ECE 461 Fall 2006Optimum Receiver for Linear Memoryless Modulation and Performance◦ Linear modulated signals are complex one-dimensional signalssm(t) =pEmejθmg(t) m = 1, 2, . . . , M, 0 ≤ t ≤ Tswhere g(t) is the unit energy pulse shaping function, which also serves as the basis for the signalspace.◦ Thus the sufficient statistic, when symbol m is sent, is given by:R = hr(t), g(t)i =pEmejθm+ W = sm+ Wwhere W is CN (0, N0).◦ Thus pm(r) ∼ CN (sm, N0), i.e.,pm(r) =1πN0exp−|r − sm|2N0◦ Assuming equal priors,ˆmMPE(r) = arg maxmpm(r) = arg min |r − sm|2Thus the optimum detector is a Minimum Distance detector, i.e., we pick the signal smthat isclosest in distance to r. We saw in class how to obtain the optimum decision regions in the (rI, rQ)space. Let Γmdenote the region in the complex plane where a decision in favor of symbol m ismade.◦ Probability of (symbol) error. The probability of error, conditioned on symbol m being sent isgiven by:Pe,m= 1 − Pc,m, with Pc,m=ZΓmpm(y)dy . (1)The average probability of error (assuming equally likely symbols) is given by:Pe=1MM−1Xm=0Pe,m. (2)For symmetric constellations, Pe= Pe,mfor all m.◦ For MPE detection, we showed in class that Pecan be calculated exactly in some special casessuch as BPSK and QPSK (also see HW 6). For BPSKPe= Q r2EsN0!and for QPSKPe= 2Q rEsN0!− Q2 rEsN0!.cV.V. Veeravalli, 2006 1◦ Union Bound on Pe. The expression for Pefor BPSK suggests the following bound on Peforgeneral linear modulation schemes.Pe,m= P[`6=m{decide `}{m sent}≤X`6=mP ({decide `}|{m sent}) = Qsd2m,`2N0where dm,`is distance between the points m and ` in the constellation.◦ Intelligent Union Bound (IUB). The Union Bound is generally too conservative. A better boundis obtained by keeping only the terms in the Union Bounds that are required to cover the errorregion.◦ Nearest Neighbor Approximation (NNA). Letdmin(m) = min`6=mdm,`and let the number of neighbors that are at this minimum distance be Ndmin(m). ThenPe,m≈ Ndmin(m)Qsd2min(m)2N0.NNA and IUB coincide in many cases since the nearest neighbors usually cover the entire errorregion.cV.V. Veeravalli, 2006