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GMD-Based Hybrid Beamforming for Large Reconfigurable Intelligent Surface Assisted Millimeter-Wave Massive MIMO
Keke Ying, Zhen Gao, Shanxiang Lyu, Yongpeng Wu, Hua Wang, Mohamed-Slim Alouini
TL;DR
Large RIS arrays make channel-estimation pilot overhead high, while uneven spatial-subchannel SNR complicates BER-oriented design. The paper combines GMD baseband processing, SOMP analog beam selection, and LoS-based RIS design; simulations report better BER than SVD and NLoS without RIS.
Problem
High pilot overhead for large RIS arrays and uneven spatial-subchannel SNR complicate channel estimation and BER-oriented MIMO design.
Source: §I · p. 2, A. GMD-Based Baseband… · p. 3Method
The scheme uses GMD for baseband beamforming, common angular-domain sparsity with SOMP for multiple analog beams, and LoS angles to design the RIS reflecting matrix.
Source: §I · p. 2, 2) Oversampling Codebook Design · p. 5, §V · p. 7Results
GMD outperforms SVD in BER; for ρ greater than 2, SOMP outperforms PCA and, above 3, approaches ideal SOMP; RIS design improves BER over NLoS without RIS.
Source: §IV · pp. 6–7Takeaways & Limitations
GMD-based beamforming can avoid complicated bit or power allocation while improving BER over SVD-based schemes.
Source: §I · p. 2
Abstract
from arXiv · showhide
Reconfigurable intelligent surface (RIS) is considered to be an energy-efficient approach to reshape the wireless environment for improved throughput. Its passive feature greatly reduces the energy consumption, which makes RIS a promising technique for enabling the future smart city. Existing beamforming designs for RIS mainly focus on optimizing the spectral efficiency for single carrier systems. To avoid the complicated bit allocation on different spatial domain subchannels in MIMO systems, in this paper, we propose a geometric mean decomposition-based beamforming for RIS-assisted millimeter wave (mmWave) hybrid MIMO systems so that multiple parallel data streams in the spatial domain can be considered to have the same channel gain. Specifically, by exploiting the common angular-domain sparsity of mmWave massive MIMO channels over different subcarriers, a simultaneous orthogonal match pursuit algorithm is utilized to obtain the optimal multiple beams from an oversampling 2D-DFT codebook. Moreover, by only leveraging the angle of arrival and angle of departure associated with the line of sight (LoS) channels, we further design the phase shifters for RIS by maximizing the array gain for LoS channel. Simulation results show that the proposed scheme can achieve better BER performance than conventional approaches. Our work is an initial attempt to discuss the broadband hybrid beamforming for RIS-assisted mmWave hybrid MIMO systems.
I. INTRODUCTION
Prior RIS beamforming work largely targets narrowband, fully digital or low-frequency settings, leaving wideband mmWave hybrid MIMO underexplored. This paper proposes a RIS-assisted wideband hybrid beamforming scheme using GMD, codebook-based SOMP, and LoS angles for RIS design.
I. INTRODUCTION
- Most prior RIS beamforming designs target narrowband, fully digital MIMO, while wideband RIS work does not consider the BS beamformer and mmWave hybrid MIMO remains unaddressed.The introduction also notes that RIS channel estimation can entail high pilot overhead, motivating designs that use partial channel information.Source: §I · pp. 1–2
- The proposed scheme uses GMD for digital beamforming and SOMP over an oversampling 2D-DFT codebook to select multiple analog beams at the BS and UE.The codebook avoids requiring all channel steering vectors as prior information.Source: §I · p. 2
- The RIS reflecting matrix is designed using only the LoS BS–RIS AoA and RIS–UE AoD, because NLoS path angles are difficult to acquire.The authors report satisfactory performance in wideband RIS scenarios and describe the work as an initial attempt at this setting.Source: §I · p. 2
II. SYSTEM MODEL
The system model is a single-user, wideband mmWave hybrid MIMO-OFDM link in which an RIS provides a reflected path around a blocked direct BS–UE link. It models frequency-selective channels, phase-only RIS control, and hybrid transmit and receive processing.
II. SYSTEM MODEL
- The single-user system places an RIS between the BS and UE to provide a signal path when buildings block their LoS link.Both endpoints use UPAs; the BS and UE have fewer RF chains than antennas, and the supported data streams do not exceed the UE RF-chain count.Source: §II · p. 2, Figure 1 · p. 3
- Each BS–RIS and RIS–UE channel is modeled as one LoS path plus NLoS scattering clusters containing multiple paths and relative delays.The two channel matrices have dimensions Nu × Nt and Nr × Nu, respectively.Source: §II · pp. 2–3
- The RIS elements independently adjust phase, and the model assumes no amplitude adjustment of incident signals.At the BS and UE, baseband processing is per subcarrier, while RF analog beamforming and combining operate around the OFDM IFFT/FFT processing.Source: §II · p. 3
III. PROPOSED SCHEME
The proposed scheme designs digital beamforming and combining, analog beams, and RIS reflection coefficients in separate stages. It assumes perfect CSI for the BS/UE designs but uses only LoS angles to configure the RIS.
