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Energy-Efficient Hybrid Analog and Digital Precoding for mmWave MIMO Systems with Large Antenna Arrays

Xinyu Gao, Linglong Dai, Shuangfeng Han, Chih-Lin I, Robert W. Heath

arXiv:1507.04592v1cs.IT

TL;DR

Large mmWave MIMO arrays need energy-efficient precoding, while fully-connected hybrid architectures require many phase shifters and sub-connected design remains difficult. The paper develops SIC-based sub-connected precoding by decomposing rate optimization across sub-arrays and avoiding SVD and matrix inversion. It reports about 10% of the complexity of spatially sparse precoding with near-optimal simulated rates.

  • Problem

    Large mmWave MIMO arrays motivate hybrid precoding, but fully-connected designs require many phase shifters and sub-connected hybrid-precoding design remains difficult under altered constraints.

  • Method

    The paper decomposes non-convex total-rate optimization into sequential sub-rate problems for individual sub-antenna arrays and develops SIC-based precoding without SVD or matrix inversion.

  • Results

    The proposed method has about 10% of the complexity of spatially sparse precoding and achieves near-optimal performance in simulations.

  • Takeaways & Limitations

    SIC-based sub-connected precoding offers a reported trade-off among achievable-rate performance, energy consumption, and computational complexity.

Abstract

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Millimeter wave (mmWave) MIMO will likely use hybrid analog and digital precoding, which uses a small number of RF chains to avoid energy consumption associated with mixed signal components like analog-to-digital components not to mention baseband processing complexity. However, most hybrid precoding techniques consider a fully-connected architecture requiring a large number of phase shifters, which is also energyintensive. In this paper, we focus on the more energy-efficient hybrid precoding with sub-connected architecture, and propose a successive interference cancelation (SIC)-based hybrid precoding with near-optimal performance and low complexity. Inspired by the idea of SIC for multi-user signal detection, we first propose to decompose the total achievable rate optimization problem with non-convex constraints into a series of simple sub-rate optimization problems, each of which only considers one sub-antenna array. Then, we prove that maximizing the achievable sub-rate of each sub-antenna array is equivalent to simply seeking a precoding vector sufficiently close (in terms of Euclidean distance) to the unconstrained optimal solution. Finally, we propose a low-complexity algorithm to realize SICbased hybrid precoding, which can avoid the need for the singular value decomposition (SVD) and matrix inversion. Complexity evaluation shows that the complexity of SIC-based hybrid precoding is only about 10% as complex as that of the recently proposed spatially sparse precoding in typical mmWave MIMO systems. Simulation results verify the near-optimal performance of SIC-based hybrid precoding.

I. INTRODUCTION

mmWave MIMO motivates hybrid precoding to reduce the energy and hardware burden of large antenna arrays, but sub-connected hybrid-precoding design remains open. The paper proposes SIC-based sub-connected precoding with low complexity and near-optimal performance.

  • Motivation: Hybrid analog and digital precoding reduces the number of RF chains needed for large-antenna mmWave MIMO systems.This addresses the energy consumption and hardware complexity associated with using one RF chain per antenna.
  • Research gap: Sub-connected architectures are expected to be more energy-efficient and practical, but their altered constraints make hybrid-precoding design an open problem.
  • Proposed approach: The paper proposes SIC-based hybrid precoding for the sub-connected architecture.
  • Proposed approach: The total non-convex rate optimization is decomposed into sequential sub-rate problems, each associated with one sub-antenna array.
  • Proposed approach: Each sub-rate problem is solved by finding a precoding vector close in Euclidean distance to the unconstrained optimum.
  • Reported outcome: 10% complexity and near-optimal performance are reported for SIC-based precoding relative to spatially sparse precoding in typical systems.The algorithm avoids SVD and matrix inversion; simulations verify near-optimal performance.
  • Contribution: The authors identify the work as the first study of hybrid-precoding design with a sub-connected architecture.

II. SYSTEM MODEL

The system model contrasts fully-connected and sub-connected hybrid-precoding architectures and specifies the narrowband mmWave channel and array-response assumptions used for analysis.

  • Architectures: Fully-connected architectures connect every RF chain to all BS antennas, whereas sub-connected architectures connect each RF chain to only a subset.
  • Architectures: A fully-connected system requires N^2M phase shifters, while each sub-connected RF chain drives only M antennas.
  • Signal model: In the sub-connected architecture, digital precoding handles N baseband streams and each RF chain applies analog weighting to its associated M-antenna sub-array.
  • Scope: The analysis treats a narrowband system as a first step, while broadband extension is left for future work.
  • Signal model: The hybrid precoder is represented as P = AD and satisfies a Frobenius-norm constraint for total transmit power.
  • Channel model: The channel follows a geometric Saleh–Valenzuela model with a limited number of effective paths and path-dependent gains and angles.
  • Array model: The array-response vectors are specified for uniform linear and uniform planar arrays using antenna spacing, wavelength, and array dimensions.

