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Linear Detection With Cross-Domain Insight

MMSE (Section 2) is fast but diversity-1. MP (Section 3) is near-ML but requires iterative computation. The low-complexity DD (LCD) detector bridges the two: linear complexity $O(MN \log(MN))$ , but with a cross-domain structure that exploits both the DD channel matrix and its TF-domain counterpart. The result is linear-complexity detection with BER close to the ML benchmark — particularly at moderate SNR where MMSE fails.

The LCD detector is the practical workhorse of high-efficiency OTFS receivers. It is deployed in recent OTFS modem designs and is the detector of choice when both throughput and latency matter.

Definition:
Low-Complexity DD (LCD) Detector

The LCD detector combines cross-domain linear equalization with iterative soft quantization:

Apply MMSE in the DD domain (Section 2) to obtain a soft estimate $\hat{\mathbf{x}}^{(0)}$ .
Iterate $T_{\text{iter}}$ $T_{iter}$ times:
- Subtract the soft-estimated ISI from the received DD grid: $\mathbf{r}^{(t)} = \mathbf{y}_{DD} - \mathbf{H}_{DD}\,\hat{\mathbf{x}}^{(t-1)}$ .
- Apply MMSE to the residual: $\Delta\mathbf{x}^{(t)} = \text{MMSE}(\mathbf{r}^{(t)})$ .
- Update: $\hat{\mathbf{x}}^{(t)} = \hat{\mathbf{x}}^{(t-1)} + \Delta\mathbf{x}^{(t)}$ .
- Soft-quantize: $\hat{\mathbf{x}}^{(t)} \leftarrow Q_{\text{soft}}(\hat{\mathbf{x}}^{(t)})$ .
Hard-quantize for final output.

The iterative refinement recovers diversity loss of the single-pass MMSE. The complexity per iteration is $O(MN \log(MN))$ (same as MMSE), and convergence is typically within 2-3 iterations.

Theorem: LCD Convergence and Performance

The LCD detector converges within $O(\log(P))$ iterations to a fixed point whose BER at high SNR matches the full-diversity (order $P$ ) slope. The per-iteration complexity is $O(MN \log(MN))$ (two 2D FFTs plus element-wise operations), and total complexity is $O(T_{\text{iter}}\,MN\log(MN))$ , where $T_{\text{iter}} = 3$ suffices in practice.

Each iteration refines the estimate by correcting for the previously-estimated ISI. After a few iterations, the residual is noise-limited rather than interference-limited, and the soft-quantized decisions are near the true symbols. This is essentially a serial interference cancellation (SIC) scheme with MMSE for parallel processing.

Proof

Iteration update

$\hat{\mathbf{x}}^{(t)} = \hat{\mathbf{x}}^{(t-1)} + \text{MMSE}(\mathbf{y}_{DD} - \mathbf{H}_{DD}\hat{\mathbf{x}}^{(t-1)})$ . For $\hat{\mathbf{x}}^{(t-1)}$ close to $\mathbf{x}$ , the residual is approximately $\mathbf{H}_{DD}(\mathbf{x} - \hat{\mathbf{x}}^{(t-1)}) + \mathbf{w}$ .

Contraction

Denote $\mathbf{e}^{(t)} = \mathbf{x} - \hat{\mathbf{x}}^{(t)}$ . The iteration reduces to $\mathbf{e}^{(t)} = \mathbf{e}^{(t-1)} - \text{MMSE}(\mathbf{H}_{DD}\mathbf{e}^{(t-1)} + \mathbf{w})$ . For the MMSE filter designed against $\mathbf{H}_{DD}$ , this is a contraction map with factor depending on SNR.

Convergence rate

The contraction factor scales as the MMSE gain of the residual. After $O(\log P)$ iterations, the error reaches the noise-limited floor.

Diversity

At convergence, the soft-quantized decisions average over all $P$ paths — equivalent to the MP result, with BER slope $P$ . $\blacksquare$

Key Takeaway

LCD is the detector sweet spot. Linear complexity per iteration, $O(MN\log(MN))$ ; only 2-3 iterations needed; BER slope = $P$ (full diversity). For $MN = 10^4$ and $P = 8$ , LCD runs in $\sim 3 \times 10^5$ ops per frame — about 3× MMSE and $\sim 1/30$ × the MP cost. For most practical OTFS receivers, LCD is the operational choice.

LCD Algorithm Pseudocode

Complexity:

O(T_{\text{iter}}\cdot MN\log(MN))

Input:

\mathbf{y}_{DD}

, channel

\{(h_i, \ell_i, k_i)\}

,

noise variance

\sigma^2

, QAM alphabet

\mathcal{X}

,

max iterations

T_{\text{iter}} = 3

Output: Detected symbols

\hat{X}_{DD}

1. Compute 2D DFT of

\mathbf{y}_{DD}

:

\tilde{\mathbf{y}} = \mathbf{F}\,\mathbf{y}_{DD}

.

2. Compute channel eigenvalues

\boldsymbol{\Lambda}

from the

P

path parameters (done once for the frame).

3. Initialize

\hat{\mathbf{x}}^{(0)} = \mathbf{F}^H((\boldsymbol{\Lambda}^*/|\boldsymbol{\Lambda}|^2 + \sigma^2)\,\tilde{\mathbf{y}})

. (MMSE)

4. for

t = 1, \ldots, T_{\text{iter}}

do

5.

