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Frequency Selective Hybrid Precoding for Limited Feedback Millimeter Wave Systems

Ahmed Alkhateeb, Robert W. Heath

arXiv:1510.00609v4cs.IT

TL;DR

Acquiring instantaneous channel state information at the transmitter is difficult in large mmWave systems because of high channel dimensionality. The paper derives optimal and efficient hybrid precoding designs, including a Gram-Schmidt-based greedy algorithm, which improve over prior work while remaining close to unconstrained perfect-channel performance.

  • Problem

    Acquiring instantaneous channel state information at the transmitter is difficult in large mmWave systems due to high channel dimensionality.

  • Method

    The paper derives optimal hybrid precoding for a given RF codebook, develops efficient analog/digital codebooks, and proposes a Gram-Schmidt-based greedy hybrid precoding algorithm.

  • Results

    The proposed designs improve over prior work and remain within a small gap from unconstrained perfect-channel performance.

  • Takeaways & Limitations

    The approach can enhance the feasibility of limited feedback in mmWave systems.

  • Takeaways & Limitations

    The supported scope centers on enhancing limited-feedback feasibility in mmWave systems.

Abstract

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Hybrid analog/digital precoding offers a compromise between hardware complexity and system performance in millimeter wave (mmWave) systems. This type of precoding allows mmWave systems to leverage large antenna array gains that are necessary for sufficient link margin, while permitting low cost and power consumption hardware. Most prior work has focused on hybrid precoding for narrowband mmWave systems, with perfect or estimated channel knowledge at the transmitter. MmWave systems, however, will likely operate on wideband channels with frequency selectivity. Therefore, this paper considers wideband mmWave systems with a limited feedback channel between the transmitter and receiver. First, the optimal hybrid precoding design for a given RF codebook is derived. This provides a benchmark for any other heuristic algorithm and gives useful insights into codebook designs. Second, efficient hybrid analog/digital codebooks are developed for spatial multiplexing in wideband mmWave systems. Finally, a low-complexity yet near-optimal greedy frequency selective hybrid precoding algorithm is proposed based on Gram-Schmidt orthogonalization. Simulation results show that the developed hybrid codebooks and precoder designs achieve very good performance compared with the unconstrained solutions while requiring much less complexity.

I. INTRODUCTION

Wideband mmWave systems need hybrid precoding and limited-feedback designs because large arrays improve received power, while hardware constraints and high-dimensional channel knowledge complicate conventional precoding. The paper derives an optimal benchmark, develops codebooks, and proposes a low-complexity greedy design for frequency-selective systems.

  • Large antenna arrays are beneficial at mmWave frequencies for guaranteeing sufficient received signal power.
  • High cost and power consumption constrain mixed-signal hardware for high-data-rate mmWave precoding.
  • Instantaneous transmitter channel-state information is difficult to acquire in large mmWave systems because of high channel dimensionality.
  • Prior hybrid precoding work largely assumed narrowband channels with perfect or partial transmitter channel knowledge, while broadband spatial multiplexing remains important.
  • The paper derives an optimal design for a quantized RF codebook, develops efficient codebooks, and proposes Gram-Schmidt greedy precoding for limited-feedback frequency-selective systems.
  • The proposed codebooks and algorithms are evaluated by numerical simulations against digital-only and unconstrained digital precoding schemes.

II. SYSTEM AND CHANNEL MODELS

The system uses OFDM hybrid precoding with a frequency-flat analog stage and subcarrier-dependent digital processing over a geometric wideband mmWave channel. Its model includes quantized constant-modulus RF hardware, RF combining, and total-power or unitary-power constraints.

  • The OFDM transmitter applies an NRF × NS digital precoder per subcarrier before RF precoding and IFFT processing.
  • The RF precoding matrix is shared across all subcarriers, whereas baseband precoders can vary by subcarrier.
  • Analog RF precoder entries have constant modulus and quantized phase-shifter angles selected from a finite set.
  • The model considers total-power and unitary-power constraints, with the latter enforcing equal power across subcarriers and data streams.
  • At the receiver, RF combining precedes cyclic-prefix removal and FFT processing, followed by subcarrier-specific digital combining.
  • The wideband channel uses clusters with delays, arrival and departure angles, multiple rays, path gains, path loss, and pulse shaping.

