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Conjugate Gradient-based Soft-Output Detection and Precoding in Massive MIMO Systems

Bei Yin, Michael Wu, Joseph R. Cavallaro, Christoph Studer

arXiv:1404.0424v2cs.IT

TL;DR

Realistic massive MIMO configurations make near-optimal detection and precoding computationally expensive. The paper uses CG methods and direct SINR tracking for soft-output detection, finding near-exact error-rate performance with lower complexity than exact inversion and better performance and complexity than Neumann-series methods.

  • Problem

    Realistic massive MIMO systems remain computationally demanding because near-optimal detection and precoding require expensive algorithms.

  • Method

    The paper applies CG methods to uplink detection and downlink precoding, computing soft-output SINR information directly within CG.

  • Results

    CG-based detection achieves similar error-rate performance to exact Cholesky detection at lower complexity across the considered configurations.

  • Takeaways & Limitations

    The proposed CG approach is well suited to soft-output detection and precoding in realistic massive MIMO systems.

Abstract

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Massive multiple-input multiple-output (MIMO) promises improved spectral efficiency, coverage, and range, compared to conventional (small-scale) MIMO wireless systems. Unfortunately, these benefits come at the cost of significantly increased computational complexity, especially for systems with realistic antenna configurations. To reduce the complexity of data detection (in the uplink) and precoding (in the downlink) in massive MIMO systems, we propose to use conjugate gradient (CG) methods. While precoding using CG is rather straightforward, soft-output minimum mean-square error (MMSE) detection requires the computation of the post-equalization signal-to-interference-and-noise-ratio (SINR). To enable CG for soft-output detection, we propose a novel way of computing the SINR directly within the CG algorithm at low complexity. We investigate the performance/complexity trade-offs associated with CG-based soft-output detection and precoding, and we compare it to exact and approximate methods. Our results reveal that the proposed method outperforms existing algorithms for massive MIMO systems with realistic antenna configurations.

I. INTRODUCTION

Massive MIMO improves wireless-system spectral efficiency, coverage, and range but remains computationally demanding in realistic configurations. The paper applies CG methods to low-complexity detection and precoding, adding direct SINR computation for soft-output detection.

  • Massive MIMO uses hundreds of base-station antennas to serve relatively few users simultaneously, promising improved spectral efficiency, coverage, and range.
  • Realistic antenna configurations remain far from the large-antenna limit, making near-optimal detection and precoding computationally expensive.
  • Truncated Neumann-series methods reduce complexity but lose error-rate performance when the base-station-to-user antenna ratio approaches one.
  • The paper proposes CG methods for uplink detection and downlink precoding, including exact and approximate SINR computation directly within CG.
  • Soft-output detection computes symbol estimates, equalized channel gains, post-equalization SINRs, and LLRs, while the proposed method avoids explicit computations of the corresponding matrices.
  • The proposed CG approach directly computes the equalized estimate, channel gains, and SINRs, significantly reducing soft-output detection complexity.

B. Downlink System Model and Precoding

The downlink model encodes users’ bits into constellation symbols and applies linear MMSE precoding before transmission. The proposed method computes the precoded vector directly rather than explicitly forming the precoding matrix.

  • The base station maps encoded downlink bits to constellation symbols and forms a transmit vector for all users.
  • Linear MMSE precoding is used to maximize received SINR and mitigate inter-user interference.
  • The precoded vector is normalized to unit power before transmission over the frequency-domain downlink channel.
  • The proposed precoding method directly computes the precoded vector without explicitly forming the precoding matrix, often significantly reducing complexity.

III. CG-BASED DATA DETECTION AND PRECODING

Explicit MMSE matrix formation requires inversion of a U × U matrix, motivating CG-based soft-output detection and precoding that avoid this inversion.

  • Existing linear soft-output detectors and some precoders explicitly form MMSE matrices, requiring inversion of a U × U matrix.
  • The proposed detection and precoding approach avoids explicit matrix inversion to reduce computational complexity in realistic massive MIMO systems.

A. Conjugate Gradient (CG) Basics

CG iteratively solves positive-definite linear systems with low per-iteration complexity and can terminate early near the exact solution. The paper extends CG with in-iteration SINR tracking for soft-output detection.

