Chapter Summary

Chapter Summary

Key Points

• 1.

  OTFS achieves full delay-Doppler diversity. For a $P$-path integer-Doppler channel with distinct $(\ell_i, k_i)$, the ML detector achieves BER $\sim C_{\mathcal{X}}\,\mathrm{SNR}^{-P}$ at high SNR. The proof (Surabhi-Chockalingam-Caire 2019) is via PEP analysis: the minimum-rank of the error-covariance matrix $\mathbf{A}(\Delta\mathbf{x})$ is $P$, giving diversity order $P$.

• 2.

  Practical detectors (MP, LCD) achieve the diversity bound. MMSE has diversity 1 (linear detection cannot exploit all paths), but message passing, LCD, and iterative detection-decoding all achieve the full $P$-fold diversity. The detector complexity is $O(P\,MN\,T_{\text{iter}})$ or $O(MN\log(MN))$ — realtime feasible for 5G NR-aligned OTFS frames.

• 3.

  Uncoded BER scales as $\binom{2P-1}{P}\,(4\rho)^{-P}$. This closed-form expression (from the chi-squared quadratic form) matches Monte Carlo simulations to within 0.5 dB. For QAM higher-order constellations, the formula extends with a factor $1/(M-1)$ reducing the effective SNR.

• 4.

  DMT of OTFS dominates OFDM at every rate-reliability point. The Zheng-Tse tradeoff for OTFS is $d^*(r) = P - r$ (piecewise linear), while OFDM's is $d^*(r) = 1 - r$. OTFS's curve is strictly above OFDM's for all $r \in [0, P]$; the area between them represents the operational gain.

• 5.

  Outage advantage: $\epsilon^{1/P}$ quantile scaling. OTFS's outage capacity is $\log_2(1 + \rho P\,\epsilon^{1/P})$ vs OFDM's $\log_2(1 + \rho \epsilon)$. For URLLC targets ($\epsilon = 10^{-5}$), OTFS with $P = 8$ delivers $\sim 10-12\times$ more reliable throughput than OFDM on the same channel.

• 6.

  Ergodic capacity: OTFS and OFDM are equal. Both waveforms are unitary and achieve the same ergodic capacity on the same channel with optimal coding. OTFS's advantage is in outage and finite-blocklength performance — driven by the DD-domain diversity that OFDM cannot exploit. This is the quantitative foundation of OTFS's candidacy for 6G URLLC and mobility.

• 7.

  Real-world gains are about half the theoretical. Imperfect channel estimation ($\sim 1$ dB loss), suboptimal detection ($\sim 1$–$2$ dB), fractional Doppler (20-40% diversity loss) aggregate. Expected real SNR gain at $\mathrm{BER} = 10^{-5}$, $P = 4$: $\sim 5$ dB. Still decisive for 6G mobility scenarios.