III. PROPOSED SCHEME
- The scheme first jointly designs the digital baseband beamformer and combiner, then searches an oversampling 2D-DFT codebook with SOMP for analog beams, and finally designs the RIS reflection matrix.The effective channel for the BS/UE design combines the BS–RIS channel, RIS phase matrix, and RIS–UE channel.Source: §III · p. 3
- Perfect CSI is assumed at the BS and UE for the digital and analog beamforming designs.Source: §III · p. 3
- Because accurate NLoS channel information is difficult to obtain, the RIS design uses only the LoS component's AoA and AoD.The passage motivates this choice by noting that dense BS and RIS deployment makes a strong LoS path at both links probable.Source: §III · p. 3
A. GMD-Based Baseband Beamformer/Combiner Design
The GMD-based baseband design targets BER without requiring complicated bit allocation across spatial streams. It gives the selected streams equal diagonal channel gains, with SIC handling the resulting inter-stream interference.
A. GMD-Based Baseband Beamformer/Combiner Design
- Waterfilling can widen gain differences among spatial subchannels, making BER sensitive to the lowest-SNR stream and requiring different modulation and coding schemes across streams.Source: A. GMD-Based Baseband… · p. 3
- The paper extends GMD-based beamforming with SIC from narrowband to wideband RIS-assisted hybrid MIMO, aiming for good BER and high SE without complicated adaptive modulation and coding.Source: A. GMD-Based Baseband… · pp. 3–4
- GMD decomposes the equivalent baseband channel so R1 is upper triangular with identical diagonal entries equal to the geometric mean of the largest Ns singular values.The corresponding semi-unitary G1 and Q1 matrices provide the receive and transmit bases used for the streams.Source: A. GMD-Based Baseband… · p. 4
- Using the GMD baseband beamformer leaves a non-diagonal equivalent channel, so the receiver uses SIC to cancel inter-stream interference and avoids bit allocation.Source: A. GMD-Based Baseband… · p. 4
B. Oversampling Codebook Based SOMP Analog Beamforming
The analog design extends OMP to wideband SOMP and combines it with GMD-based hybrid beamforming. It uses an oversampling codebook to select beams without requiring the full set of LoS and NLoS steering vectors.
B. Oversampling Codebook Based SOMP Analog Beamforming
- SOMP extends OMP-based spatially sparse beamforming from narrowband MIMO to the wideband setting.The paper adopts this extension for its analog beamformer and combiner design.Source: B. Oversampling Codebook Based SOMP… · p. 4
- The proposed mixed SOMP- and GMD-based scheme uses an oversampling codebook to select analog beams when full-path steering vectors are difficult to acquire.The design focuses on the transmitter analog beamformer; the receiver analog combiner can be obtained similarly.Source: B. Oversampling Codebook Based SOMP… · p. 4
- Conventional OMP-based hybrid beamforming selects the Ns subchannels with the largest singular values from L multipath steering vectors.The baseband beamformer is then designed by least squares to approach fully digital performance.Source: B. Oversampling Codebook Based SOMP… · p. 4
1) Problem Formulation:
The formulation represents analog beamforming by selecting steering vectors from a sensing matrix to approximate the fully digital beamformer across subcarriers. Because the ideal matrix is impractical to acquire, the design uses a codebook and sparse recovery.
1) Problem Formulation:
- The sensing matrix contains BS steering vectors that can span the fully digital beamformer at each subcarrier, and its constant-modulus columns form the analog beamformer.The target Fopt[k] consists of the first Ns right singular vectors of Heff[k].Source: 1) Problem Formulation · p. 4
- A selection matrix T chooses Mt columns from the sensing matrix to form the analog beamformer as FRF = AtT.Mt and Mr are assumed smaller than the number L of candidate steering vectors.Source: 1) Problem Formulation · p. 4
- The analog-beamforming objective fits Fopt[k] across subcarriers using a sparse selection matrix, enabling recovery methods such as SOMP.The formulation exploits common angular-domain sparsity across subcarriers.Source: 1) Problem Formulation · p. 4
2) Oversampling Codebook Design:
An oversampling codebook supplies finer angular candidates than a conventional DFT codebook. SOMP searches these candidates across subcarriers to match the fully digital beamformers and form the analog beamformer.