\quad

Soft-quantize:

\tilde{\mathbf{x}}^{(t-1)} = Q_{\text{soft}}(\hat{\mathbf{x}}^{(t-1)})

(soft QAM demap — closest QAM symbol, weighted by likelihood)

6.

\quad

Compute residual:

\mathbf{r}^{(t)} = \mathbf{y}_{DD} - \mathbf{H}_{DD}\,\tilde{\mathbf{x}}^{(t-1)}

(sparse multiply,

O(P\,MN)

)

7.

\quad

Apply MMSE to residual:

\Delta\mathbf{x}^{(t)} = \text{MMSE}(\mathbf{r}^{(t)})

8.

\quad

Update:

\hat{\mathbf{x}}^{(t)} = \tilde{\mathbf{x}}^{(t-1)} + \Delta\mathbf{x}^{(t)}

9. end for

10. Hard-quantize final:

\hat{\mathbf{x}} = Q_{\text{hard}}(\hat{\mathbf{x}}^{(T_{\text{iter}})})

.

11. Return

\hat{X}_{DD} = \text{reshape}(\hat{\mathbf{x}})

.

The soft quantization $Q_{\text{soft}}$ returns an expected value under the QAM posterior — effectively a "soft decision" that preserves uncertainty. Hard quantization is applied only at the end. This preserves diversity across iterations.

Uncoded BER: MMSE, LCD, MP, ML-Bound

Plot uncoded BER vs SNR for all four detectors on the same $P$ -path OTFS channel. The ML bound is the lower envelope (diversity $P$ ). MP tracks the ML bound within $\sim 0.5$ dB. LCD is close to MP (within 1-2 dB). MMSE is visibly inferior (diversity 1 slope). This is the master comparison plot for OTFS detection.

Parameters

P

4

M

32

N

16

Min SNR (dB)5

Max SNR (dB)30

OTFS Detector Landscape: Complexity vs Performance

Detector	Complexity	Diversity	SNR gap to ML (typical)
MMSE	$O(MN \log(MN))$	1	5–8 dB
LCD (3 iter)	$O(MN \log(MN))$	$P$ (full)	1–2 dB
MP (10 iter)	$O(P \cdot MN)$	$P$ (full)	0.5–1 dB
ML (sphere decoding)	$O(MN^3)$ average	$P$	0 dB
ML (brute force)	$O(\|\mathcal{X}\|^{MN})$	$P$	0 dB

Example: LCD for QPSK on $P = 4$ , MN = 1024

An OTFS receiver with $MN = 1024$ and $P = 4$ paths uses LCD with 3 iterations. Channel SNR is 20 dB. Estimate the detection BER and the number of flops.

Solution

BER estimate

Full diversity $P = 4$ + moderate SNR: BER scales as $1/\mathrm{SNR}^4 = 10^{-8}$ . Very low uncoded BER.

Flops per iteration

Per iteration: 2 × (2D FFT of size $MN = 1024$ ) + $O(MN)$ linear: $\sim 10^4 \log_2(1024) \approx 10^5$ ops.

Total

3 iterations: $3 \cdot 10^5 = 3 \times 10^5$ ops. Compared with MMSE ( $10^5$ ): 3× more. Compared with MP ( $10$ iter $\times P\cdot MN = 4 \cdot 10^4 \cdot 10 = 4 \times 10^5$ ): marginally better.

Operational choice

LCD achieves MP-like BER at MMSE-like complexity. For this deployment, LCD is the clear winner.

LCD Detector Flowchart — LCD detector flowchart: start with MMSE output, iterate between residual computation (subtract soft-quantized ISI) and MMSE refinement. Three iterations typically suffice. Soft-quantization preserves diversity across iterations; hard-quantization is applied only at the final output stage.

⚠️Engineering Note

LCD in Practical OTFS Receivers

LCD is favored in deployment because:

Simplicity: two 2D FFTs + element-wise operations. Easy to implement on standard DSP hardware.
Deterministic latency: fixed number of iterations (3) gives predictable computation time — critical for URLLC applications.
Full diversity: at moderate QAM orders, LCD's BER slope is indistinguishable from ML within the operating SNR range.
Graceful degradation: at low SNR, LCD's performance is dominated by MMSE but remains better than MMSE alone.
Compatible with existing 5G NR silicon: uses the same 2D FFT kernels as OFDM equalization.

As of 2024, LCD is the detector of choice for pilot OTFS deployments (including the CommIT cell-free OTFS testbed). MP is reserved for research and cases where the 1-2 dB extra gain is decisive.

Practical Constraints

•
3 LCD iterations = deterministic latency
•
Works on standard OFDM silicon
•
Adequate BER performance for QPSK to 64-QAM

Common Mistake: Don't Over-Iterate LCD

Mistake:

Using $T_{\text{iter}} \geq 10$ in LCD, expecting better BER at higher iterations. In fact, LCD typically converges in 3 iterations and further iterations slightly degrade BER due to round-off and soft-quantization noise accumulation.

Correction:

Use exactly 3 iterations in LCD. Measure convergence by the norm of the residual; if it plateaus after 2-3 iterations, halt. Spending more iterations is wasteful without benefit. Some implementations use 2 iterations with minor BER loss (< 0.5 dB) and linear compute savings.

The Low-Complexity DD (LCD) Detector