SD-Turbo: Iterative SDR Detection for LDPC MIMO

• SD-Turbo is an iterative turbo receiver for LDPC-coded MIMO systems that integrates FEC constraints directly into the SDR detection problem to achieve near-ML performance.
• It employs a joint MAP-SDR framework with decoder-provided LLRs and candidate list-based LLR extraction, enabling efficient turbo iterations.
• The single-SDR turbo variant significantly reduces computational complexity by 3–10× while maintaining error performance within 0.2 dB of the full multi-SDR approach.

SD-Turbo refers to the class of iterative turbo receivers for LDPC-coded @@@@1@@@@ systems based on semi-definite relaxation (SDR) detection, in which code constraints from the forward error correction (FEC) code are directly integrated into the SDR detection problem and soft information is iteratively exchanged with an LDPC decoder. This methodology achieves near maximum-likelihood (ML) symbol detection performance in polynomial time, while supporting efficient turbo iterations and substantial error-rate gains over traditional separate detection and decoding. The central constructs are joint SDR detection incorporating LDPC constraints, transition to maximum a posteriori (MAP) SDR using decoder-provided LLRs, iterative LLR list-based turbo processing, and a single-SDR simplification that maintains most of the error performance while greatly reducing computational cost (Wang et al., 2018, Wang et al., 2018).

1. System Model and Standard SDR Detection

Consider a spatial-multiplexing MIMO system with $N_t$ transmit and $N_r$ receive antennas over $K$ time slots. The real-valued system model is

$$\mathbf{y}_k = \mathbf{H}_k \mathbf{x}_k + \mathbf{n}_k,\quad \mathbf{x}_k \in \{\pm1\}^{2N_t},\ k=1,\dots,K,$$

where $\mathbf{y}_k \in \mathbb{R}^{2N_r}$, $\mathbf{H}_k \in \mathbb{R}^{2N_r \times 2N_t}$, and $$\mathbf{n}_k \sim \mathcal{N}(\mathbf{0},\sigma_n^2\mathbf{I})$$. Classical ML detection is formulated as: $$\min_{\mathbf{x}_k \in \{\pm1\}^{2N_t}} \sum_{k=1}^K \|\mathbf{y}_k - \mathbf{H}_k \mathbf{x}_k\|^2.$$ This problem is NP-hard in $N_t$. The SDR formulation introduces lifting via rank-1 matrices $$\mathbf{X}_k = [\mathbf{x}_k; t_k][\mathbf{x}_k^T\ t_k]$$, where $t_k\in\{\pm1\}$, and cost matrices

$$\mathbf{C}_k = \begin{bmatrix}\mathbf{H}_k^T\mathbf{H}_k & \mathbf{H}_k^T\mathbf{y}_k \ -\mathbf{y}_k^T\mathbf{H}_k & \|\mathbf{y}_k\|^2 \end{bmatrix},$$

leading to the relaxation: $$\min_{\{\mathbf{X}_k\succeq0\}} \sum_{k=1}^K \mathrm{tr}(\mathbf{C}_k\,\mathbf{X}_k),\quad \mathrm{tr}(\mathbf{A}_i \mathbf{X}_k) = 1,\ i = 1,\ldots, 2N_t+1.$$ This “disjoint” SDR sets the stage for integration of code constraints.

2. Joint SDR Detection with LDPC Constraints

To exploit the LDPC code in detection, “joint ML-SDR” incorporates code constraints on the bit variables $f_n \in [0,1]$, with forbidden-set (FS) parity-check inequalities for each LDPC check node $m$: $$\sum_{n\in \mathcal{F}} f_n - \sum_{n\in \mathcal{N}_m\setminus\mathcal{F}} f_n \leq |\mathcal{F}|-1,\quad \forall\ \mathcal{F}\subseteq \mathcal{N}_m,\,|\mathcal{F}| \text{ odd},$$ and box constraints $0 \leq f_n \leq 1$. The Gray symbol-to-bit mapping is enforced by linear constraints tying $\mathbf{X}_k$ to $f_n$: $$\mathrm{tr}(\mathbf{B}_i\mathbf{X}_k) = 1 - 2f_{2N_t(k-1)+2i-1},\ \mathrm{tr}(\mathbf{B}_{i+N_t}\mathbf{X}_k) = 1 - 2f_{2N_t(k-1)+2i}.$$ The resulting joint ML-SDR optimization is

$$\min_{\{\mathbf{X}_k, f_n\}} \sum_{k=1}^K \mathrm{tr}(\mathbf{C}_k\,\mathbf{X}_k),$$

subject to the positive semidefinite, diagonal, mapping, FS, and box constraints. This coupling yields significant improvement over standard SDR, as LDPC parity relationships are integrated at the detection stage (Wang et al., 2018, Wang et al., 2018).

