Exam2Solutions

Exam2Solutions - ECEN 661 -Modulation Theory Spring 2008...

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ECEN 661 -Modulation Theory Spring 2008 Midterm Exam #2 Problem 1. (40 points) A certain 8-ary quadrature amplitude modulation scheme sends one of the following signals: , , , where , and is a constant. The signal is received over an AWGN channel so the the received signal is , where is zero-mean Gaussian noise with a (two-sided) PSD of . (a) Describe how the ML receiver works. That is, what processing does it do on the received sig- nal and how does it make a decision. (b) Evaluate the exact probability of error of the ML receiver. Express your answer in terms of . (c) Derive a union bound on the probability of error of the ML receiver. Express your answer in terms of . (d) Find the asymptotic efficiency of this signalling format. Solution: Choose as basis functions: and . Based on these basis functions the vector representation of the signals is . The signal space diagram is shown below. s i t () A i ω c t cos B i ω c t sin + = 0 tT s < i 12 8 ,, , = A i a ± 3 a ± , {} B i a ± a rt s i t nt + = N o 2 E b N o E b N o f 1 t 2 T s ---- ω c t cos = f 2 t 2 T s ω c t sin = s i A i 2 T s 2 ----------- B i 2 T s 2 ,    = * * * * * * * * r 1 r 2 b 3 b b 3 b b b b a 2 T s 2 ---------- =
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(a) Project the received signal onto the basis functions to form the received vector: , where , . Then choose which signal was sent based on the decision regions as shown in the above figure.
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This note was uploaded on 02/21/2012 for the course ECEN 611 taught by Professor Miller during the Spring '08 term at Texas A&M.

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Exam2Solutions - ECEN 661 -Modulation Theory Spring 2008...

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