ECE 461 Fall 2006October 25, 2006HOMEWORK ASSIGNMENT 7Reading: Text, Chapter 5Due Date: November 2, 2006 (in class)1. “Semi-Orthogonal” Signal Set. Consider the signal set with M = 2N signals given by:sm(t) =(√Egm(t) m = 1, . . . , Nj√Egm−N(t) m = N + 1, . . . , M.where {gk(t)}Nk=1are real-valued orthonormal functions. Clearly this signal set satis-fies: Re[ρk,`] = 0, for k 6= `.(a) Argue that rk= hr(t), gk(t)i, k = 1, . . . , N, form sufficient statistics for optimaldecision making at the receiver for an AWGN channel.(b) Now define the M real-valued statisticsym=(rm,Im = 1, . . . , Nr(m−N),Qm = N + 1, . . . , M.Show that the MPE decision rule is given byˆmMPE= arg maxmym(c) Find an expression for Pefor the MPE decision rule2. Gray coding for QPSK. Consider the following two bit assignments for QPSK10(2)(1)000101 0011 1011(a) Show that assignment (1), which corresponds to Gray coding, results in an averagebit error probability of Pb= Q(√2γb).(b) Show that under assignment (2), the first bit (from the left) sees an averageprobability of error of Q(√2γb), whereas the second bit sees an average probabilityof error of 2Q(√2γb)[1 −Q(√2γb)]. ThusPb=12Q(p2γb) + Q(p2γb)[1 −Q(p2γb)]cV.V. Veeravalli, 2006 13. Competing QAM Constellations. Consider the three 8-ary QAM constellations shownbelow (from Exam 1):(3)(1) (2)(a) For each constellation, determine whether you can label the signal points usingthree bits so that nearest neighbors differ by at most one bit (Gray coding). Ifso, find such a labeling. If not, state why not and find a labeling that minimizesthe bit transitions between neighbors.(b) For the labelings found in part (a), compute the nearest neighbor approximationfor the average bit error probability Pbas a function of the bit SNR γb= Eb/N0.Evaluate these approximations for γb= 10 dB.4. Noncoherent Demodulation of Linearly Modulated Signals. The received signal for onesymbol period for linear memoryless modulation on an ideal AWGN channel is givenby:r(t) =pEmejθmg(t)ejφ+ w(t)where the phase offset φ is due to the delay introduced by channel. If φ is known at thereceiver, we can correct for it (by projecting y(t) on g(t)ejφto produce the sufficientstatistic) and suffer no loss in detection performance. However, if φ is not known, wemay project y(t) on g(t) to get the sufficient statisticR =pEmejθmejφ+ Wwhere W ∼ CN(0, N0). Since φ is not of direct interest to the receiver, we treat itas a nuisance parameter. As we saw in class, there are two ways to deal with suchparameters.(a) Assume that φ ∈ [0, 2π], and find ˆmJML(r) using the joint ML approach. Interpretyour answer.(b) Now assume that φ is a random variable that is uniformly distributed on [0, 2π],and find ˆmMAP(y). Simplify your answer as much as possible (note that youranswer can be written in terms of the Bessel function I0).Note: You should see from this problem that noncoherent demodulation of linearlymodulated signals is not a very good idea.cV.V. Veeravalli, 2006