Dispersion-limited digital-fibre link length ($L_{\max }\sim T_r/[|{\partial }_f(1/v_g)|\Delta f]$)

A pulse's frequency components accumulate a differential group delay $\Delta t=L|{\partial }_f(1/v_g)|\Delta f$. If that spread approaches the pulse repetition period $T_r$, neighbouring symbols overlap. The resulting length estimate depends on bandwidth, initial pulse width, chirp and the receiver's allowable intersymbol interference. For $n(f)=n_0+\tau (f-f_0)$, the group index is $n_0+\tau (2f-f_0)$ and ${\partial }_f(1/v_g)=2\tau /c$: differentiating the phase index alone loses a factor of two.

A digital repeater can recover the symbol sequence and generate new pulses with restored amplitude, shape and timing. This resets waveform degradation between spans, provided decisions remain reliable. An amplifier alone increases the broadened pulse amplitude and also adds noise; it does not undo group velocity dispersion. optical dispersion compensation can instead reverse an accumulated frequency-dependent delay within its design bandwidth.