III. SIC-BASED HYBRID PRECODING FOR MMWAVE MIMO SYSTEMS

The paper presents a low-complexity SIC-based hybrid-precoding algorithm intended to achieve near-optimal performance and evaluates its computational complexity against existing solutions.

  • Algorithm and evaluation: The proposed SIC-based hybrid precoding targets near-optimal performance with low computational complexity.The section also evaluates its complexity relative to current solutions.

A. Structure of SIC-based hybrid precoding

The proposed SIC-based design decomposes the constrained total-rate problem into sequential sub-rate optimizations, exploiting the block-diagonal structure of sub-connected precoding. Each sub-antenna array is optimized in turn while updating the matrix that captures prior arrays' effects.

  • Architecture: The hybrid precoding matrix is block diagonal, with one non-zero precoding vector per sub-antenna array.This follows from the sub-connected architecture and the factorization P = AD.
  • Architecture: Each precoding column has equal-amplitude non-zero elements, reflecting the fixed-amplitude analog precoder and diagonal digital precoder.The digital coefficients provide power allocation while the analog elements have fixed amplitude.
  • Constraints: The Frobenius norm constraint on P enforces the total transmit-power limit.The constraint is ∥P∥F ≤ N.
  • SIC decomposition: Because different sub-antenna arrays are independent under the block structure, the total-rate optimization can be decomposed into one sub-rate problem per array.The original constraints remain non-convex, but the decomposition produces simpler sequential subproblems.
  • SIC decomposition: The SIC procedure optimizes arrays sequentially, updating T_n after each array until the final array is processed.The achievable sub-rate of the nth array appears as the second term in the rate decomposition, while the remaining term retains the previous form.

B. Solution to the sub-rate optimization problem

The sub-rate solution approximates each unconstrained optimum with a feasible constant-modulus precoding vector, then reuses the resulting update for the next sub-antenna array. A lower-complexity implementation avoids the SVD and large matrix inversions required by the direct solution.

  • Sub-rate formulation: The nth sub-rate problem designs only the nth precoding vector while holding the preceding-array structure in the effective problem.The feasible set incorporates the three precoder constraints, and the nth vector occupies only its associated antenna block.
  • Unconstrained solution: The unconstrained optimum is the first right singular vector v1 of the corresponding effective matrix, but it generally violates the equal-amplitude constraint.Thus v1 cannot be used directly as a practical precoding vector.
  • Feasible approximation: Maximizing the sub-rate is equivalent to finding a feasible precoding vector with minimum Euclidean distance to v1.This converts the constrained rate problem into a distance-minimization problem.
  • Feasible approximation: The feasible analog vector matches the phases of v1, while the digital scalar supplies the associated amplitude scaling.The resulting solution has the form described by the phase vector of v1 and the corresponding digital coefficient.
  • Feasibility: The constructed precoders satisfy the total transmit-power constraint across all sub-antenna arrays.The paper states that the resulting solutions obey the required Frobenius-norm bound.
  • Complexity reduction: The direct per-array solution requires an SVD and large matrix inversion, motivating a low-complexity algorithm that avoids both operations.After obtaining one array's solution, the relevant matrix is updated and the same procedure is reused for the next array.

C. Low-complexity algorithm to obtain the optimal solution

The paper replaces explicit SVD and large matrix operations with power iteration and an efficient update scheme to obtain the quantities needed for SIC-based precoding.

  • Power iteration: Power iteration computes the largest singular value and first right singular vector of the Hermitian matrix used in each SIC step.Because the matrix is Hermitian, its singular values and right singular vectors equal its eigenvalues and eigenvectors.
  • Power iteration: The algorithm initializes u(0), repeatedly forms z(s)=Ḡ_n−1u(s−1), extracts the largest-amplitude element, and updates u(s).The iteration runs for a predefined maximum of S iterations.
  • Convergence: When the two largest singular values differ, the iterates converge to the largest singular value and its corresponding right singular vector.The convergence guarantee is stated for Σ1 ≠ Σ2.
  • Acceleration: When Σ1≈Σ2, power iteration may converge slowly, so the method adds Aitken acceleration to increase convergence speed.The paper identifies the singular-value gap as the condition governing convergence rate.
  • Efficient updates: Proposition 2 updates Ḡ_n using a vector outer product rather than complicated matrix-to-matrix multiplication and matrix inversion.This reuses Σ1 and v1 obtained by the power-iteration procedure.

D. Summary of the proposed SIC-based hybrid precoding

Algorithm 2 sequentially optimizes each sub-antenna array using power-iteration outputs, then updates the effective problem for the next array.