III. PROBLEM STATEMENT

The paper formulates limited-feedback wideband hybrid precoding as codebook selection that maximizes achievable mutual information under RF and power constraints. It derives optimal baseband designs for a given RF codebook, reducing the coupled design to RF precoder selection in key cases.

  • The paper focuses on a downlink system with limited feedback, perfect channel knowledge at the mobile station, and nearest-neighbor decoding based on the received signal.
  • The objective is to construct RF and baseband precoding codebooks that maximize achievable mutual information in limited-feedback wideband hybrid architectures.
  • For a given RF codebook, the paper derives optimal baseband precoders under total and unitary power constraints.
  • The design is non-trivial because RF hardware imposes non-convex constraints and the analog and digital precoders are coupled through the power constraint.
  • The resulting mutual information is determined only by RF precoder design, so exhaustive RF-codebook search is sufficient to find the maximum achievable rate.

B. Unitary Power Constraint

Under a unitary power constraint, the paper derives the optimal hybrid precoding structure for any RF codebook and uses it to benchmark heuristic designs and guide codebook construction.

  • The optimal baseband precoder decomposes into an RF-dependent matrix and a semi-unitary equivalent baseband precoder.
  • Achieving the optimum requires the RF codeword index solving the outer problem and the optimal equivalent baseband matrix for each subcarrier.
  • The optimal hybrid precoding rate for any given RF codebook provides a benchmark for evaluating heuristic or iterative algorithms and estimating possible improvement.
  • The optimal mutual information depends only on the RF codebook, which directly informs RF codebook design.
  • The proposed greedy frequency-selective algorithm uses Gram-Schmidt orthogonalization to avoid exhaustive RF-codebook search while providing near-optimal performance.

V. CODEBOOK DESIGN FOR FREQUENCY SELECTIVE HYBRID PRECODING

The paper develops limited-feedback RF and hybrid codebooks for wideband frequency-selective mmWave channels by exploiting optimal-precoder structure and minimizing Grassmannian distortion.

  • NS = NRF: For NS = NRF, the hybrid codebook design can be reduced to an RF codebook design problem using the optimal hybrid precoder structure.
  • Distortion criterion: The codebook criterion minimizes average squared chordal distance between a representative Grassmannian point and frequency-dependent dominant subspaces.
  • RF codebook construction: RF hardware constraints make exact codebook optimization non-trivial because constant-modulus entries and phase quantization impose non-convex constraints.
  • RF codebook construction: Algorithm 1 is a Lloyd-type procedure that first minimizes wideband channel distortion without RF constraints, then approximates the resulting codewords under RF hardware constraints.
  • RF codebook construction: The algorithm requires channel-parameter statistics rather than closed-form channel-matrix distributions to construct the RF codebook.

B. Case 2: NS < NRF

When NS < NRF, quantizing the equivalent baseband precoders preserves the optimal precoder structure while requiring sequential RF and baseband codebook design.

  • When NS < NRF, mutual information depends on the equivalent baseband precoders, so RF and baseband precoders must both be quantized.
  • The proposed design quantizes equivalent baseband precoders instead of the original baseband precoders, following the optimal structure from the earlier analysis.

2 F∗ RFH∗[k]

The codebook design minimizes an upper bound on limited-feedback distortion by separately designing RF and equivalent baseband codebooks. The resulting construction uses generalized chordal-distance criteria and a unitary baseband structure.

