A digital optical fibre system operates at a bit rate of $\backslash (2\backslash$mathrm$\{$~Gb$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The system uses Gaussian shaped optical pulses with the FWHM pulse power width equal to the bit period, and an rms spectral width of $1$ nm at $1550$ nm $.$

The system uses non$-$zero dispersion shifted fibre $($NZDSF$)$ as the transmission medium. This fibre has a dispersion value of $\backslash (3\backslash$mathrm$\{$ps$\}/(\backslash$mathrm$\{$nm$\}.\backslash$mathrm$\{$km$\}\backslash ))$ at $1550$ nm $.$

$(1).$ Estimate the fibre length where the dispersion power penalty due to pulse broadening equals $\backslash (\backslash$mathbf$\{3$ ~ d B$\}\backslash )($please fill in the blank with corresponding letter as e$.$g$.$a$,$ b$,$ c or d$)$

$($a$)173$ km

$($b$)206$ km

$($c$)122$ km

$($d$)97$ km

$(2).$ A length of dispersion compensating fibre $($DCF$)$ with a dispersion value of $-16\backslash (\backslash$mathrm$\{$ps$\}/(\backslash$mathrm$\{$nm$\}.\backslash$mathrm$\{$km$\})\backslash )$ is to be used to compensate for chromatic dispersion in the system. Estimate the length of DCF required to compensate for the amount of dispersion over the fibre length found in part $(1)$

$($a$)32\mathrm{.}4$ km

$($b$)44\mathrm{.}3$ km

$($c$)22\mathrm{.}9$ km

$($d$)38\mathrm{.}6$ km

$(3).$ The dispersion limited transmission distance for NRZ signals at $\backslash (10\backslash$mathrm$\{$~Gb$\}/\backslash$mathrm$\{$s$\}\backslash )$ in standard singlemode optical fibre with $\backslash (\backslash$mathbf$\{$D $=17\}\backslash$mathbf$\{$~ p s $/($ n m $.$ k m$\})\backslash )$ is $60$ km $.$ What would you expect the dispersion limited transmission distance to be using non$-$zero dispersion shifted fibre with $\backslash (\backslash$mathbf$\{$D$\}=\backslash$mathbf$\{6\}\backslash )$ ps$/($nm$.$km$)?$

$($a$)130$ km

$($b$)150$ km

$($c$)190$ km

$($d$)170$ km