Three messages $m_1$, $m_2$ and $m_3$ are to be transmitted over an AWGN channel with noise power spectral density $N_0/2$. The messages are
$s_1(t) = egin{cases} A, & 0 le t le T, \ 0, & ext{otherwise}, end{cases}$
$s_2(t) = -s_3(t) = egin{cases} -A, & 0 le t le T/2, \ +A, & T/2 < t le T, \ 0, & ext{otherwise}, end{cases}$
where A is a constant.
(a) What is the dimensionality of the signal space ?
(b) Find a basis for the signal space.
(c) Give a vector representation of the signals and draw the signal constellation for this problem.
(d) Assuming equiprobable signals, sketch the optimal decision regions for $m_i$, $i = 1,2,3$.
(e) Prove that the message $m_1$ is more vulnerable to errors. Hint: Consider $P( ext{error}|m_i ext{ transmitted})$, $i = 1,2,3.$