PaperlayerView exact revision on arXiv

Source-linked AI summary

Dynamic Subarrays for Hybrid Precoding in Wideband mmWave MIMO Systems

Sungwoo Park, Ahmed Alkhateeb, Robert W. Heath

arXiv:1606.08405v1cs.IT

TL;DR

Wideband frequency-selective mmWave systems require hybrid precoding beyond the narrowband and fully-connected settings emphasized by prior work. The paper derives closed-form designs for fully-connected and partially-connected architectures, then constructs dynamic subarrays from long-term channel statistics; the designs approach fully-digital spectral efficiency, and dynamic subarrays outperform fixed structures.

  • Problem

    Prior hybrid precoding focused mainly on narrowband channels, although mmWave systems are expected to be wideband and frequency selective.

  • Method

    The paper derives a near-optimal closed-form solution for fully-connected and partially-connected OFDM-based hybrid precoding, then designs dynamic subarrays using long-term channel statistics.

  • Results

    The proposed designs approach fully-digital spectral efficiency in wideband mmWave channels, while dynamic subarrays outperform fixed subarray architectures.

  • Takeaways & Limitations

    Dynamic subarray structures are promising for wideband mmWave systems because their performance exceeds that of fixed subarray structures in the reported conditions.

Abstract

from arXiv · show

Hybrid analog/digital precoding architectures can address the trade-off between achievable spectral efficiency and power consumption in large-scale MIMO systems. This makes it a promising candidate for millimeter wave systems, which require deploying large antenna arrays at both the transmitter and receiver to guarantee sufficient received signal power. Most prior work on hybrid precoding focused on narrowband channels and assumed fully-connected hybrid architectures. MmWave systems, though, are expected to be wideband with frequency selectivity. In this paper, a closed-form solution for fully-connected OFDM-based hybrid analog/digital precoding is developed for frequency selective mmWave systems. This solution is then extended to partially-connected but fixed architectures in which each RF chain is connected to a specific subset of the antennas. The derived solutions give insights into how the hybrid subarray structures should be designed. Based on them, a novel technique that dynamically constructs the hybrid subarrays based on the long-term channel characteristics is developed. Simulation results show that the proposed hybrid precoding solutions achieve spectral efficiencies close to that obtained with fully-digital architectures in wideband mmWave channels. Further, the results indicate that the developed dynamic subarray solution outperforms the fixed hybrid subarray structures in various system and channel conditions.

I. INTRODUCTION

The paper addresses wideband frequency-selective mmWave hybrid precoding, extending beyond prior narrowband and fully-connected assumptions with closed-form designs and dynamic subarrays. Its simulations show performance approaching fully-digital precoding, while dynamic subarrays outperform fixed structures.

  • Motivation and prior work: Prior hybrid precoding work primarily targeted narrowband channels, whereas mmWave systems are expected to operate over frequency-selective wideband channels.
  • Proposed designs: The paper develops near-optimal closed-form hybrid precoding for fully-connected and partially-connected architectures in OFDM-based wideband mmWave systems.The design relaxes the mutual-information maximization problem and obtains baseband and RF precoders.
  • Proposed designs: The proposed solution has exactly the same spectral efficiency as unconstrained fully-digital precoding when the number of channel paths is below the number of RF chains.
  • Dynamic subarrays: The paper proposes grouping antennas into subarrays using a criterion tied to a proxy for system spectral efficiency and long-term channel statistics.A greedy algorithm reduces the complexity of finding antenna partitions compared with exhaustive search.
  • Results: Simulation results show wideband hybrid designs approach fully-digital spectral efficiencies for fully-connected and fixed-subarray architectures.
  • Results: Dynamic subarrays outperform fixed subarray structures across the reported system and channel conditions.The proposed structure adapts antenna groupings according to long-term channel statistics.

