### Homework 9 (Due: April 13)

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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 9
Due: April 13, 2017

Problems
1. Hypothesis Testing with Laplacian Noise
A random variable N is said to be Laplacian distributed if its probability density function is given by

Consider the following decision problem involving the observed random variable Z:

where the two hypotheses are equally likely.

1. Provide expressions for the probability density function for Z for each of the two hypotheses.
2. Show that the maximum likelihood decision rule can be simplified to

Determine the value of the optimum threshold γ.

3. Compute the probability of error for this decision rule.

For the remainder of the problem, consider a two-dimensional random vector =

 N1 N2
with independent Laplacian distributed components, i.e.,

with =

 x1 x2
.

Consider the following decision problem involving the observed random vector =

 Z1 Z2
:

where again the two hypotheses are equally likely.

4. Provide expressions for the probability density function for for each of the two hypotheses.
5. Show that the maximum likelihood decision rule can be simplified to

6. The absolute values in the decision rule induce three distinct intervals for Z1 (Z1 < -2, -2 Z1 2, and Z1 > 2) and three intervals for Z2 (Z2 < -2, -2 Z2 1, and Z2 > 1).

Consider all nine regions formed by combinations of these intervals (e.g., the region with Z1 < -2 and Z2 < -2) and simplify the decision rule for each of these combinations.

7. Draw a two-dimensional signal-space diagram with axes Z1 and Z2. Mark the locations of E[|Hi] for the two hypotheses. Then, draw the decision boundary formed by the optimal decision rule using the results from part (f).
2. Binary Signal Sets
The following signal set is employed to transmit equally likely signals over an additive white Gaussian noise channel with spectral height .

1. Sketch and accurately label the block diagram of a receiver that minimizes the probability of error.
2. Compute the energy of each of the two signals.
3. Compute the probability of error for your receiver from part (a).
4. Consider now the following receiver:

r>](input)--(integrator);\pathr¿] (integrator) – (decision); r>](decision)--(output);\end{signalflow}\ end{center}Findtheconditionaldistributionoftherandomvariable$R$attheoutputoftheintegratorforeachofthetwosignals$s_ 0(t)$and$s_1(t)$.\itemComputetheprobabilityoferrorachievedbythesuboptimumreceiver.\itemComparetheprobabilityoferrorforthesuboptimumreceivertothatoftheoptimumreceiver. Expressyouranswerintheform:‘‘toachievethesameprobabilityoferrorastheoptimumreceiver, thesuboptimumsystemrequires$a$timesmoreenergy. ’’Determinethefactor$a$.\itemAssumenowthatthereceivedsignaliscorruptedbyaninterferingsignal$x(t) =\frac{A}{2}$,for$-\frac{T}{2}\leqt\leq\ frac{T}{2}$sothatthereceivedsignalunderthe$i$-thhypotesis($i=0,1$)isgivenby\ Hi: R˙t = s˙i(t) + +Nt for - t Compute the probability of error by the optimum receiver in the presence of the interfering signal. Explain your result in part (g).