  • A. Conjugate Gradient (CG) Basics: CG solves linear systems with positive-definite matrices iteratively, avoiding direct computation of A^-1b.
  • A. Conjugate Gradient (CG) Basics: CG can solve the systems underlying linear detection and precoding, but regular CG lacks the SINR information required for LLR computation.
  • A. Conjugate Gradient (CG) Basics: The proposed method computes the necessary SINR information directly within each CG iteration for soft-output detection.
  • B. CG-D: CG-Based Soft-Output Data Detection: Algorithm 1 recursively updates CG vectors and outputs the MMSE estimate together with per-user equalized gains and SINRs.

B. CG-D: CG-Based Soft-Output Data Detection

CG computes MMSE detection iteratively and can provide soft-output information by recursively tracking equalized gains and post-equalization SINRs within the CG procedure. The exact tracker reaches the MMSE quantities at U iterations, while early termination provides lower-complexity approximations.

  • CG-Based Soft-Output Data Detection: Regular CG iteratively solves the MMSE detection problem, producing an estimate that is exact after K = U iterations.Early termination yields an estimate close to the exact solution with lower computational complexity.
  • CG-Based Soft-Output Data Detection: Regular CG alone is unsuitable for soft-output detection because LLR computation requires post-equalization gains and SINRs.Hard-output detection is straightforward but typically causes significant error-rate loss in coded communication systems.
  • CG-Based Soft-Output Data Detection: The proposed method computes the equalized estimate and recursively obtains the CG-equivalent equalization matrix and intermediate soft-output quantities during each iteration.The recursion extracts µi|k and νi|k on the fly from Lk.
  • CG-Based Soft-Output Data Detection: The tracked quantities provide µi|k and ρi|k for computing LLR values at each CG iteration.The post-equalization SINR is formed as ρi|k = µ2_i|k/ν2_i|k.
  • CG-Based Soft-Output Data Detection: At k = U, the SINR-tracking scheme is exact; terminating earlier approximates the LLRs at often significantly lower computational complexity.The exact method recovers µi = µi|U and ρi = ρi|U.

2) Approximate SINR Computation:

The approximate SINR method exploits diagonal dominance of the regularized Gram matrix in massive MIMO to replace matrix recursions with diagonal operations. This reduces the per-iteration computation to U multiplications while retaining an approximation to the soft-output quantities.

  • 2) Approximate SINR Computation:: The exact SINR method requires a U×U matrix multiplication per iteration, motivating a lower-complexity approximation.The proposed approximation is described as very accurate for massive MIMO systems.
  • 2) Approximate SINR Computation:: For massive MIMO, the regularized Gram matrix is diagonally dominant, so its main diagonal D approximates A.The approximation replaces A with D in the recursion for Lk.
  • 2) Approximate SINR Computation:: Because the approximate recursion remains diagonal, it requires only U multiplications per CG iteration.The initialization uses eL1 = α1IU and eLk = 0U×U for k < 1.
  • 2) Approximate SINR Computation:: The diagonal approximation supplies approximate µi|k and ν2_i|k values for estimating the post-equalization SINR.The quantities use diagonal entries of G and eLk.
  • 2) Approximate SINR Computation:: The approximation's computational complexity does not depend on the iteration index k.

D. CG-P: CG-Based Linear Precoding

CG applies directly to downlink precoding by solving the regularized linear system and forming the precoded vector without requiring post-equalization SINR computation. It is exact after U iterations and can approach optimal performance with few iterations in massive MIMO.

  • D. CG-P: CG-Based Linear Precoding: CG solves the downlink precoding system and forms the precoded vector as q = HH_d v.Unlike soft-output detection, precoding via CG is described as straightforward.
  • D. CG-P: CG-Based Linear Precoding: CG precoding requires lower complexity than CG soft-output detection because it does not require post-equalization SINR computation.
  • D. CG-P: CG-Based Linear Precoding: At K = U, the CG algorithm performs exact MMSE precoding.
  • D. CG-P: CG-Based Linear Precoding: For K = 1, CG precoding becomes matched-filter precoding, and few iterations are sufficient for near-optimal performance in massive MIMO systems.The large-antenna limit is cited as making matched-filter precoding optimal.

