ECE 461 Fall 2006December 6, 2006HOMEWORK ASSIGNMENT 10Reading: Notes on wireless channelsDue Date: none1. Useful Result. Show thatZ∞0Q(√x)e−x/γγdx =121 −rγ2 + γ.2. MPSK in Rayleigh fading. Based on the expression for Pefor MPSK (without fading)derived in problem 3(b) of HW#6, show that the average symbol error probability Pefor Rayleigh fading is given in closed form byPe=1 −1M−1√1 + a212+1πtan−1cot π/M√1 + a2,whereγbis the average bit SNR and a2=1γblog2M sin2π/M.Hint: You may need to use the following integralZθ2θ11cosec2θ + a2dθ =1a21√1 + a2tan−1cotθ√1 + a2−π2− θθ2θ1for 0 ≤ θ1≤ θ2≤ π/2.3. Diversity. Consider BPSK with channel gain a, i.e., the received signal isr(t) = ±a√Eg(t) + w(t), 0 ≤ t ≤ T,where {w(t)} is a zero-mean complex WGN process with PSD N0, g(t) is a unit energysignal, and the channel gain a is random with probability mass functionP{a = 0} = 0.1 and P{a = 2} = 0.9.(a) Determine the average probability of error Pefor MPE detection.(b) What value does Peapproach as E/N0approaches infinity?(c) Suppose the same signal is transmitted on two statistically independent channelswith gains a1and a2, whereP{a1= 0} = P{a2= 0} = 0.1 and P{a1= 2} = P{a2= 2} = 0.9.The additive noises on the two channels are also independent and identicallydistributed. The demodulator employs a matched filter for each channel andadds the two filter outputs to form the decision variable (which is compared to 0for decision-making). Determine Pein this case.cV.V. Veeravalli, 2006 1(d) For the case in part (c), what value does Peapproach when E/N0approachesinfinity?4. Optimality of maximal-ratio combining scheme for coherent detection with diversity:Consider BPSK signaling on an L-th order diversity channel. Each channel introducesa fixed attenuation and phase shift so that the received signal at the output of the `-thchannel is:r`(t) = ±α`ejφ`√E g`(t) + w`(t)where the processes w`(t) are independent complex WGN processes with PSD N0.The receiver uses the decision statisticR =LX`=1β`hr`(t), g`(t)iwhere the {β`} are complex weighting factors to be determined. A decision in favor of+1 (“symbol 1”) is made if rI> 0 and −1 (“symbol 2”) otherwise.(a) Determine the p.d.f. of RIwhen +1 is transmitted.(b) Show that the probability of bit error Pbis given by:Pb= Qr2EN0PL`=1Re{β`α`ejφ`}qPL`=1|β`|2.(c) Determine the values of {β`} that minimize Pb.Hint: Use the Cauchy-Schwarz inequalitycV.V. Veeravalli, 2006