Capacity-Region-Achieving Sparse Regression Codes for MIMO Multiple-Access Channels

1. Problem Statement

The paper addresses the challenge of achieving the capacity region of MIMO Multiple-Access Channels (MIMO-MAC) using practical coding schemes.

• Context: While Sparse Regression (SR) codes have been proven to achieve the capacity of point-to-point AWGN channels and Gaussian MACs using Approximate Message Passing (AMP) or Orthogonal AMP (OAMP) decoders, their application to MIMO-MAC has been limited.
• Limitations of Existing Solutions:
  • Standard AMP-type algorithms require observation matrices with specific statistical properties (e.g., i.i.d. Gaussian or right-unitary invariance) that are rarely met in practical MIMO channels (which often involve multipath and static block structures).
  • Existing capacity-achieving schemes for MIMO often rely on Superposition Coding Modulation (SCM), which requires designing multiple layers of distinct codes and performing Successive Interference Cancellation (SIC). This approach is complex and difficult to implement.
  • There is a lack of a coding scheme that uses a single SR code per user while decoding via Parallel Interference Cancellation (PIC) in MIMO environments to achieve the full capacity region.

2. Methodology

The authors propose a novel coding framework that integrates Sparse Regression (SR) codes with a Multiple-Access Orthogonal AMP (MA-OAMP) receiver.

A. System Model

• Transmitter: Each of the $U$ users transmits an SR codeword $x_u$. The codeword is generated by applying a random semi-unitary dictionary matrix $\Xi_u$ to a position-modulated signal $s_u$. The signal $s_u$ consists of independent sections with exactly one non-zero entry per section (power allocation $p_{u,l}$).
• Channel: The received signal is $y = \sum H_u x_u + n$, where $H_u$ represents the MIMO channel matrix. The dictionary matrices are chosen (e.g., Haar, permutation invariant) to ensure the equivalent channel $A_u = H_u \Xi_u$ belongs to a universality class where AMP-type algorithms achieve Bayes-optimal performance.
• Receiver: The MA-OAMP receiver is employed. Unlike standard OAMP which tracks a single scalar state, MA-OAMP tracks a $U$-dimensional vector of variances, enabling robust multi-user detection.

B. Theoretical Framework

1. State Evolution (SE) & Decoupling: The authors utilize the state evolution of MA-OAMP to establish a "decoupling principle." This implies that the effective channel seen by each user behaves like an additive white Gaussian noise (AWGN) channel with a specific noise variance $\tau_u$.
2. Achievable Rate Suprema: They derive an analytical integral representation for the maximum achievable rate ($R^{sup}_u$) for each user. This rate is determined by the intersection of:
   • The Variational Transfer Function (VTF) of the MA-OAMP receiver.
   • The Minimum Mean Square Error (MMSE) function of the input signal (specifically, the inverse of the Gaussian MMSE function).
3. Optimal Coding Principle: To achieve the capacity region, the paper establishes a principle where the signal's MMSE function must be designed such that it avoids fixed points with the VTF. Specifically, the inverse MMSE function $\hat{\phi}^{-1}_u(\xi)$ must satisfy:
   $$\hat{\phi}^{-1}_u(\xi) \lesssim \min(F_u(\xi), \hat{\phi}^{-1}_{Gau}(\xi))$$
   This ensures asymptotically error-free detection.

C. Code Design via Power Allocation

The core contribution is the design of SR codes that satisfy the optimal coding principle through power allocation over the position-modulated signals:

• Asymptotic Regime: An iterative power allocation scheme (Theorem 1) is derived to match the target rate tuple on the dominant face of the capacity region. This involves solving for power levels $p_{u,k}$ that align the signal's MMSE curve with the VTF.
• Finite-Length Optimization: Recognizing that asymptotic results may not hold for finite block lengths, the authors propose a numerical optimization (Problem 19) to minimize the Maximum A Posteriori (MAP) Section Error Rate (SER). This optimization adjusts power allocation to operate slightly above the limit SNR to ensure reliability in practical scenarios.
• Capacity Region Achievement: By combining these codes with time sharing, the framework can achieve:
  • The Sum Capacity.
  • The vertices of the dominant face (via specific variance paths/SIC ordering).
  • The entire capacity region (via convex combinations of vertices).

3. Key Contributions

1. Single-Code Framework: Proposes the first capacity-region-achieving scheme for MIMO-MAC that uses a single SR code per user without complex multi-layer superposition coding.
2. MA-OAMP Integration: Demonstrates that MA-OAMP, when paired with random semi-unitary dictionaries, enables Bayes-optimal Parallel Interference Cancellation (PIC) in MIMO settings.
3. Theoretical Characterization: Establishes the Optimal Coding Principle (Eq. 13) linking the signal's MMSE function to the receiver's VTF, proving that codes satisfying this principle achieve the entire capacity region.
4. Practical Power Allocation: Develops both an asymptotic iterative power allocation algorithm and a finite-length optimization strategy to realize these theoretical gains in practice.

4. Numerical Results

The authors validate their framework through simulations on a MIMO-MAC with Tapped Delay Line (TDL-A) channels:

• Setup: 2-user 2x2 MIMO-MAC, 5G-NR LDPC codes used as a benchmark.
• Performance:
  • The proposed Optimized Power Allocation scheme achieves a Bit Error Rate (BER) below $10^{-6}$ within 2.8 dB of the theoretical limit SNR.
  • Comparison: Conventional schemes (Average PA, Exponential PA) and 5G-NR LDPC codes failed to decode successfully under the same asymmetric channel conditions.
  • Robustness: The proposed SR codes significantly outperform 5G-NR LDPC codes in MIMO-MAC environments, demonstrating near-capacity-region-achieving performance.

5. Significance

• Bridging Theory and Practice: This work bridges the gap between theoretical capacity limits and practical coding for complex MIMO-MAC environments, moving beyond the restrictive assumptions of i.i.d. Gaussian channels.
• Simplification: By eliminating the need for multi-layer superposition coding and complex SIC structures, the proposed framework offers a more implementable path to high spectral efficiency in future wireless systems (e.g., 6G).
• Universality: The use of semi-unitary dictionaries allows the coding scheme to be robust across various channel realizations, making it highly relevant for real-world multipath MIMO scenarios.
• Benchmarking: The results suggest that SR codes with MA-OAMP decoding could potentially outperform current 5G standards (LDPC) in multi-user MIMO scenarios, offering a strong candidate for next-generation communication standards.