University of Illinois at Urbana-ChampaignECE 461: Communications IISpring 2005Final ExamSaturday, May 7, 8-11 a.m., Room 169 Everitt LaboratoryName:• You have three hours for this exam. The exam is closed book and closed note, except you mayconsult both sides of three 8.500× 1100sheets 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 justification will receivevery little credit. If you need extra space, use the back of the previous page.• There are four problems with a total of 25 parts. Each of the parts is worth four points, for atotal of 100 points.Score:1. (28 pts.)2. (20 pts.)3. (28 pts.)4. (24 pts.)Total: (100 pts.)11. (28 points) Concisely answer each of the following seven questions.(a) Name one advantage of using a raised cosine function as a system waveform (in comparison to asinc waveform) and one disadvantage.(b) Name one advantage of using the duobinary function as a system waveform (in comparison to asinc waveform) and one disadvantage.(c) Name one advantage of noncoherent detection over coherent detection, and one disadvantage.(d) Explain one advantage of using an interleaver (specify the context) and one disadvantage.2(e) What is an advantage of using fractional spacing (such as τ =T2) in an equalizer? What is adisadvantage?(f) What sort of modulation would you suggest for low rate communication to a satellite in deep space,and why?(g) What sort of modulation would you suggest for high speed data communication over a one kilometerlong pair of copper twisted wires (s uch as a short central office to home telephone line), and why?32. (20 points) Suppose that the signal sm(t) = um(t) cos(2πfct+φ) is used to transmit a bit m ∈ {0, 1},where the waveforms u1and u0are pictured. The signal is subjected to AWGN with two-sided p owerspectral densityN02. Assume that fcT >> 1.−AATTA−Au (t)u (t)01(a) ExpressEbNoin terms of A, T , and No.(b) Carefully sketch the minmax optimal receiver in matched filter form. Specify the impulse responsefunction and threshold value. Include the carrier tracking loop, but you needn’t indicate the componentsof the carrier tracking loop.(c) Give the error probability for the receiver of part (b). What is the effective loss in energy efficiency,expressed in dB, for not using an antipodal pair of signals?4(Problem 2 continued. For convenience the figure is repeated.)−AATTA−Au (t)u (t)01(d) Consider the suboptimal receiver in which the received waveform is correlated with the signals1(t) and the correlator output is compared with a threshold. What is the minmax optimal choice ofthreshold in the suboptimal receiver? What is the resulting value of the probability of error? Whatis the total effective loss in energy efficiency, expressed in dB, for not using an ideal coherent detectorand for not using an antipodal pair of signals?(e) Consider the noncoherent receiver shown below.u (T −t)cos(2 f t )oTdetectorEnvelope 0 1τr(t)VlcπoGiven that m = 1 is sent, describe the distribution of V . Identify parameter values in terms of A,T ,and No.53. (28 points) Consider a QPSK system with transmitted signals(t) =Xn[Aan,cgT(t − n) cos(2πfct + φ) − Aan,sgT(t − n) sin(2πfct + φ)]such that gT(t) = sin(πt)I[0,1](t), the data symbols an,cand an,sare all in {1, −1}, and fc>> 1. Notethat the symbol duration is one.(a) Sketch the signal s over the time interval [0, 3], assuming whatever symbol values you’d like. Thenidentify the lowpass equivalent signal sl. (Hint: It is the complex-valued baseband signal such thats(t) = Re(sl(t)ej2πfct). Note that this part of the problem is independent of the res t.)Suppose the signal s is corrupted by AWGN with two-sided power spectral densityNo2to produce thereceived signal r, which is fed into the receiver indicated.ϕ ππccϕcos(2 f t + )s(t)WGNaa00+1−1+1−1hht=n+1t=n+1r(t)hn,cn,s−sin(2 f t + ) The receive filter h is given by h(t) = e−tfor t ≥ 0 and h(t) = 0 for t < 0, and perfect phase trackingis assumed.(b) Ignoring the effects of intersymbol interference, find the average probability of error. Roughly howmany dB loss is there compared to that of the ideal matched filter receiver? Hint:R10e−tsin(πt)dt ≈0.40.pe=dB loss over optimal receiver =6Problem 3 continued. For the remainder of this problem, the effects of intersymbol interference are tobe taken into account. The block diagram is repeated for convenience.ϕ ππccϕcos(2 f t + )s(t)WGNaa00+1−1+1−1hht=n+1t=n+1r(t)hn,cn,s−sin(2 f t + ) (c) Do the bits in the quadraphase signal interfere with the bits in the inphase signal? Explain.(d) Identify the worst case bit sequence for receiving bit a0,c, and find the corresponding maximumerror probability. (Hint: e−1≈ 0.38.)Worst case bit pattern =pmax=7Problem 3 continued.(e) Suppose a three-tap equalizer is inserted after the filter in each branch of the receiver, and thatthe vector of tap weights (c−1, c0, c1) is chosen to minimize the peak relative distortion. Find the tapweights.(c−1, c0, c1) =(f) Repeat part (d), but now with the equalizer installed. (Hint: Account for the effect of the equalizeron the noise.)Worst case bit pattern =pmax=(g) Of the answers to parts (b), (d), and (f), which is the largest error probability? Which is smallest?Largest error probability:Smallest error probability:84. (24 points) This problem focuses on using trellis coded modulation using the following 8-ary QAMconstellation, where the coordinates are taken relative to an orthonormal basis for the signal space.2AA3AASuppose the (n = 2, k = 1) convolutional code with generators g1= [1 0 1] and g2= [1 1 1] is usedto produce two coded bits per data bit to select one of 4 subconstellations, and then an additionaluncoded data bit is used to specify a symbol w ithin the selected subconstellation. This scheme has thesame bandwidth requirement as an uncoded QPSK system.(a) Express the energy per data bit, Eb, in terms of A. (Remembe r that there are two data bits persymbol.)(b) Apply the method of set partitioning to the signal constellation. As in the text, label the originalconstellation A, the two subconstellations below it B0and B1, and the four constellations below those,C0, C2, C1, and C3.9Problem 4 continued(c) Show a