II. SYSTEM AND CHANNEL MODELS

The paper models wideband hybrid precoding for a mmWave MIMO-OFDM link, combining frequency-flat analog precoding with per-subcarrier digital precoding over a geometric, limited-scattering channel.

  • System Model: The transmitter uses N_TX antennas, N_RF RF chains, and S streams to communicate with a mobile user having N_RX antennas.
  • System Model: The transmitted signal at subcarrier k is x[k] = F_RF F_BB[k] s[k], where F_RF is wideband and F_BB[k] is subcarrier-specific.
  • System Model: Analog precoding is fixed across the bandwidth, whereas digital baseband precoding and receiver combining operate per subcarrier.
  • System Model: The model imposes a total transmit-power constraint and assumes perfect synchronization, with cyclic-prefix removal before subcarrier processing.
  • Channel Model: The channel follows a geometric wideband model with N_CH paths, path gains, delays, and transmit/receive azimuth and elevation angles.
  • Channel Model: The frequency response H[k] is constructed from transmit and receive array responses and frequency-dependent path contributions.

III. PROBLEM FORMULATION

The paper formulates hybrid precoding as a mutual-information maximization under a total-power constraint, then reduces the design to optimizing the RF precoder while solving digital precoding per subcarrier.

  • III. PROBLEM FORMULATION: The objective is to design the base-station hybrid analog and digital precoders for the wideband channel.
  • III. PROBLEM FORMULATION: The formulation maximizes mutual information for Gaussian transmitted symbols subject to a total power constraint.
  • III. PROBLEM FORMULATION: The main difficulty is coupling between the RF and baseband precoders in the power constraint.
  • H [k] F_RF(F∗: A change of variables produces an equivalent relaxed problem that avoids direct optimization over F_RF.
  • III. PROBLEM FORMULATION: For a fixed RF precoder, the digital precoder is obtained from the effective-channel SVD and water-filling power control.
  • III. PROBLEM FORMULATION: Because the optimal baseband precoders depend only on H[k] and F_RF, the optimization can be rewritten over F_RF alone.
  • H [k] F_RF(F∗: The relaxed problem is non-convex and difficult to solve, so the paper maximizes the sum of squared singular values of the effective channels.
  • H [k] F_RF(F∗: The relaxation is evaluated for practical wideband mmWave parameters, while hardware constraints such as constant modulus and quantized phases are addressed separately.

IV. WIDEBAND HYBRID PRECODING DESIGN FOR FULLY-CONNECTED ARCHITECTURES

For fully-connected architectures, the paper derives a closed-form RF precoder from the dominant covariance eigenvectors and shows that it can match fully-digital spectral efficiency when enough RF chains are available.

  • Architecture: The fully-connected architecture connects every RF chain to all N_TX antennas.
  • Closed-Form Solution: The relaxed objective is expressed using the sample covariance matrix R and its dominant eigenvectors.
  • Closed-Form Solution: The optimal RF precoder has the form F_RF = V_R A, where A is any full-rank NRF × NRF matrix and unitary factors reflect invariance.
  • Optimality: The resulting hybrid solution achieves the same spectral efficiency as the fully-digital architecture when NRF ≥ NCH.
  • Optimality: For sparse mmWave channels, the covariance rank is at most the number of channel paths, which is typically smaller than the antenna count.
  • Optimality: NRF ≥ NCH lets the hybrid precoders represent the fully-digital SVD solution using F_RF = V_R A and a corresponding subcarrier-dependent baseband precoder.
  • RF Constraints: Under RF constraints, the unconstrained design is approximated by retaining each RF coefficient’s phase, motivated by DFT-like singular vectors for large uniform arrays.
  • RF Constraints: Simulations assess whether this constrained approximation is effective for wideband mmWave channels with practical parameters.

V. WIDEBAND HYBRID PRECODING DESIGN FOR FIXED SUBARRAY ARCHITECTURES

For fixed subarray architectures, each RF chain serves a disjoint antenna subset, and the optimal analog beam for that subset is determined by its dominant covariance singular vector.

