Optimum Receiver Design in AWGN

A binary digital communication system employs the signals $s(t)=\{s_0(t),s_1^{\text{'}}(t)\}$ where,

$s_0(t)=0,0tT,$

$s_1(t)=A,0tT,$

for transmitting the information. The demodualator cross$-$correlates the received signal $r(t)$ with $s(t)$ and samples the output of the correlator $t+T.$

$($a$)$ Determine $($i$)$ the optimum detector for an AWGN channel and $($ii$)$ the optimum threshold assuming that the signals are equiprobable.

$($b$)$ Given that The noise power spectral density is $\frac{N_0}{2},$ determine the probability of symbol error as a function of SNR$.$

$($c$)$ How does this signaling scheme compare with the antipodal signaling $?$ Hint: Compare the energy required to achieve the same error performance.

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Please dont put AI generated solution.