  • Algorithm overview: SIC-based hybrid precoding first computes Σ1 and v1 of Ḡ_n−1 with Algorithm 1.These quantities provide the information needed to form the optimal precoding vector for the current sub-antenna array.
  • Sequential optimization: For each sub-antenna array, the algorithm obtains its optimal precoding vector and updates Ḡ_n before proceeding to the next array.The procedure is summarized as Algorithm 2.
  • Extension to combining: The SIC-based approach can also be extended to hybrid combining by alternately determining precoding and combining matrices from the effective channel.The paper notes that a Ping-pong iteration can jointly explore the precoding-combining pair, leaving further discussion for future work.

E. Complexity evaluation

The proposed SIC-based precoding has lower computational requirements than spatially sparse precoding while retaining a substantially simpler update structure.

  • Complexity components: The complexity evaluation counts complex multiplications and divisions across four algorithmic components.These components include initialization, power iteration, optimal-vector acquisition, and effective-matrix updates.
  • Operation counts: SIC-based precoding approximately requires M^2(NS+K) multiplications and 2NS divisions.This expression summarizes the proposed method’s overall operation count.
  • Comparison: 10% as complex as spatially sparse precoding is the reported SIC-based complexity ratio in typical mmWave MIMO systems.The comparison is based on the operation counts reported for the two methods.
  • Typical system: 4×10^3 multiplications and 10^2 divisions are required by SIC-based precoding in the typical N=8, M=8, K=16, L=3 setting with S=5.The iteration count S=5 is stated as sufficient to guarantee the reported performance.
  • Typical system: 5×10^4 multiplications and 10^3 divisions are required by spatially sparse precoding in the same typical setting.The paper compares these counts directly with the SIC-based method.

IV. SIMULATION RESULTS

Simulations show that SIC-based hybrid precoding achieves near-optimal rates while using a sub-connected architecture with favorable energy and complexity trade-offs. Its performance remains close to fully connected benchmarks and can improve with larger antenna dimensions.

  • Achievable-rate performance: 99% of the optimal unconstrained sub-connected precoding rate is achieved in the 64 × 16 system with N = 8 RF chains.The proposed method also outperforms conventional analog precoding across the simulated SNR range.
  • Achievable-rate performance: More than 90% of the spatially sparse precoding rate is achieved at SNR = 0 dB in both simulated configurations.The method is also close to optimal unconstrained and spatially sparse precoding with fully connected architectures.
  • Antenna scaling: Increasing the numbers of BS and user antennas improves performance while keeping the number of RF chains fixed at N = 8.The paper states that this requires much lower energy consumption than increasing energy-intensive RF chains.
  • Antenna scaling: When NM = 64 and N = 8, SIC-based precoding at K = 30 matches spatially sparse precoding at K = 20.The comparison is presented as compensation for performance loss caused by the sub-connected architecture.
  • Overall trade-off: The proposed approach is concluded to offer a favorable trade-off among performance, energy consumption, and computational complexity.Its sub-connected implementation uses fewer phase-shifter connections than fully connected alternatives.

V. CONCLUSIONS

The paper develops SIC-based hybrid precoding for sub-connected mmWave MIMO systems by decomposing the non-convex rate problem and approximating each subproblem through proximity to an unconstrained solution. The resulting algorithm avoids SVD and matrix inversion while retaining near-optimal performance and substantially lower complexity.

  • Proposed approach: The total achievable-rate optimization is decomposed into sub-rate problems, each involving one sub-antenna array.This transforms the original problem with non-convex constraints into a sequence of simpler optimizations.
  • Proposed approach: Each sub-rate problem is equivalent to finding a precoding vector sufficiently close to the unconstrained optimal solution in Euclidean distance.The derivation relates maximizing projection onto the dominant singular vector to minimizing Euclidean distance.
  • Algorithm and complexity: The SIC-based algorithm realizes sub-connected hybrid precoding without SVD or matrix inversion.The construction uses successive interference cancellation to obtain a low-complexity implementation.
  • Algorithm and complexity: The proposed method achieves near-optimal performance and about 10% of the complexity of recently proposed spatially sparse precoding.The comparison is stated for typical mmWave MIMO systems.

APPENDIX B PROOF OF PROPOSITION 2

Appendix B derives an efficient update for the inverse-related matrix quantity used in the SIC procedure. The proof replaces repeated inversion with a rank-one update and uses the proximity of the selected precoding vector to the dominant singular vector.

  • Matrix update: Partitioning P_n as [P_{n−1} p_n] rewrites the matrix requiring inversion into a form suitable for a rank-one update.This reformulation prepares the application of the Sherman–Morrison formula.
  • Matrix update: The Sherman–Morrison formula is used to derive the updated inverse-related expression without directly recomputing a full matrix inverse.The resulting expression provides the recursive quantity needed to obtain Ḡ_n.
  • Approximation: The previously obtained optimal vector p̄^opt_n is approximated by the dominant singular vector v_1 because the two vectors are sufficiently close.This approximation simplifies the subsequent expression for the matrix update.
  • Approximation: The derivation uses the Hermitian property of Ḡ_{n−1} to express its factorization through the dominant singular component.The appendix identifies the corresponding algebraic equality in the proof.
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