Homework 11

ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Practice Problems

1. Amplitude Variations
   In realistic communication systems, the amplitude of signals in a signal set used for digital communication may be known by the transmitter but are usually scaled by an unknown amount prior to reception. Performance can become poor when the received signals are small and are contaminated by additive white Gaussian noise.

   Assume that a binary, orthogonal signal set having equally likely, equal energy components is chosen for this channel. The received signal is of the form
   

   $$R_t = A s_i(t) +N_t, \; 0 \leq t \leq T, \; i=1,2,$$
   
   
   where $\vert\vert s_i(t)\vert\vert^2 = E$.
   1. Find the minimum probability of error receiver for this signal set.
   2. A clever engineer wants to estimate the unknown amplitude $A$ by evaluating
      
      $$\hat{A} = \int_0^T r(t)g(t)\,dt$$
      
      
      when the signal $s_1(t)$ is present. What single choice for $g(t)$ both
      • minimizes the percentage error of the estimate as expressed by the coefficient of variation (defined as the ratio of the standard deviation of the estimate to its mean) and
      • results in an unbiased estimate, i.e., $E[\hat{A}] = A$.
   3. The engineer decides that the receiver should ``shut off'' when the estimate of $A$ suggests that errors are occurring too frequently. What threshold value for $\hat{A}$ should be used in an attempt to guarantee that the error rate is less than $10^{-2}$. Assume $E/N_0$ equals unity in this part.

2. Jamming and Football
   For those who remember the days of the bandwagon, the secret to the Washington Redskins success in the 1991/92 season was the installation of a new digital communication system for relaying messages from the press box to the field. A former ECE 630 student designed the following binary signal set:
   
   $$\begin{array}{cc} s_0(t) = \sqrt{2f_0} \cos(2 \pi f_0t) & s_1(t) = - \sqrt{2f_0} \cos(2 \pi f_0t). \end{array}$$
   
   
   The duration of each transmission interval is $1/f_0$ and the frequency is $f_0$ is 1 MHz. The communication channel is modeled as an additive, white Gaussian noise channel with spectral amplitude 1. Assume the signals are equally likely.
   1. What is the minimum probability of error that any receiver can achieve when this presumably well designed communication system is used.
   2. When visiting RFK stadium, the Houston Oilers decided to jam this system and nearly managed to beat the Redskins. (I know this was a long time ago, but you may recall that they missed the game-winning field goal with time running out.) What they did is to transmit a constant amplitude cosine wave of frequency $f_0$. In the presence of this jammer, the received signal can be modeled as
      
      $$R_t = s_i(t) + \sqrt{2f_0} \cos (2 \pi f_0 t) + N_t.$$
      
      
      The noise has the same characteristics as described above. What is the probability of error when the receiver from part (a) is used.
   3. In this game, the famous Redskins' half-time adjustments included a redesign of the receiver for the communication system. Find the optimum receiver for communication in the presence of the jamming signal and the corresponding probability of error.
   4. The (then lowly) Dallas Cowboys on their visit to RFK stadium used a smarter jammer. They transmitted a cosine wave whose amplitude alternates between $+\sqrt{2f_0}$ and $-\sqrt{2f_0}$. This jamming signal has a fixed amplitude over each signalling interval and the amplitude is equally likely to be positive or negative. Assuming the sign of the jamming signal during a particular transmission interval is statistically independent of the sign in any other interval and the noise characteristic is as before, explain why the Cowboys brought the Redskins winning streak to an end despite all efforts to adjust at halftime.

3. Diversity Channels
   Diversity signalling is the transmission of the same message over $N$ distinct channels simultaneously to a receiver. If the statistical characteristics of each channel are independent of each other, potentially an improvement in performance can be obtained.

   Assume that an on-off signalling scheme is used over each of N Rayleigh channels. The received signals are of the form:
   

   $$\begin{array}{cll} H_0: & R_t^{(i)} = N_t^{(i)} & 0 \leq t \... ... N_t^{(i)} & 0 \leq t \leq T,\\ & & i=1,\ldots,N. \end{array}$$
   
   
   The amplitude $A_i$ are statistically independent Rayleigh random variables having variance $\sigma^2$ and the phases $\theta_i$ are statistically independent uniform random variables, distributed over $[-\pi,\pi)$. The additive noise $N_t^{(i)}$ on channel $i$ is white Gaussian noise of spectral height $N_0/2$; the noise in one channel is independent of the noise in other channels. The energy of each of the signals $m_i(t)$ is $E/N$.
   1. Find the optimum receiver which uses the output of only one channel.
   2. Find the optimum receiver which uses the output of all channels.
   3. Find an expression for the probability of error when the hypotheses are equally likely for each receiver.
      Warning: This problem is very hard.
   4. Compare the results for the first receiver ($N=1$) and the second receiver ($N>1$). Does diversity signalling result in improved performance?

4. Neural Networks as Receivers
   Artificial Neural Networks are being explored in a variety of problem areas. In this problem we will demonstrate that a neural network can be used as a vector receiver in communication applications. (An important feature of neural networks is their ``learning'' ability, which we do not consider here.)

   In a vector communication problem, the vector receiver observes a vector $\underline{X}$ of $N$ random variables and tries to determine which of several hypotheses, $H_i: \underline{X} = \underline{S}_i + \underline{N}$, is most likely to have produced this observation.

