Chapter Summary

Chapter Summary

Key Points

• 1.

  OFDM is optimal under LTI. With a cyclic prefix longer than the channel memory, OFDM diagonalizes the channel matrix: each subcarrier becomes an independent scalar AWGN channel. This is what made OFDM the foundation of 4G/5G.

• 2.

  Under Doppler, OFDM develops ICI. A normalized Doppler $\delta = f_D T_s$ leaks power to neighboring subcarriers according to the sinc leakage formula $|\Phi_{m m'}(\delta)|^2$. The ICI-to-signal ratio scales as $(\pi\delta)^2/3$ for small $\delta$ and saturates to non-negligible values at moderate mobility. The SINR ceiling at zero AWGN is $\sim 3/(\pi\delta)^2$ — no transmit power can overcome it.

• 3.

  An OFDM symbol is a full Doppler column in the DD plane. The SFFT of a single OFDM data symbol $X_{TF}[n_0, m_0]$ is a pattern at fixed delay $\ell = -m_0 \bmod M$ with uniform magnitude across all $N$ Doppler bins. This spreading is the DD-geometric reason for ICI: every OFDM symbol overlaps the Doppler support of every path of the channel.

• 4.

  OFDM wastes DD sparsity. Three concrete losses: (i) diversity order 1 per cell (not $P$ as in OTFS), so deep fades destroy symbols OTFS would protect; (ii) pilot overhead $5$–$10\%$ vs $1$–$3\%$ for OTFS, a direct spectral-efficiency tax; (iii) ICI ceiling under high Doppler vs OTFS's essentially ICI-free operation. All three stem from signaling on the TF grid instead of the DD grid.

• 5.

  OTFS is the "right waveform" in DD. OFDM data lives on the TF grid; OTFS data lives on the DD grid. Because the DD grid is the channel's natural coordinate system (Chapter 4), OTFS's signaling lattice matches the sparse DD convolution structure — one data symbol is one DD point, not a spread stripe. This match is the structural reason for OTFS's diversity and pilot advantages.

• 6.

  OTFS does not replace OFDM everywhere. For low-mobility scenarios, OFDM remains optimal: ICI is negligible, pilot overhead is modest, equalization is nearly free. The OTFS advantages are concrete only when $\delta \gtrsim 0.05$ — which is where 6G's demanding use cases (V2X, HST, mmWave, LEO) live. 5G NR will likely add OTFS-like signaling as an option, not replace OFDM wholesale (see Chapter 19).