  • Architecture: The fixed subarray architecture partitions the antennas into subsets, with each RF chain connected only to its assigned subset.
  • Architecture: The analog RF precoding matrix is block diagonal, with one N_sub × 1 beamforming vector for each RF chain.
  • Comparison: Unlike the fully-connected architecture, the fixed design restricts each RF chain to its predefined antenna subset.
  • Subarray Channels: The overall channel is represented through the channel matrices of the individual subarrays.
  • Optimal Structure: Proposition 2 derives the optimal hybrid precoder structure for the fixed subarray architecture.
  • Optimal Structure: For subarray r, the RF beam is proportional to the largest singular vector of its associated channel covariance matrix.
  • Effective Channel: The resulting effective channel and objective decompose according to the RF-chain-specific subarray beams.

H [k] FRF(F∗

The paper derives optimal analog beamforming structures for fixed subarrays and formulates dynamic subarray selection as a combinatorial optimization problem. Because exhaustive partition search and exact singular-value computation are costly, it develops a tractable approximation and a low-complexity relocation algorithm.

  • Fixed subarray precoding: Each RF-chain analog beamforming vector is proportional to the largest singular vector of its associated covariance submatrix.
  • Fixed subarray precoding: The maximum objective for fixed subarrays equals the sum of the largest singular values of the NRF covariance submatrices.
  • Dynamic subarray formulation: Unlike fully-connected precoding, the fixed-subarray objective depends on how antennas are partitioned among the RF chains, motivating dynamic subarrays.
  • Dynamic subarray formulation: Dynamic partitioning assigns every antenna to exactly one nonempty subset, while allowing different RF-chain subarrays to have different cardinalities.
  • Complexity and approximation: The proposed method replaces numerical largest-singular-value evaluation with a normalized Minkowski ℓ1 approximation and repeatedly relocates antennas when the approximate objective increases.

VII. SIMULATION RESULTS

The simulations evaluate wideband hybrid precoding under clustered frequency-selective mmWave channels using realistic angular, delay, antenna, and OFDM settings. The study compares the proposed dynamic-subarray algorithm across simulated channel and array configurations.

  • Evaluation scope: The simulations evaluate the proposed wideband hybrid precoding design and the dynamic subarray algorithm under the Section II channel model.
  • Channel model: The evaluation uses a clustered channel model with 8 clusters and multiple subrays per cluster, following 3GPP 3D-MIMO and WINNER II-like delay and angle distributions.
  • Channel model: Azimuth angles are uniform over [−180°, 180°], elevation angles over [−90°, 90°], and each cluster contains 10 subrays with 5° Laplacian angular spread.
  • Simulation setup: Both ULA and UPA arrays are simulated with antenna spacing of 0.5λ.

A. Evaluating the Relaxation of the Optimum Criterion

This section tests whether the relaxed optimization criterion accurately approximates the original criterion. The approximation becomes increasingly close as the transmit-to-receive antenna ratio grows, including in sparse mmWave channels.

  • Evaluation setup: The simulations use the relaxed optimum criterion instead of the exact optimum criterion.
  • Bound behavior: Jensen’s upper bound is tight at low SNR and can also be tight at high SNR when transmit antennas outnumber receive antennas.
  • IID Rayleigh channel: With two receive antennas, the gap between the bound and exact value decreases as the transmit-to-receive antenna ratio increases.
  • Sparse mmWave channel: The gap is approximately 1 bps/Hz when more than 16 transmit antennas are deployed, supporting the relaxed problem for large MIMO mmWave systems.
  • Sparse mmWave channel: The approximation is also characterized for an 8-cluster, 10-subray sparse mmWave channel model.

B. Wideband Hybrid Precoding over Frequency Selective Channels

The paper evaluates wideband hybrid precoding against fully digitalized baseband precoding in rich-scattering and sparse mmWave channels. Performance differs sharply by channel structure: substantial loss appears in IID Rayleigh fading, whereas the gap is negligible in the sparse mmWave case with eight RF chains.

