5 2 a communication system transmits 3 bits using one of eight equally

Question

5.2 A communication system transmits 3 bits using one of eight equally likely signals in two dimensions, as shown in Figure 5.26. That
is, the signals are
so(t) = Apo(t)
$₁(t) = A$(1) + A$1 (1)
$₂(t) = A$1(1)
$3(t) = −A$(1) + A$1 (1)
$4(t) = -Apo(t)
$5(t) = -A$(1) - A$1 (1)
$6(1) = -A$1(1)
$7(1) = A$(1) - A$1(1).
The signals (1) and 1(1) are orthonormal. The received signal is the transmitted signal plus WGN.
(a) Draw a block diagram of the optimal receiver.
(b) Determine (draw) the optimal decision regions (to minimize the average error probability).
(c) Determine the probability of error for signal s₁(t).
(d) Determine a bound on the error probability for signal so(t).
281