3. MAP-SDR and Iterative Turbo Receiver Framework

The “MAP-SDR” further exploits a priori information from the decoder via a posteriori LLRs $L_{A1}$, modifying the cost function as

$$\sum_{k=1}^K \mathrm{tr}(\mathbf{C}_k\,\mathbf{X}_k) + 2\sigma_n^2 L_{A1}^T f.$$

The turbo receiver performs iterative exchange of soft information (LLRs) between the MAP-SDR detector and the LDPC decoder. For each turbo iteration:

• S1: Solve the joint MAP-SDR, using current $L_{A1}$.
• S2: Extract max-log extrinsic LLRs for each bit (see Section 4).
• S3: Pass extrinsic LLRs to the LDPC decoder; obtain updated a priori LLRs; iterate.

Stopping criteria include successful parity check satisfaction or reaching a maximum iteration count. This iterative turbo principle yields rapid convergence, as evidenced by sharper EXIT curve “open tunnels” and higher extrinsic mutual information at early iterations compared to conventional full-list turbo algorithms (Wang et al., 2018, Wang et al., 2018).

4. List-Based LLR Extraction and the Single-SDR Turbo Receiver

Classical computation of extrinsic LLRs for each bit is exponentially complex in $N_t$. SD-Turbo resolves this by generating a candidate list $\overline{\mathcal{L}_k}$ based on the SDR hard decision $\mathbf{b}_k^*$ and a small Hamming radius $P$: $$\overline{\mathcal{L}_k} = \{\mathbf{b}: d(\mathbf{b}, \mathbf{b}_k^*) \leq P\}.$$ LLRs are computed via max-log sum over this list. In the single-SDR turbo receiver variant, only one (joint ML-SDR) solve per codeword is performed; subsequent iterations update LLRs solely via list generation and recomputation using the combined LLR

$L_{comb} = L_{E1}^{init} + L_{A1}.$

This substantially reduces complexity while incurring only minor performance loss. Simulation results confirm that the single-SDR approach is within 0.2 dB of the full multi-SDR at BER $=10^{-4}$, achieving most of the coding gain and convergence benefits at a fraction of the computational cost (Wang et al., 2018, Wang et al., 2018).

5. Complexity, Performance, and Comparisons

The computational complexity per SDR is polynomial, estimated as $O(N_t^{3.5}\text{–}4.5)$ (depending on the solver and SDP implementation). For $I$ turbo iterations:

• Multi-SDR turbo: $I$ SDPs per codeword, $I \times O(N_t^{3.5})$ operations.
• Single-SDR turbo: 1 SDP per codeword, $O(N_t^{3.5})$, with negligible cost for list-based LLR updates.

Performance benchmarks for $N_t=N_r=4$, QPSK, $(256,128)$ or $(1024,512)$ LDPC:

• Joint ML-SDR (hard-decision + SPA) achieves $≈2$ dB gain over the disjoint SDR baseline at BER $=10^{-4}$.
• Joint MAP-SDR turbo (multi-SDR) matches or outperforms conventional full-list turbo after 2–3 iterations.
• Single-SDR turbo is within $≈0.2$ dB of multi-SDR after 2–3 iterations, at a fraction of computational cost.
• Compared to other list-based SDR schemes ([11] in the source), SD-Turbo delivers $0.5$–$1$ dB BER improvement at $10^{-4}$ and $3$–$10\times$ reduction in computation time, especially at low SNR (Wang et al., 2018, Wang et al., 2018).

6. Special Properties and Generalization

The SD-Turbo methodology’s integration of LDPC parity constraints in the detection stage, followed by iterative LLR exchange, produces a turbo detector whose early iterations often yield higher extrinsic information than conventional methods. EXIT chart analysis demonstrates steeper detection–decoding tunnels, implying faster and more reliable iterative convergence.

The single-SDR turbo principle generalizes to any iterative receiver employing candidate-list–based soft-output detection. By decoupling the main joint optimization from subsequent iterative LLR updating, its benefits apply equally to tree-search, randomized, or other list-generation–based MIMO decoding schemes (Wang et al., 2018).

7. Summary of Simulation and Practical Insights

Summarized characteristics of SD-Turbo implementations (for $N_t=N_r=4$, QPSK, regular LDPC):

Variant	Main Detection	BER Gain at $10^{-4}$	Complexity Reduction
Joint ML-SDR + SPA	Hard, joint	$\sim2$ dB over SDR	–
MAP-SDR turbo (multi)	MAP, each it.	≳full-list turbo	–
Single-SDR turbo	MAP, once	$\leq0.2$ dB loss	$3$–$10\times$ lower

Simulation highlights demonstrate efficient convergence with only one heavy SDR solve per codeword, strong robustness to SNR, and computation-time savings even at modest antenna array sizes, suggesting scalability for modern high-throughput coded MIMO systems (Wang et al., 2018, Wang et al., 2018).