  • Hybrid Codebook Construction: Regardless of the RF codebook, the optimal equivalent baseband codebook is unitary under a unitary hybrid precoding constraint.This statement applies to the case k=1 identified in the cited passage.
  • Hybrid Codebook Design Criterion: The distortion decomposes into RF-codebook loss and additional equivalent-baseband quantization loss.This decomposition decouples the two codebook effects for design.
  • Hybrid Codebook Construction: RF-codebook distortion is bounded using generalized chordal distance between subspaces of different dimensions.The bound considers the N_S dominant right singular vectors of the channel and large-mmWave-MIMO approximations.
  • Hybrid Codebook Design Criterion: The codebook objective minimizes upper bounds on the combined distortion D(F_RF) + D(G_BB | F_RF).The bounds are derived separately for RF and equivalent baseband quantization.
  • Hybrid Codebook Construction: The RF and equivalent baseband codebooks are designed sequentially using modified versions of Algorithm 1.RF design replaces chordal distance with generalized chordal distance, while baseband construction omits the RF-approximation step.
  • Hybrid Codebook Construction: The sequentially designed hybrid codebooks achieve good performance compared with the perfect-channel-knowledge case.The cited passage states that this result is shown in Section VII.

VI. GRAM-SCHMIDT BASED GREEDY HYBRID PRECODING

The section develops a greedy frequency-selective hybrid precoding approach to avoid the exhaustive RF-codebook search required by direct greedy hybrid precoding. Its Gram-Schmidt-based version can match the direct algorithm exactly.

  • Motivation and Direct Greedy Design: The optimal RF precoder requires exhaustive search over RF-codebook matrix codewords.This search can be highly complex for large antenna systems.
  • Gram-Schmidt Based Design: The proposed greedy frequency-selective hybrid precoding algorithm is based on Gram-Schmidt orthogonalization.It is motivated by the optimal baseband precoder structure.
  • Motivation and Direct Greedy Design: Direct greedy hybrid precoding selects RF beamforming vectors iteratively to maximize mutual information.The algorithm performs N_RF iterations while choosing unique codewords from the RF vector codebook.
  • Optimality Result: Proposition 4 states that the direct greedy and Gram-Schmidt-based hybrid precoding algorithms achieve exactly equal mutual information.The paper presents this equality as an optimality statement for the proposed algorithm.

A. Gram-Schmidt Based Greedy Hybrid Precoding

The Gram-Schmidt hybrid precoder orthogonalizes candidate RF beams before selecting them, preserving the direct greedy algorithm’s mutual information while enabling lower-complexity updates.

  • Gram-Schmidt Selection: The mutual-information gain from a candidate RF beam comes from its component orthogonal to the existing RF precoding matrix.This motivates explicit orthogonalization during greedy selection.
  • Gram-Schmidt Selection: The GS-HP algorithm projects candidate beamforming codewords onto the orthogonal complement of the previously selected RF subspace.The projection is incorporated at each iteration.
  • Complexity Reduction: Gram-Schmidt orthogonalization allows the iteration matrix to be represented through the transformed candidate subspace.The resulting eigenvalue calculation can be updated from the previous iteration.
  • Complexity Reduction: The eigenvalues are calculated as a rank-1 update of the previous iteration’s eigenvalues, reducing overall complexity.This update is used in the approximate Gram-Schmidt implementation.
  • Optimality Result: The achieved mutual information of GS-HP is exactly equal to that of direct greedy hybrid precoding.Proposition 4 formally states equality between the two algorithms.

B. Approximate Gram-Schmidt Based Greedy Hybrid Precoding

The approximate GS-HP algorithm replaces the exact greedy calculation with maximum-projection selection and sequentially designs RF and baseband precoders. It provides a low-complexity, near-optimal design with reduced feedback in a special case.

  • Approximate GS-HP Design: Approximate GS-HP selects RF beams using a simple maximum-projection step.The algorithm sequentially builds the RF and baseband precoding matrices in separate stages.
  • Approximate GS-HP Design: The RF beams are selected first, after which the baseband precoder is optimally designed according to the optimal baseband structure.This sequential design reduces complexity relative to joint RF/baseband approaches.
  • Performance: Algorithm 2 achieves a significant gain over prior solutions and performance very close to the optimal solution.The paper characterizes it as a near-optimal low-complexity frequency-selective hybrid-precoding design.
  • Feedback Overhead: When N_S = N_RF, the equivalent baseband precoder has a unitary structure and requires no baseband feedback bits.In this case, spectral efficiency is invariant to the equivalent baseband precoder.
  • Feedback Overhead: For the proposed greedy design, RF feedback consists of the selected codeword index for each RF beamforming vector.The total is B_RF = N_RF log2 |F_RF^v| bits.

