ECE 461 Fall 2006November 2, 2006HOMEWORK ASSIGNMENT 8Reading: Text, Chapter 5Due Date: November 9, 2006 (in class)1. BPSK versus QPSK with Phase Error. We saw in the previous HW assignment thatnoncoherent detection of linearly modulated signals is not a good idea. In this problemwe consider the situation where the phase is estimated at the receiver (say, using aphase-locked loop) but there is a residual phase error φ, which is not known at thereceiver. As before, the sufficient statistic for decision making when symbol m is sentis given by:R =pEmejθmejφ+ WThe receiver does not know that there is a phase error, and so it makes decisionsassuming that φ is equal to zero. It is intuitively clear that the phase error shouldincrease the symbol/bit error probability at the output of the detector. We investigatethis deterioration in performance f or the special cases of BPSK a nd QPSK.(a) Show that for BPSKPb= Qp2γbcos2φ(b) Show that for QPSK with Gray codingPb=12Qq4γbsin2(φ + π/4 )+12Qp4γbcos2(φ + π/4 )(c) Compare Pbfor BPSK and QPSK for γb= 10 and φ = 0, 0.1, 0.2 , and 0.3 radians(use Matlab or any o t her program of your choice to compute these values). Whichmodulation scheme is more sensitive to phase errors?2. Noncoherent Orthogonal Signaling as M → ∞.(a) Using the union bound, show that the following bound on Peholds for M-aryorthogonal modulation with noncoherent detection:Pe<M2exp−γblog2M2(b) Now show that Pe→ 0 as M → ∞ as long as γb> 2 ln 2.(Note: It is interesting that this asymptotic result is identical to the result weshowed in class for coherent detection. This indicates that noncoherent detectionsuffers no loss relative to coherent detection as M → ∞)cV.V. Veeravalli, 2006 13. Closed-Form expression for Pefor Noncoherent Orthogonal Signaling. In class weshowed that the probability of correctly demodulating the transmitted symbol is givenby:Pc=Z∞0x1 − exp−x22M−1exp−x22+ γsI0xp2γsdx(a) Use the binomial expansion on the second term in the integrand, and the factthat a Ricean p df integrates to 1, to show that the symbo l error probability isgiven byPe=M−1Xn=1(−1)n+1M − 1n1n + 1exp−nγs(n + 1).(b) Use part (a) to find an expression for Pbin terms of γb.(c) For each of M = 2 , 4, 8, plotPbversus γb(ranging from 5 to 40 dB). Plot allcurves in one figure so you can compare them. You may use Matlab or any otherprogram of your choice (please include your co de).cV.V. Veeravalli, 2006