Problem $3\mathrm{.}2$

For the Binary Phase Shift Keying $($BPSK$),$ we have the following two signals

$2$Eb

s$1($t$)=$ T cos$($ct$)$

b

$2$Eb

s$2($t$)=$ T cos$($ct$)$

b

for $0$ t $$ Tb and all signals are $0$ outside this interval. The signal is transmitted over

an AWGN channel with a PSD of N$0/2.$ If the carrier at the receiver has a phase shift

of $.$ Derive an expression for the bit error rate $($BER$)$ as a function of $($assume that

fc $=$ m for some integer m where c $=2$fc$).$ What will be the value of the BER for Tb

$=0$ and $=$ $/2.$ Comment on your results.