VII. SIMULATION RESULTS

Simulations validate the analytical results and evaluate the proposed codebooks and hybrid precoding designs under wideband mmWave system settings. The proposed designs achieve performance close to unconstrained or optimal solutions, while the Gram-Schmidt algorithm provides a low-complexity near-optimal alternative.

  • Optimal hybrid precoders: For very sparse channels, total power constraints provide greater gain over unitary constraints, but this gain becomes very small beyond 4–5 clusters.The simulations compare optimal hybrid precoders under total, per-subcarrier total, and unitary power constraints.
  • Hybrid codebook designs: Despite relatively small codebook sizes, the designed codebooks achieve performance close to the unconstrained SVD solution for both three-stream and two-stream configurations.The two-stream setup uses an RF codebook of size 128 and an equivalent baseband precoder codebook of size 8.
  • Greedy hybrid precoding: The direct greedy and Gram-Schmidt algorithms have exactly the same performance and are almost equal to the optimal solution, validating Proposition 4.Algorithm 2 is described as low-complexity while achieving very close performance to the optimal hybrid precoding design.
  • Greedy hybrid precoding: The proposed algorithm remains close to digital SVD at small and large stream counts, while its difference from unconstrained precoding varies with the number of streams.At larger stream counts, sparse channels and equal power allocation can assign power to less important multipath components.

C. Gain of RF Chains

Increasing the number of RF chains beyond the number of data streams yields diminishing spectral-efficiency gains while substantially increasing feedback overhead. Using NRF = NS can therefore reduce feedback requirements in limited-feedback wideband mmWave systems.

  • Feedback overhead: When NS = NRF, only the RF precoder index needs feedback because the optimal baseband precoder depends on the RF precoder and a unitary matrix.The optimal baseband structure is an RF-precoder-dependent matrix multiplied by a unitary matrix.
  • Feedback overhead: Feedback bits scale linearly with the number of subcarriers when NS < NRF.
  • Spectral-efficiency gain: The spectral-efficiency gain from adding RF chains saturates after a few RF chains.Figure 8 evaluates the proposed Gram-Schmidt greedy hybrid precoding for NS = 1, 2, and 3 streams with quantized RF and baseband precoders.
  • Spectral-efficiency gain: NRF = 2NS achieves less than 20% gain while requiring much more feedback overhead.
  • Spectral-efficiency gain: For NS = 2, 6 RF chains require 3102 bits for 9.7 bps/Hz, whereas NRF = NS = 2 requires only 10 bits for 8.6 bps/Hz.These values are listed for the required feedback overhead and achievable spectral efficiency in the corresponding comparison.
  • Algorithmic performance: The proposed Gram-Schmidt greedy algorithm achieves performance similar to the optimal hybrid design while avoiding exhaustive RF-codebook search.Its sequential RF and baseband design achieves the same performance as more sophisticated joint-design algorithms.

APPENDIX A

Appendix A proves equivalence between the paper’s optimization formulation and a standard hybrid-precoding problem, then establishes matching beam selections for the GS-HP and DG-HP algorithms.

  • Proof of Proposition 1: The mapped optimization is a standard hybrid-precoding problem whose optimal baseband precoder is obtained through the stated matrix correspondence.The appendix identifies G[k] as the equivalent baseband precoder and connects the resulting optimum to F⋆[k].
  • Proof of Proposition 1: The proof uses a one-to-one mapping between the equivalent baseband-precoder formulation and the feasible set of the original problem.The mapping covers the original domain because the RF precoder has linearly independent columns and the associated matrix is non-singular.
  • Equivalence of GS-HP and DG-HP: The GS-HP and DG-HP algorithms select the same RF beamforming vector at every iteration when they search the same RF codebook.The argument proceeds by induction: identical first-iteration exhaustive searches imply identical subsequent selections under the shared preceding RF precoder.
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