# Coherence bandwidth/delay spread in OFDM

Say we want to design an OFDM system and we have some parameters given to us, i.e. the coherence time $T_c$ and the max delay spread $T_m$.

It's clear to me why we need to ensure $T_s(N+N_{cp}) \ll T_c$ where $N_{cp}$ is the cyclic prefix in samples - we want the channel to stay coherent during the entire symbol. What is not clear is what happens to the constraint $T_s(N+N_{cp}) \gg T_m \Leftrightarrow B_N \ll B_c$. Apparently, in OFDM it suffices with $T_s N_{cp} \geq T_m$.

It makes intuitive sense in a way, but I'm still not sure exactly how the coherence bandwidth interacts with the subchannel bandwidths. Why don't we need to ensure $B_N \ll B_c$?

Indeed, the constraint $T_s(N+N_{cp}) \gg T_m$ should be derived as $T_s(N+N_{cp}) \gg T_s N_{cp} \geq T_m$.
Thus, $B_N << B_c$ is a consequence of the prior design criterion. However, if subcarrier spacing $B_N$ is much smaller than coherence bandwidth $B_c$, the channel estimation is facilitated by adding several pilots in $B_c$. For info, in LTE, two pilot are separated by 45 kHz while the rms delay spread is about 1us gives $B_{c,50\%} \approx 200kHz$ and $B_{c,90\%} \approx 20kHz$; thus 45kHz pilot spacing is good compromise between pilot overhead and estimation