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ECE 630: Statistical Communication Theory
Prof. B.-P. Paris
Homework 9
Due: April 13, 2017
Consider the following decision problem involving the observed random variable Z:
where the two hypotheses are equally likely.
Determine the value of the optimum threshold γ.
For the remainder of the problem, consider a two-dimensional random vector =
N_{1} |
N_{2} |
with =
x_{1} |
x_{2} |
Consider the following decision problem involving the observed random vector =
Z_{1} |
Z_{2} |
where again the two hypotheses are equally likely.
Consider all nine regions formed by combinations of these intervals (e.g., the region with Z_{1} < -2 and Z_{2} < -2) and simplify the decision rule for each of these combinations.
Suboptimum Receiver
[input] (input) /Users/pparis/Home/Courses/ece630/R_{t}; [filter] (integrator) [right from=input] /Users/pparis/Home/Courses/ece630/∫ _{-T∕2}^{T∕2}dt; [label=above:R] (int˙out) [right from=integrator] ; [filter] (decision) [right from=int˙out] 0; [output] (output) [right from=decision] ;
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\ H_{i}: R˙t = s˙i(t) + +N_{t} for - ≤ t ≤ Compute the probability of error by the optimum receiver in the presence of the interfering signal. Explain your result in part (g).