$($Topic: Optimum Threshold$)$ For binary on$-$off signaling with a rectangular pulse shape, consider the following receiver structures. For each case, derive the optimum threshold to minimize PE$.$ Assume the signals are equi$-$probable.

$($a$)$ The receiver is an integrate and dump receiver $($Ideal integrator followed by a sampler at every Tb sec$,$ where Tb is the bit duration.$)$

$($b$)$ The receiver is an RC integrator with h$($t$)=1$ Tb e$$ t Tb u$($t$)$ followed by a sampler at every Tb sec$.$