   The fundamental element of artificial neural networks is the neuron, typically depicted as:

   
   
   

   A neuron receives $N$ inputs $X_1,\ldots,X_N$ and yields the single output $Y$ by computing
   

   $$Y = \mbox{sign} \left( \sum_{n=1}^N w_n X_n - \Theta \right),$$
   
   
   where $\{w_n\}$ is a set of weights and $\Theta$ is a threshold. The output of this neuron is bi-valued (+1 or -1) and can thus be used to indicate which of two signal vectors, $\underline{S}_0$ or $\underline{S}_1$, is present.
   1. Find the optimum set of weights and threshold used by a neuron to distinguish between equally likely signals, $\underline{S}_0$ and $\underline{S}_1$, when they are presented to the neuron in additive Gaussian noise with uncorrelated components.
   2. One problem with neural networks is unknown signal amplitudes: they can be sensitive to scaling. Under what conditions will the single neuron be insensitive to the size of the signal?
   3. How can an optimum receiver for ternary signal sets be constructed from neurons? Hint: You will have to interconnect neurons into layers. The first layer is responsible for distinguishing between each possible pair of hypotheses and the second layer combines the results from the first layer.
5. The following signal set is used to transmit three equally-likely symbols.
   
   
   

   The channel adds white Gaussian noise of spectral height $\frac{N_0}{2}$ to the transmitted signal.

   1. Draw and accurately label a block diagram of the optimum receiver, i.e., the receiver that minimizes the probability of a wrong decision.
   2. Find the appropriate signal space and indicate the decision regions of the optimum receiver.
   3. Compute the minimum probability of error attainable with this signal set.
   4. Repeat part (b) under the assumption that the amplitude of signal $s_1(t)$ is increased to $\sqrt{2}$.
   5. Find another signal set with $s_0(t) = A \cos(2\pi f t)$ that achieves the same performance as the signal set sketched above.
6. M-ary Signal Sets
   The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$,
   
   $$s_{i,j} = \sqrt{\frac{2E}{T}}(i \cdot \cos(2\pi f_c t) + j \... ...n(2\pi f_c t)) \;\;\mbox{for $0 \leq t \leq T$, $i,j=-1,0,1$.}$$
   
   
   Thus, this signal set consists of $M=9$ signals.
   1. Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.
   2. Compute the probability of error achieved by the optimum receiver.
   3. Assume that the energy of the transmitted signal can never exceed $2E$. Is it possible to modify the above signal set in such a way that the probability of error is reduced without exceeding the limit on the signal energy? Explain why or why not.
7. M-ary Signal Sets
   The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$,
   
   $$s_{i_1} = \sqrt{\frac{2E}{T}} \cdot i_1 \cdot \cos(\frac{20 \pi t}{T}) \;\;\mbox{for $0 \leq t \leq T$, $i_1=-1,0,1$.}$$
   
   
   Thus, this signal set consists of $M=3$ signals.
   1. Draw and accurately label a block diagram for the optimum receiver for this signal set.
   2. Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver. Then, compute the probability of error achieved by the optimum receiver.
   3. Repeat part (b) for the following signal set
      
      $$\begin{array}{ll} s_{i_1,i_2} = \sqrt{\frac{2E}{T}} \cdot (&... ...\;\;\mbox{for $0 \leq t \leq T$, $i_1,i_2=-1,0,1$.} \end{array}$$
      
      

   4. Repeat part (b) for the following signal set
      
      $$\begin{array}{ll} s_{i_1,i_2,i_3} = \sqrt{\frac{2E}{T}} \cdo... ...\mbox{for $0 \leq t \leq T$, $i_1,i_2,i_3=-1,0,1$.} \end{array}$$
      
      

   5. Derive a general expression for the probability of error of the N-dimensional signal set
      
      $$s_{i_1,\ldots,i_N} = \sum_{n=1}^{N} \sqrt{\frac{2E}{T}} \cdo... ...20n\pi t}{T}) \;\;\mbox{for $0 \leq t \leq T$, $i_n=-1,0,1$.}$$
      
      

8. M-ary Signal Sets
   The following signal set is used to transmit equally likely messages over an additive white Gaussian noise channel with spectral height $\frac{N_0}{2}$,
   
   $$s_{i,j}(t) = \sqrt{\frac{2E}{T}}(i \cdot \cos(\omega t) + j \cdot \sin(\omega t))$$
   
   
   for $0 \leq t \leq T$, $i=-1,0,1$ and $j=-2,-1,0,1,2$. Thus, this signal set consists of $M=15$ signals.
   1. Draw and accurately label the signal constellation in an appropriately chosen signal space and indicate the decision boundaries formed by the optimum receiver.
   2. Compute the probability of error achieved by the optimum receiver.
   3. Assume now that signal $s_{00}$ is removed from the above signal set. Draw and accurately label the new signal constellation in an appropriately chosen signal space and indicate the new optimum decision boundaries.
   4. Can you still express the resulting probability of error in terms of the Q-function? If your answer is yes, compute the probability of error; if it is no, explain why not and indicate whether the probability of error of the reduced signal set is larger or smaller than that of the original set.