(40 points) We consider a AWGN channel model. Let s[n] be a 4-level symbol sent
by a transmitter, i.e., s[n] ? C = {a,b,c,d} where a,b,c,d ? IR. Assuming that an
ideal shaping filter, the input-output relationship of the channel is given by
y[n] = s[n] + v[n] for n? {1,2,..., N},
where v[n] is noise and its distribution is Gaussian, i.e., v[n] ~ N (0,?²) and N is an
integer value large enough.
(a) (10 points) Design the four constellation points C = {a,b,c,d} such that E[s[n]] =
0 and E[|s[n]|²] = 1.
(b) (5 points) Define signal-to-noise ratio (SNR).
(c) (10 points) The receiver performs maximum likelihood (ML) detection to es-
timate the transmitted symbol using the received signals as discussed in the
lecture. Find the decision boundaries for the ML detection.
(d) (15 points) With the ML detection rule, derive the average symbol error proba-
bility in terms of SNR. (Use a Q-function)
