Consider equiprobable binary orthogonal signaling with signals $s_1=(\sqrt[\phantom{2}]{E},0)$ and $s_2=(0,\sqrt[\phantom{2}]{E}).$

When these signals are sent over the additive white Gaussian noise $($AWGN$)$ channel, the received

signal vector can be expressed as

$r=(r_1,r_2)=s_m+n,$ for $m=1,2$

where $n=(n_1,n_2)$ is the noise vector whose components are independent and identically dis$-$

tributed Gaussian random variables with mean zero and variance $\frac{N_0}{2}.$

The optimal detection rule in this setting is formulated as

$f(r|s_1){}_{s_2}^{s_1}f(r|s_2)$

where $f(r|s_m)=\frac{1}{N_0}{e}^{-\frac{(r_1-s_m{)}^2+(r_2-s_{m2}{)}^2}{N_0}}$ and $s_m=(s_{m1},s_{m2}).$ The above detection rule indicates

that $s_1$ is detected if $f(r|s_1)>f(r|s_2).$ Otherwise, $s_2$ is detected.

a$)$ Show that the above detection rule can be simplified to

$r_1{}_{s_2}^{s_1}{r}_2.$

b$)$ Assume that $s_1=(\sqrt[\phantom{2}]{E},0)$ was sent. Show that the conditional error probability is given by

$Pr(e|s_1$ sent $)=Q(\sqrt[\phantom{2}]{\frac{E}{N_0}})$

where $Q-$function is defined as $Q(a)=Pr(x>a)$ where $x$ is a Gaussian random variable with

mean zero and variance $1.$