Problem $5(30$ points$)$

Consider the following constellations: Constellation $1$ defined by the symbols $($A$,$ A $)$ and $(-$A$,A$; and Constellation $2$ defined by the symbols $(0,0)$ and $(-2A,-2A).$ The symbols are transmitted through an AWGN channel with mean zero and $PSD{S}_n(f)=\frac{N_0}{2}.$

For $1)$ to $3),$ we consider the case in which $($for both constellations$)$ both symbols are equally likely.

Represent the two constellations, indicating what the decision regions that minimize the probability of symbol error are.

Without calculating the probability of symbol error, explain which constellation has lower probability of error i$)$ as a function of A and ii$)$ as a function of the energy per symbol $E_s.$ Which constellation would you choose in a practical system?

Calculate the probability of symbol error for both constellations as a function of $\frac{E_s}{N_0}.$

Denote the probabilities of symbol $(A,A)$ in the first constellation and symbol $(0,0)$ in the second constellation as $P_{AA}$ and $P_{00},$ respectively, and consider the case of $P_{AA}\frac{1}{2},P_{00}\frac{1}{2}.$

For each constellation, calculate the equation of the line separating the two decision regions that minimize the probability of symbol error, and give an explanation of the differences you observe compared with the decision regions in $1).$

Calculate the probability of symbol error for both constellations as a function of $\frac{E_s}{N_0}.$