A digital optical fiber communication link operates at a data rate of $2\mathrm{.}5$ Gbps with a required BER of $101210^\{-12\}.$ The system uses a PIN photodiode with a responsivity of $0\mathrm{.}9$ A$/$W and operates at a wavelength of $1550$ nm$.$ The e$$ective noise bandwidth is $3$ GHz$,$ and the noise figure of the receiver is $5$ dB$.$ Calculate the minimum received optical power $($receiver sensitivity$)$ required to achieve the desired BER. Provide detailed calculations and assumptions.