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A homework assignment for the statistical signal processing course (ece 567) offered in spring 2008. The assignment includes five problems related to maximum likelihood testing, neyman-pearson testing, and minimizing energy consumption for signal transmission in the presence of gaussian noise. Students are expected to find solutions for problems such as finding the ml test to distinguish between two hypotheses, determining the neyman-pearson test for a given pfa, and calculating the optimal transmission amplitude and number of pulses.
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Homework Assignment # Due: 6 February 2008
p(x|H 0 ) =
exp(−|x|)
p(x|H 1 ) =
exp(−|x − A|)
when A > 0. What is PD and PF A?
H 0 : p(x) =
exp(−|x|)
H 1 : p(x) =
2 π
exp(−
x^2 ).
(a) Plot ln(Λ(x)) versus x. (b) Determine the form of the NP test.
H 0 : x(n) = w(n), n = 0,... , N − 1 H 1 : x(n) = A + w(n), n = 0,... , N − 1
where H 0 represents the situation with no signal and H 1 the situation with a trans- mission. Assume that each w(n) ∼ N (0, σ^2 ), and that the {w(n)}N n=0^ −^1 are i.i.d. with σ^2 = 1. We want to minimize N A^2 (the energy to transmit A over N samples) while achieving PD of at least 0.9 at the receiver while PF A is no greater than 0.01. What N and A do you choose?