E. What About CG Least Squares?

CGLS avoids forming the regularized Gram matrix and can be used for both uplink detection and downlink precoding. Its complexity may be lower than CG, but its advantage is limited to very few iterations where performance is typically sub-optimal.

  • E. What About CG Least Squares?: CGLS handles nonsquare matrices, avoiding computation of the regularized Gram matrix.This enables CGLS for uplink detection and also for precoding.
  • E. What About CG Least Squares?: CGLS can potentially require lower complexity than CG because it avoids Gram-matrix computation.
  • E. What About CG Least Squares?: CGLS is advantageous over CG only for a very small number of iterations, when both methods typically deliver sub-optimal error-rate performance.

F. Other Low-Complexity Detection and Precoding Methods

The paper reviews Cholesky decomposition as an exact, computationally efficient matrix-inversion method for uplink detection and downlink precoding. It obtains the inverse through Gram-matrix factorization followed by forward and backward substitution.

  • Cholesky decomposition exactly computes the matrix inverse required for uplink detection and downlink precoding.It is described as computationally efficient relative to general inversion procedures.
  • The regularized Gram matrix is formed separately for the uplink or downlink case before factorization.The matrix is denoted A after omitting the uplink and downlink subscripts.
  • The factorization writes A = MMH, with M lower-triangular.This structure supports the subsequent substitution steps.
  • Forward and backward substitution yield A−1 = (MMH)−1 without directly performing a general matrix inversion.

2) Approximate Matrix Inversion:

The evaluation compares truncated Neumann-series inversion with CG- and CGLS-based soft-output detection and CG-based precoding. Across realistic antenna configurations, CG-based methods offer lower-complexity operation while retaining stronger error-rate performance than Neumann inversion.

  • Approximate Matrix Inversion: For K ≤3, the Neumann approximation requires very low complexity.
  • Computational Complexity Analysis: In 32×8 and 128×8 systems, CG-based soft-output detection is less complex than Cholesky inversion when k ≤5.
  • Computational Complexity Analysis: In 32×16 and 128×16 systems, CG-based detection is less complex than Cholesky inversion when k ≤12.
  • Computational Complexity Analysis: At matching performance, CG-based detection has lower complexity than exact inversion, while CGLS and Neumann are cheaper only for small iteration counts.The comparison counts real-valued multiplications and exploits Hermitian structure and zero entries.
  • Block Error-Rate (BLER) Performance: CG-based detection approaches Cholesky BLER at K = 5, 3, 8, and 4 for 32×8, 128×8, 32×16, and 128×16 configurations, respectively.Each case also has lower associated complexity than exact inversion; CGLS has the same outputs and performance as CG.
  • Block Error-Rate (BLER) Performance: The Neumann approach exhibits a high error floor in the considered configurations but works well for 128×8, where the BS-to-user antenna ratio is large.
  • Performance/Complexity Trade-offs: CG-based precoding achieves lower complexity at similar error-rate performance to the exact Cholesky-based precoder.

C. Performance/Complexity Trade-offs

The proposed CG approach is evaluated as a low-complexity solution for soft-output detection and precoding. It approaches exact-method error-rate performance with lower complexity and improves on the approximate Neumann approach in the considered massive MIMO settings.

  • Performance/Complexity Trade-offs: CG-based soft-output detection achieves lower complexity and similar error-rate performance to exact Cholesky detection across all considered cases.The trade-off uses the minimum SNR required to achieve 10% BLER.
  • Performance/Complexity Trade-offs: Complexity savings at equal SNR performance are more pronounced for configurations with U = 16 user antennas.
  • Performance/Complexity Trade-offs: CG-based precoding likewise achieves lower complexity with similar error-rate performance to exact Cholesky-based precoding.
  • Conclusions: The proposed approach combines CG methods with post-equalization SINR tracking to support low-complexity soft-output detection.The conclusions identify this tracking scheme as crucial for the detector.
  • Conclusions: For reasonably large BS-to-user antenna ratios, CG detection and precoding quickly converge toward exact-method performance while requiring often significantly lower complexity.
  • Conclusions: The CG approach outperforms the approximate Neumann-series inversion in both performance and complexity.
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