How does the receiver from part (a) have to be modified to accommodate this channel if the density of $A$ is one sided exponential,
      
      $$p_A(a) = \left \{ \begin{array}{cl} \mbox{e}^{-a} & \mbox{for $a \geq 0$} \\ 0 & \mbox{for $a < 0$?} \end{array}\right.$$
      
      

   3. Find an expression for the probability of error of your receiver.
   4. Repeat parts (b) and (c) for the case when the distribution of the amplitude $A$ is Gaussian with zero mean and unit variance.
2. Digital Interference
   One potential problem in digital communication is interference from other digital transmitters as well as from the channel noise. Assume that transmitter A is using signal set A,
   
   $$\begin{array}{ll} s_0^A(t) = \left\{ \begin{array}{cl} \sqrt... ...q t \leq 1$}\\ 0 & \mbox{else} \end{array} \right. \end{array}$$
   
   
   and transmitter B uses signal set B,
   
   $$\begin{array}{ll} s_0^B(t) = \left\{ \begin{array}{cl} \sqrt... ...q t \leq 1$}\\ 0 & \mbox{else} \end{array} \right. \end{array}$$
   
   
   Assume the signals in each set are equally likely. The receiver trying to pay attention to transmitter A receives the signal
   
   $$R_t = s^A(t) + s^B(t) + N_t,$$
   
   
   where $N_t$ is white Gaussian noise. Assume that the transmitters A and B are synchronized so that the bit intervals coincide. The signals sent by each transmitter are statistically independent.
   1. Determine the minimum probability of error receiver for the reception of transmitter A's signals in this situation.
   2. What is the resulting probability of error in this situation?
   3. How does your answer for part (a) change if if transmitter B uses the signal set
      
      $$\begin{array}{ll} s_0^B(t) = \left\{ \begin{array}{cl} \sqrt... ...q t \leq 1$}\\ 0 & \mbox{else} \end{array} \right. \end{array}$$
      
      
      Sketch decision regions in the signal space spanned by $s^A(t)$ and $s^B(t)$.
3. Binary Phase Channel
   A modulated antipodal signal set is used over a channel which changes the phase of the transmitted signal by $\frac{\pi}{2}$ or leaves the phase unchanged. This phase shift changes randomly from bit-to-bit and is equally likely to occur or not. The transmitted signals are equally likely to occur.
   1. Find the optimum receiver for this channel.
   2. Calculate the resulting probability of error for your receiver.
4. Channel Measurement Signal Sets
   One method of communicating over a channel in which parameters vary slowly compared with a bit interval is to precede the information-bearing portion of the bit interval with a known probe signal. This signal can then be used to provide some information about the channel which can be used to aid in the detection problem.

   Assume a modulated signal set is used over a random phase channel. The baseband probe signal $m_0(t)$ is always transmitted over the first half of the bit interval. The baseband message signal $m_1(t)$ is used to transmit equally-likely information in the second half. The received signal is of the form.
   

   $$\begin{array}{ll} H_0: & R_t = \sqrt{2}(m_0(t)+m_1(t)) \cos(... ...(m_0(t)-m_1(t)) \cos(2 \pi f_c t + \Theta) + N_t\\ \end{array}$$
   
   
   where $N_t$ is white Gaussian noise and $E_0$ and $E_1$, the energies of probe and message signals, are equal.
   1. Assume the phase $\Theta$ is a known constant. Show that the optimum receiver ignores the probe portion of the received signal.
   2. Now assume that $\Theta$ is a random variable uniformly distributed over the interval $[-\pi,\pi)$. Find the minimum probability of error receiver.
   3. Show that this receiver can be put in the form of a phase discriminator where the phases of the probe and message portions of the received signal are compared. The discriminator announces $H_1$ if the phase difference is greater than $\frac{\pi}{2}$ in magnitude and $H_0$ otherwise.
   4. Find the probability of error of the optimum receiver.
   5. How does the performance of this signal set compare with that when no probe signal is used?