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Learning Sparsifying Transforms for mmWave Communication via $\ell^4$-Norm Maximization

Sueda Taner, Christoph Studer

TL;DR

This work develops a complex-unitary sparsifying-transform learning framework for mmWave beamspace channels by maximizing the $\ell^4$-norm of transformed channel vectors. It extends the real-valued framework of $\ell^4$-norm maximization to the complex domain and introduces two learning algorithms, MSP and Coordinate Ascent (CA), with rigorous analysis of fixed points and local optimality. The authors show that for multipath models the DFT can be a fixed point under MSP, and for LOS it can be a fixed point under CA, while the DCT is not generally optimal for real sinusoid data. Numerical results on synthetic and real-world measured channels reveal that learned transforms can substantially improve sparsity and BER performance in hardware-impaired beamspace scenarios, though the DFT often remains a strong sparsifier for idealized channels. These results suggest learned unitary transforms can offer meaningful gains in practical mmWave systems with non-ideal hardware, while preserving tractable, noise-statistics-consistent processing in the beamspace domain.

Abstract

The high directionality of wave propagation at millimeter-wave (mmWave) carrier frequencies results in only a small number of significant transmission paths between user equipments and the basestation (BS). This sparse nature of wave propagation is revealed in the beamspace domain, which is traditionally obtained by taking the spatial discrete Fourier transform (DFT) across a uniform linear antenna array at the BS, where each DFT output is associated with a distinct beam. In recent years, beamspace processing has emerged as a promising technique to reduce baseband complexity and power consumption in all-digital massive multiuser (MU) multiple-input multiple-output (MIMO) systems operating at mmWave frequencies. However, it remains unclear whether the DFT is the optimal sparsifying transform for finite-dimensional antenna arrays. In this paper, we extend the framework of Zhai et al. for complete dictionary learning via $\ell^4$-norm maximization to the complex case in order to learn new sparsifying transforms. We provide a theoretical foundation for $\ell^4$-norm maximization and propose two suitable learning algorithms. We then utilize these algorithms (i) to assess the optimality of the DFT for sparsifying channel vectors theoretically and via simulations and (ii) to learn improved sparsifying transforms for real-world and synthetically generated channel vectors.

Learning Sparsifying Transforms for mmWave Communication via $\ell^4$-Norm Maximization

TL;DR

This work develops a complex-unitary sparsifying-transform learning framework for mmWave beamspace channels by maximizing the -norm of transformed channel vectors. It extends the real-valued framework of -norm maximization to the complex domain and introduces two learning algorithms, MSP and Coordinate Ascent (CA), with rigorous analysis of fixed points and local optimality. The authors show that for multipath models the DFT can be a fixed point under MSP, and for LOS it can be a fixed point under CA, while the DCT is not generally optimal for real sinusoid data. Numerical results on synthetic and real-world measured channels reveal that learned transforms can substantially improve sparsity and BER performance in hardware-impaired beamspace scenarios, though the DFT often remains a strong sparsifier for idealized channels. These results suggest learned unitary transforms can offer meaningful gains in practical mmWave systems with non-ideal hardware, while preserving tractable, noise-statistics-consistent processing in the beamspace domain.

Abstract

The high directionality of wave propagation at millimeter-wave (mmWave) carrier frequencies results in only a small number of significant transmission paths between user equipments and the basestation (BS). This sparse nature of wave propagation is revealed in the beamspace domain, which is traditionally obtained by taking the spatial discrete Fourier transform (DFT) across a uniform linear antenna array at the BS, where each DFT output is associated with a distinct beam. In recent years, beamspace processing has emerged as a promising technique to reduce baseband complexity and power consumption in all-digital massive multiuser (MU) multiple-input multiple-output (MIMO) systems operating at mmWave frequencies. However, it remains unclear whether the DFT is the optimal sparsifying transform for finite-dimensional antenna arrays. In this paper, we extend the framework of Zhai et al. for complete dictionary learning via -norm maximization to the complex case in order to learn new sparsifying transforms. We provide a theoretical foundation for -norm maximization and propose two suitable learning algorithms. We then utilize these algorithms (i) to assess the optimality of the DFT for sparsifying channel vectors theoretically and via simulations and (ii) to learn improved sparsifying transforms for real-world and synthetically generated channel vectors.
Paper Structure (45 sections, 15 theorems, 100 equations, 3 figures)

This paper contains 45 sections, 15 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Relevant Prior Art
  4. Notation
  5. Paper Outline
  6. System Model
  7. Uplink System Model
  8. Channel Model
  9. Beamspace Representation
  10. $\ell^4$-Norm as a Measure of Sparsity
  11. Problem Setup
  12. Why the TEXT-Norm?
  13. Learning a Sparsifying Transform
  14. MSP: Matching, Stretching, and Projection
  15. Analysis of the MSP Algorithm
  16. ...and 30 more sections

Key Result

Lemma 1

For any unitary matrix $\mathbf{A}\xspace\in U(N;\mathbb{C}\xspace)$, we have that $\|\mathbf{A}\xspace\|_4^4\in[1,N]$. The upper bound is achieved, i.e., $\|\mathbf{A}\xspace\|_4^4=N$, if and only if $\mathbf{A}\xspace\in \mathrm{CP}(N)$.

Figures (3)

  • Figure 1: Unit $\ell^1$-, $\ell^2$-, and $\ell^4$-spheres in $\mathbb{R}\xspace^2$.
  • Figure 2: Uncoded BER for beamspace processing with the DFT or learned transform (LT) using synthetic QuaDRiGa channels with LMMSE and the sparsity-exploiting LE detector, $B=256$ BS antennas, and $U=16$ UEs.
  • Figure 3: Uncoded BER for beamspace processing with the DFT or learned transform (LT) using measured channel vectors gauger2020massive with LMMSE and the LE detector, $B=64$ BS antennas (with eight malfunctioning), and $U=1$ UE.

Theorems & Definitions (20)

  • Remark 1
  • Lemma 1: Extrema of the $\ell^4$-Norm Over Unitary Group
  • Remark 2
  • Lemma 2: Approximate Maxima of $\ell^4$-Norm Over Unitary Group
  • Lemma 3
  • Lemma 4
  • Lemma 5: Projection Onto the Unitary Group
  • Lemma 6: Fixed Point of alg:msp_yless
  • Lemma 7: Convergence of PGA with Arbitrary Step Size
  • Lemma 8: Cubic Convergence Rate around Global Maximizers
  • ...and 10 more