binary communication system uses equiprobable signals s$1($t$)$ and s$2($t$)$

s$1($t$)=2$Eb $1($t$)$ cos$(2$ f c t$)$

s$2($t$)=2$Eb $2($t$)$ cos$(2$ f c t$)$

for transmission of two equiprobable messages. It is assumed that $1($t$)$ and $2($t$)$ are

orthonormal. The channel is AWGN with noise power spectral density of N$0/2.$

$1.$ Determine the optimal error probability for this system, using a coherent detector.

$2.$ Assuming that the demodulator has a phase ambiguity between $0$ and $(0$ $$ $)$

in carrier recovery, and employs the same detector as in part $1,$ what is the resulting

worst$-$case error probability?

$3.$ What is the answer to part $2$ in the special case where $=$ $/2$