A BPSK scheme operating over an AWGN channel with noise power spectral density of $$\frac{{{N_0}}}{2},$$  uses equiprobable signals
$${s_1}\left( t \right) = \sqrt {\frac{{2E}}{T}} \sin \left( {{\omega _c}t} \right)$$     and $${s_2}\left( t \right) = \sqrt {\frac{{2E}}{T}} \sin \left( {{\omega _c}t} \right)$$
over the symbol internal (0, T). If the local oscillator in a coherent receiver is ahead in phase by 45° with respect to the received signal, the probability of error in the resulting system is