$3.(20$ points$)$

The four signals shown in the figure above are used for communication of four equiprobable messages over an additive white Gausian noise channel with zero mean and power spectral density $\backslash ($ N$\_\{0\}/2\backslash ).$

$($a$)$ Find the smallest orthonormal basis set for representation of signals.

$($b$)$ Calculate the average symbol energy and the average bit energy.

$($c$)$ Plot the constellation.

$($d$)$ On the constellation you plotted, determine and draw the optimal decision regions.