Chapter Summary

Chapter Summary

Key Points

• 1.

  ML is a benchmark, not a practical detector. The search space $|\mathcal{X}|^{MN}$ is exponential in the frame size; for a 5G-NR-aligned OTFS frame with $MN = 10^4$ and QPSK, this is $\sim 10^{6000}$. Sphere decoding can reduce average complexity for small frames but remains exponential worst-case.

• 2.

  Linear MMSE is $O(MN\log(MN))$ via the 2D DFT. Because $\mathbf{H}_{DD}$ is block-circulant, it diagonalizes under the 2D DFT, reducing MMSE to two 2D FFTs plus element-wise Wiener filtering. Essentially free to compute. However, MMSE has diversity order 1 — per-cell, deep fades corrupt symbols that other detectors could have saved through DD-domain averaging.

• 3.

  Message-passing on the DD factor graph is near-ML. The DD factor graph has $MN$ variable nodes and $MN$ factor nodes, each factor connected to exactly $P$ variables. Gaussian BP converges in $O(\log(MN))$ iterations to a fixed point with BER slope $P$ (full diversity). Per-iteration complexity: $O(P\,MN)$. Total: $O(P\,MN\,T_{\text{iter}}) \sim 10^6$ ops for typical frames — readily realtime.

• 4.

  LCD is the practical sweet spot. Linear MMSE initialization, then 3 iterations of residual-MMSE-soft-quantize refinement. At each iteration, $O(MN\log(MN))$ via 2D FFT. Achieves full diversity $P$ at 3× MMSE cost; BER within 1-2 dB of ML. Deployed in current OTFS research receivers and the CommIT cell-free testbed.

• 5.

  Iterative detection-decoding (IDD) closes the ML gap. Outer LDPC/Turbo decoder exchanges extrinsic LLRs with the LCD or MP detector. EXIT-chart analysis predicts convergence. With 3 outer iterations, the receiver achieves BER within 0.5 dB of joint ML decoding at total complexity $O(T_{\text{outer}}\,MN\log(MN)) \sim 10^6$ ops.

• 6.

  Cross-domain detection combines TF and DD strengths. Fuses per-subcarrier TF detection (useful when $|H(f, t)|$ is large) with DD-domain detection (useful when channel is sparse). Wins in channels that are both multipath-rich and fade-prone. Niche for most deployments; baseline is LCD + LDPC IDD.