University of Illinois at Urbana-ChampaignECE 461: Communications IISpring 2005Exam IMonday, February 28, 7:00-8:15 p.m., 165 Everitt LaboratoryName:• You have 75 minutes for this exam. The exam is closed book and closed note, except you mayconsult both sides of one 8.500× 1100sheet of notes in ten point font size or larger, or equivalenthandwriting size.• Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.• Write your answers in the spaces provided.• Please show all of your work. Answers without appropriate justificat ion will receivevery little credit. If you need extra space, use the back of the previous page.Score:1. (10 pts.)2. (10 pts.)3. (10 pts.)Total: (30 pts.)11. Consider a baseband binary communication system with possible transmitted signals s1(t) =AI[0,T ](t) and s0(t) =AtTI[0,T ](t). Here I[0,T ](t) is the function that is one if 0 ≤ t ≤ T and iszero otherwise. Suppose s1is sent with probability π1=13and s0is sent with probability π0=23. Thereceived signal is r(t) = sm(t) + n(t), where m ∈ {0, 1} and n is white Gaussian noise with two-sidedpower spectral densityNo2.(a) Sketch the Bayes optimal receiver in correlator form. Be sure to indicate the value of the thresholdin the decision device.(b) Sketch the Bayes optimal receiver using a matched filter and sampler. Be sure to indicate thefilter’s impulse response function.22. Consider a binary communication system with possible transmitted signals s1and s0, wheresm(t) = A (amcos(2πfct + φ) + β sin(2πfct + φ)) 0 ≤ t ≤ T, m ∈ {0, 1}where a1= 1 and a0= −1 and β is a positive constant. As usual, assume that A, T, fcare posi-tive constants with fcT >> 1, and assume the signal is corrupted by additive white Gaussian noisewith 2-sided power spectral densityNo2. The purpose of the sine carrier term is to help the receiverrecover the phase φ. Suppose that the receiver is able to recover the phase φ perfectly, and can there-fore correlate the received signal over [0, T ] with ψ1(t) =q2Tcos(2πfct + φ) to produce Z1and withψ2(t) =q2Tsin(2πfct + φ) to produce Z2.(a) Sketch the signal coordinates relative to the basis {ψ1, ψ2}, and sketch the maximimum likelihooddecision regions.(b) Express the probability of error in terms of A, T , No, and the Q function.(c) What is the cost in dB of using the sine carrier term? In other words, if a perfect phase trackingloop could be implemented without the sine carrier term, what would be the effective difference inrequired energy per transmitted bit, in dB? (Hint: the answer is not zero.)33. Consider an 8 − ary QAM system with the following signal coordinates relative to an orthonormalbasis {ψ1, ψ2}: (±A, 0), (±3A, 0), (±A, ±2A). The signals are sent with equal probability, and arecorrupted by AWGN with two-sided power spectral densityNo2.(a) Sketch the maximum likelihood decision regions on the following signal coordinate diagram:2AA3AA(b) Express the average symbol energy and the average energy per bit in terms of A2.Es=Eb=(c) Find a union bound on the probability of symbol error given that the signal with coordinates (A, 2A)is transmitted. Use the minimum number of terms required.(d) Express the maximum symbol error probability exactly in terms of A, No, and the Q function, forthe maximum likelihood receiver. (The maximum symbol error probability is the maximum, over alleight possible transmitted symbols, of the symbol error probability.