  • Evaluation setup: The study measures average mutual information per subcarrier versus SNR for 16 transmit antennas and 4 receive antennas.
  • IID Rayleigh channel: In the IID Rayleigh channel, the proposed hybrid precoding has substantial loss relative to fully digitalized baseband precoding even with eight RF chains.
  • Sparse mmWave channel: The sparse-channel comparison uses the 8-cluster, 10-subray mmWave model.
  • Sparse mmWave channel: When each cluster has one ray and the number of clusters does not exceed the number of RF chains, hybrid and fully-digital precoding have the same performance.
  • Sparse mmWave channel: With 5° angular spread and multiple subrays, the performance gap from fully-digital precoding is negligible when eight RF chains are used.

C. Wideband Hybrid Precoding with Dynamic Subarray Structures

The proposed dynamic subarray technique selects hybrid subarray structures using channel characteristics and outperforms fixed alternatives across ULA and UPA settings. Its advantage grows with larger arrays and RF-chain counts, while the best fixed structure depends on angular channel distributions.

  • ULA systems: The dynamic subarray algorithm approaches exhaustive-search performance with much lower complexity in the 9-antenna ULA setting.The comparison uses 9 antennas and 3 RF chains at the base station, with adjacent and interlaced fixed subarrays as baselines.
  • ULA systems: Among the two fixed ULA structures, adjacent subarrays outperform interlaced subarrays under correlated channels.The paper attributes this to larger singular values for adjacent subarrays.
  • UPA systems: For fixed UPA structures, the squared type is best and the vertical type is second; the proposed dynamic algorithm’s gain increases with array and RF-chain size.The fixed-structure ranking is also reflected in the selection ratios of the predefined-set dynamic method.
  • Impact of channel parameters: The dynamic subarray gain increases as the range of azimuth and elevation angles becomes wider.The objective function is the sum of dominant singular values across subarrays.
  • Impact of channel parameters: The best fixed structure varies with angular spread: horizontal is best for narrow azimuth ranges, vertical for narrow elevation ranges, and squared for larger ranges.These trends follow changes in the dominant singular values of horizontal-row and vertical-column covariance matrices.

VIII. CONCLUSIONS

The paper develops wideband hybrid precoding solutions and a dynamic subarray technique based on long-term channel statistics. Simulations show performance approaching fully-digital precoding and outperforming fixed subarray architectures.

  • The paper develops hybrid analog/digital precoding designs for wideband mmWave MIMO-OFDM systems with frequency selectivity.
  • A near-optimal closed-form solution is derived for fully-connected and partially-connected hybrid architectures by relaxing the mutual information maximization problem.
  • The proposed wideband hybrid precoding designs achieve spectral efficiencies approaching those of fully-digital precoding.
  • A dynamic subarray technique adapts antenna partitions according to long-term channel statistics, using a criterion and antenna partitioning algorithm.
  • Grouping more correlated antenna elements into each subarray normally leads to an efficient subarray structure.
  • Dynamic subarrays outperform fixed subarray architectures in the reported simulations, while their spectral-efficiency and energy trade-off remains future work.

APPENDIX A

The appendix derives bounds and approximations for the largest singular value using correlation magnitudes, then relates the approximate value to exact-value bounds.

  • The appendix normalizes the diagonal elements of R_S and defines γ_k as magnitudes of selected off-diagonal entries.
  • The lower and upper bounds for the exact quantity are rewritten using the defined correlation magnitudes.
  • An approximate largest singular value is represented and compared with the exact value through a lower-bound relationship.
  • The bound relationship uses γ_k ≥ 0, which follows directly from the definition of γ_k as a magnitude.
  • The appendix separately analyzes the ratio between the upper bound of the exact value and the approximate singular-value quantity.
Loading 1606.08405v1…