2) Oversampling Codebook Design:
- The oversampling codebook provides more refined angular candidates, reducing quantization error between true angles and their nearest candidates.The oversampling factor divides the vertical and horizontal angular spaces into ρNy and ρNz discrete angles, respectively.Source: 2) Oversampling Codebook Design · pp. 4–5
- The codebook D comprises UPA steering vectors for combinations of grid angles in the elevation and azimuth sets.The same oversampling factors are assumed at the BS and UE.Source: 2) Oversampling Codebook Design · p. 4
- Algorithm 1 generates the codebook and initializes the residuals to Fopt[k] before selecting the analog beamformer.Its stated inputs include the optimal beamformers, number of RF chains, and oversampling factor.Source: 2) Oversampling Codebook Design · p. 5
- With FBB[k] fixed, SOMP searches the oversampling codebook for multiple beams that match Fopt[k] across subcarriers.The receiver codebook and analog combiner are designed in the same way.Source: 2) Oversampling Codebook Design · p. 5
C. Reflection Matrix Design for RIS
The RIS reflection matrix is designed using only the estimated LoS BS–RIS angle of arrival and RIS–UE angle of departure. Under fixed-angle, LoS-only assumptions, the design maximizes the LoS array gain subject to unit-modulus reflection coefficients.
C. Reflection Matrix Design for RIS
- The RIS phase design uses the LoS BS–RIS AoA and RIS–UE AoD rather than complete CSI, addressing the difficulty of training full-dimensional RIS channels.The passage states that associated angle estimates can be obtained using existing solutions –.Source: C. Reflection Matrix Design for RIS · p. 5
- The optimization assumes the LoS angles are well estimated, ignores NLoS components, and holds the angles fixed during data transmission.Under these assumptions, the equivalent channel’s LoS component is considered.Source: C. Reflection Matrix Design for RIS · p. 5
- The reflection matrix maximizes the modulus of the LoS array-gain scalar γ while constraining each diagonal reflection coefficient to unit modulus.The paper states that γ is common across subcarriers because the subchannels share angular-domain sparsity.Source: C. Reflection Matrix Design for RIS · p. 5
- Because the product terms in the expression for γ have constant absolute value, the authors state that the phase optimization has a direct solution.The phase-setting expression is given for the diagonal elements of Φ.Source: C. Reflection Matrix Design for RIS · p. 5
IV. SIMULATION RESULTS
The simulations compare GMD and SVD baseband beamforming, analog beamforming alternatives, and RIS designs using BER and spectral efficiency. Reported results favor GMD over SVD, oversampled SOMP over PCA, and RIS assistance over the NLoS comparison scenario.
IV. SIMULATION RESULTS
- The simulations use 28-GHz channels with 64 subcarriers, 16-QAM, three data streams, and 8×8 BS, 4×4 UE, and 16×16 RIS arrays.The assumed LoS path power is 23 dB above the NLoS components.Source: §IV · pp. 5–6
- GMD-based beamforming outperforms SVD in hybrid and fully digital MIMO, with a smaller performance gap between the structures.The comparison uses the same ideal SOMP analog beamforming for both baseband schemes in the hybrid structure.Source: Figure 2 · p. 6, §IV · p. 6
- For oversampling factor ρ greater than 2, codebook-based SOMP beats PCA in BER, and for ρ greater than 3 it nearly matches ideal SOMP.The compared schemes use the same GMD baseband beamforming and proposed RIS reflection matrix.Source: Figure 3 · p. 6, §IV · p. 6
- The designed RIS reflection matrix achieves better BER than the NLoS scenario without RIS assistance.The NLoS comparison establishes a direct multipath link while the LoS path is blocked.Source: Figure 4 · p. 6, §IV · pp. 6–7
- At high SNR, ρ = 2 and ρ = 3 SOMP with RIS outperform PCA with RIS in SE, while ρ greater than 3 nearly matches ideal SOMP.The same GMD baseband beamforming is applied across the analog schemes; the comparison also reports improved SE with RIS assistance.Source: Figure 5 · p. 7, §IV · p. 7
- At 20 dB SNR, ρ = 3 SOMP with RIS maintains high, stable SE when NCPE is below 0 dB.NCPE models channel-estimation error, CSI quantization, and outdated CSI; more delicate RIS reflection design also gives better SE.Source: Figure 6 · p. 7, §IV · p. 7
V. CONCLUSIONS
The paper proposes wideband RIS-assisted mmWave hybrid beamforming using GMD for baseband design, SOMP for analog beamforming, and an LoS-based RIS reflection matrix. Simulations report good wideband performance, including improved BER and SE from the RIS design.
V. CONCLUSIONS
- GMD-based baseband beamforming for each subcarrier avoids complicated bit allocation and is reported to provide better BER than traditional SVD-based schemes.The design is intended to give parallel spatial-domain data streams the same channel gain.Source: §V · p. 7
- SOMP with an oversampling 2D-DFT codebook designs BS and UE analog beamforming for wideband MIMO without requiring all steering vectors.The approach addresses difficulties of wideband MIMO beamforming and a prerequisite of traditional sparse beamforming schemes.Source: §V · p. 7
- The RIS reflection matrix uses AoA and AoD from the LoS BS-RIS and RIS-UE channels to harvest array gain for improved BER and SE.Source: §V · p. 7
- Simulations report that the proposed scheme achieves good performance in wideband RIS-assisted hybrid MIMO systems.Source: §V · p. 7