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).

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