Chapter Summary

Chapter Summary

Key Points

• 1.

  Ideal OTFS pulses don't exist. Time-frequency uncertainty $\sigma_t \sigma_f \geq 1/(4\pi)$ bounds any real pulse. No pulse can be simultaneously time- and frequency-compact. Practical OTFS accepts bounded ISI and ICI, controlled by pulse choice.

• 2.

  Bi-orthogonality is the OTFS Nyquist condition. Transmit and receive pulses form a bi-orthogonal pair when their cross- ambiguity is a Dirac comb at the DD grid. Critical sampling $T_s W_s = 1$ is the design sweet spot where unique bi-orthogonal pairs exist.

• 3.

  RRC pair is the workhorse: bi-orthogonal, tunable via roll-off $\gamma$. Typical values: $\gamma = 0.2$-$0.35$. Zero ISI at perfect sync; controlled ICI. Cost: $\gamma \cdot 100\%$ excess bandwidth.

• 4.

  Window selection matters for OOBE. Sidelobe level × main lobe width is bounded. 5G NR OTFS needs Blackman ($-58$ dB SLL) or Nuttall for compliance at $M = 256$ subcarriers. Hamming ($-43$ dB) is insufficient.

• 5.

  Commercial defaults: Cohere uses Kaiser $\beta = 6$; 5G-NR- compatible OTFS uses Blackman. CommIT recommends RRC with $\gamma$ matched to mobility ($0.15$ static, $0.25$ vehicular, $0.35$ HST). LEO uses Gaussian.

• 6.

  Filter-bank OTFS for multi-service: $K$ parallel filters enable simultaneous sensing + data + URLLC streams. 15% capacity penalty for flexibility. Hermite basis provides orthogonal families. Post-Rel. 22 research direction.

• 7.

  Design rule: single-pulse OTFS for standardization (simpler, faster rollout). FB-OTFS for specialized ISAC or multi-service scenarios. 6G Rel. 21 spec is single-pulse. FB-OTFS post-Rel. 22.