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Low-Complexity and Power-Efficient Precoding Codebook Design on Sparse Grassmannian

Joe Asano, Yuto Hama, Hiroki Iimori, Chandan Pradhan, Szabolcs Malomsoky, Naoki Ishikawa

TL;DR

Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs.

Abstract

We propose a sparse Grassmannian design for precoding codebooks. Due to their sparse structure, our proposed codebooks achieve low peak-to-average power ratio (PAPR), low complexity of precoder multiplication, and low storage cost, while demonstrating performance comparable to the optimal codebook. Specifically, we introduce a method for constructing codebooks based on Schubert cell decomposition on the Grassmann manifold. Designing an optimal Grassmannian precoding codebook generally requires high computational complexity. In the proposed approach, by exploiting its sparsity, the objective function can be simplified, and the search space can also be significantly reduced compared to state-of-the-art codebooks. Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs. These results confirm that the proposed sparse codebook can be a practical and power-efficient alternative to conventional codebooks for a wide range of uplink transmission scenarios.

Low-Complexity and Power-Efficient Precoding Codebook Design on Sparse Grassmannian

TL;DR

Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs.

Abstract

We propose a sparse Grassmannian design for precoding codebooks. Due to their sparse structure, our proposed codebooks achieve low peak-to-average power ratio (PAPR), low complexity of precoder multiplication, and low storage cost, while demonstrating performance comparable to the optimal codebook. Specifically, we introduce a method for constructing codebooks based on Schubert cell decomposition on the Grassmann manifold. Designing an optimal Grassmannian precoding codebook generally requires high computational complexity. In the proposed approach, by exploiting its sparsity, the objective function can be simplified, and the search space can also be significantly reduced compared to state-of-the-art codebooks. Numerical simulations in uplink systems demonstrate that the proposed sparse codebook asymptotically approaches the optimal codebook and outperforms the codebook currently adopted in 5G NR, in terms of achievable rate under uncorrelated Rayleigh fading channels, while maintaining substantially lower PAPR than conventional dense designs. These results confirm that the proposed sparse codebook can be a practical and power-efficient alternative to conventional codebooks for a wide range of uplink transmission scenarios.
Paper Structure (25 sections, 34 equations, 9 figures, 3 tables)

This paper contains 25 sections, 34 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Grassmannian Codebooks
  3. Mathematical Definition bendokat2024grassmannedelman1999geometry
  4. Conventional Grassmannian Codebooks
  5. Manopt endo2024boostinggohary2009noncoherent
  6. Exp-Map kammoun2003newkammoun2007noncoherent
  7. 5G NR Codebook 3gpp_ts_38211_v1820
  8. System Model
  9. Proposed Design Method
  10. General Design Method
  11. Schubert Cell Decomposition konishi2022novelhansen2007schubert
  12. Sparsity Pattern Selection
  13. Embedding Complex Variables
  14. MCD Maximization
  15. Design Method for $T=2M$
  16. ...and 10 more sections

Figures (9)

  • Figure 1: Overview of the proposed codebook design method that consists of four steps.
  • Figure 2: Number of real variables required for optimization when applying the proposed method with the number of transmit antennas $T$ and data streams $M$, where $(T,M)=(4,2),~(6,3),$ and $(8,4)$.
  • Figure 3: Distribution of the time-domain Nyquist rate samples of OFDM with $512$ subcarriers and 4-QAM modulation, where $\ell$ denotes the number of nonzero elements in each row of the precoder
  • Figure 4: Distribution of the time-domain Nyquist rate samples of DFT-s-OFDM with $512$ subcarriers and 4-QAM modulation when varying $\ell$.
  • Figure 5: PAPR dependency in OFDM and DFT-s-OFDM with $512$ subcarriers and 4-QAM modulation when varying the number of nonzero elements in each row of the precoder $\ell=1$, $2$, $3$ and $4$.
  • ...and 4 more figures