Question

4.24 Three equiprobable messages mi, m2, and mz are to be transmitted over an AWGN channel with noise power spectral density No. The messages are $i(t) = 1o otherwise ( 1 0 <t<T si 0<i<T Sy(t) = -53(t) = -1 T<tst 10 otherwise 1. What is the dimensionality of the signal space? 2. Find an appropriate basis for the signal space. 3. Draw the signal constellation for this problem. 4. Derive and sketch the optimal decision regions R1, R2, and R3. 5. Which of the three messages is most vulnerable to errors and why? In other words, which of Perror |m; transmitted), i = 1, 2, 3, is largest?

Answer 1

I since malty = - malt) the dimensionality of the signal space is two. 2) As a basis of the signal space we consider the functions with a s t ostsT withŞ osts tl2 ute To otherwise 1 7 / 1 t <t IT O otherwise The vector representation of the signals is m, =[FF, o] My = [O, VT] 1 m₂ = [0,- 3) The signal constellation is depicted in the next fiqure CONF) CO-TA) 4) The three possible outputs of the matched filters, corresponding to the three possible transmitted signals are (8,182) = (F7 +0, ,3), ( n +2) and (ni-VT+12), whoxe n,, Bade zeso - mean Gaussian Sandom vastables with variance No. If all the signals are equiprobable the optimen deci- sion sule selects the signal that maximizes the metric C(x.mp) = 28.m-1moja of since Imp/? is the same for all a , c't. maj) = x.me Thus the optimal decision region Rre for me is the set of points 8,82), such that (8,082) - m,> (8,8). me and (80 %).m, > (8,1%). m3 since (8,8).m, = 158,, (8,1%) • m2 = 5% and (81 182). m3 = -F82, the previous conditions are written as 8,782 and 8,7-82 similasly we find that he is the set of points 18,,83) that satisfy 8, 70, 878, and Ro is the region such that 8, 40 and 8<-s. The regions R,, R₂ and R3 are shown in the next flouse

» If the signals are equiprobable then, Prelm) - PC18-,12318-31m) + P(x-mo) >18-9731210,0) when me is transmitted then 8= [ffen, ing] and therefore, Ple/m) is written as plelm) = P(92-,> 87) + P(hitz<-JF) since , n, ne are zero-mean statistically independent Gaussian anden variables, each with variance No, the random variables x=0,- and y=n, the are Zedo-mean Gaussian wett variance No. Hence, Plelm) - 1 ano de no dze + mano de sody When my is transmitted then * = [n 1, 2+ ] and therefore, - Plelme) = P(1, -2>F) + P(22KiT) = a[ + af Similarly from the symmetry of the problem, we obtain" Plelme) = PCelma) = 21+ [ since Q[o] is momononically decreasing, we obtain a 2 1 2 and therefore, the probability of eosor PCelm.) is larger than Pleme) and plelmz) · Hence, the message me is more vulnerable to extors.