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Hybrid Precoder and Combiner Design with Low Resolution Phase Shifters in mmWave MIMO Systems
Zihuan Wang, Ming Li, Qian Liu, A. Lee Swindlehurst
TL;DR
High-resolution phase shifters make mmWave hybrid beamforming costly, motivating practical low-resolution designs. The paper proposes iterative analog and digital precoder–combiner algorithms, including a quadratic-complexity one-bit method and multiuser extension. Simulations show performance advantages over existing low-resolution designs, particularly for one-bit phase shifters.
Problem
Existing mmWave hybrid beamforming designs often assume infinite- or high-resolution phase shifters, despite low-resolution PSs being used to reduce hardware cost and power consumption.
Method
The paper successively designs low-resolution analog precoder–combiner pairs for each data stream, computes digital beamformers from the effective baseband channel, and develops a quadratic-complexity one-bit algorithm.
Results
Simulations show that the proposed algorithms outperform existing low-resolution hybrid beamforming designs, particularly for one-bit PSs, while multiuser designs remain competitive with one-bit resolution.
Takeaways & Limitations
The proposed designs provide effective low-resolution hybrid beamforming for point-to-point and multiuser mmWave MIMO systems, especially under one-bit PS constraints.
Abstract
from arXiv · showhide
Millimeter wave (mmWave) communications have been considered as a key technology for next generation cellular systems and Wi-Fi networks because of its advances in providing orders-of-magnitude wider bandwidth than current wireless networks. Economical and energy efficient analog/digial hybrid precoding and combining transceivers have been often proposed for mmWave massive multiple-input multiple-output (MIMO) systems to overcome the severe propagation loss of mmWave channels. One major shortcoming of existing solutions lies in the assumption of infinite or high-resolution phase shifters (PSs) to realize the analog beamformers. However, low-resolution PSs are typically adopted in practice to reduce the hardware cost and power consumption. Motivated by this fact, in this paper, we investigate the practical design of hybrid precoders and combiners with low-resolution PSs in mmWave MIMO systems. In particular, we propose an iterative algorithm which successively designs the low-resolution analog precoder and combiner pair for each data stream, aiming at conditionally maximizing the spectral efficiency. Then, the digital precoder and combiner are computed based on the obtained effective baseband channel to further enhance the spectral efficiency. In an effort to achieve an even more hardware-efficient large antenna array, we also investigate the design of hybrid beamformers with one-bit resolution (binary) PSs, and present a novel binary analog precoder and combiner optimization algorithm with quadratic complexity in the number of antennas. The proposed low-resolution hybrid beamforming design is further extended to multiuser MIMO communication systems. Simulation results demonstrate the performance advantages of the proposed algorithms compared to existing low-resolution hybrid beamforming designs, particularly for the one-bit resolution PS scenario.
I. INTRODUCTION
MmWave hybrid beamforming reduces the impractical RF-chain burden of full-digital systems, but conventional designs often assume costly high-resolution phase shifters. This paper develops low-resolution analog and digital precoder–combiner designs, including one-bit PSs and multiuser extensions.
- Motivation: MmWave communications offer wide bandwidth but suffer high propagation loss, motivating large antenna arrays and beamforming.The smaller wavelength permits many antennas in a compact form factor.
- Motivation: Full-digital precoding is impractical for mmWave systems because it requires many expensive, power-intensive RF chains, ADCs, and DACs.Hybrid analog/digital architectures are pursued for lower cost and energy consumption.
- Motivation: Existing hybrid designs commonly assume infinite- or high-resolution PSs, whose hardware energy consumption and complexity make them impractical at mmWave frequencies.Low-resolution PSs are therefore needed to mitigate beamforming-accuracy loss under practical hardware constraints.
- Contributions: The paper successively designs a low-resolution analog precoder–combiner pair for each data stream to conditionally maximize spectral efficiency, then computes digital beamformers from the effective baseband channel.The approach targets point-to-point mmWave MIMO while maintaining low computational complexity.
- System model: The system uses constant-amplitude, quantized analog phases controlled by B-bit PSs, with digital precoding providing amplitude and phase adjustment.The transmitted symbols are digitally precoded before analog precoding over the antenna array.
- System model: Increasing PS resolution improves potential performance but also increases hardware complexity and power consumption.The paper specifically studies practical low-resolution settings such as B = 1 and B = 2.
RF HFRF FBBs + WH
The system jointly designs quantized analog precoding and combining under low-resolution PS constraints, optimizing spectral efficiency while deferring digital beamformer design to the resulting effective channel.
- System objective: Low-resolution PSs are considered to reduce power consumption and hardware complexity in the analog precoder and combiner.The practical focus includes very low resolutions such as B = 1 and B = 2.
- System objective: The achievable spectral efficiency defines the joint design objective for the digital and low-resolution analog beamformers.The analog matrices are constrained to discrete phase sets.
- Optimization challenge: The joint optimization is non-convex and NP-hard, so the paper seeks a low-complexity suboptimal solution.The difficulty arises from simultaneously optimizing the analog and digital precoder–combiner components under quantized constraints.
- Solution structure: The design decomposes the problem into analog and digital stages, computing digital beamformers from the effective baseband channel after analog design.This decomposition is intended to further maximize spectral efficiency.
- Solution structure: Under a high-SNR assumption, the achievable spectral efficiency is approximated to simplify the analog beamformer design.The approximation is justified when post-combining SNR is sufficiently high.
RF HHWRF WBB
The proposed solution further decomposes analog design into streamwise subproblems and alternates quantized phase updates for each precoder–combiner pair. This yields a convergent iterative phase-matching procedure for low-resolution beamformers.
- Problem decomposition: The analog precoder and combiner are designed one RF-chain pair at a time by decomposing the difficult joint optimization.Each pair corresponds to one data stream.
- Problem decomposition: The sparse mmWave channel is approximated using its Ns strongest singular components, where effective channel rank limits supported data streams.The SVD provides the reduced channel representation used in subsequent design.
- Iterative phase matching: For each stream, the method alternates conditionally optimal quantized phase updates for the analog precoder and combiner.The interference-included channel Q_l is updated as the remaining stream components change.
- Iterative phase matching: The iterative procedure is guaranteed to converge to at least a locally optimal solution because each design step makes the objective non-decreasing.The phase updates are applied until convergence.
- Algorithm: Algorithm 1 summarizes the joint low-resolution analog precoder and combiner design procedure.The procedure repeatedly quantizes precoder and combiner phases and returns to the streamwise design step until convergence.
B. Digital Precoder and Combiner Design
After the analog precoder and combiner pairs are determined, the paper forms the effective baseband channel and applies SVD-based digital processing to enhance spectral efficiency.
- The effective baseband channel is obtained after all analog precoder-combiner pairs are determined.
- SVD of the effective baseband channel provides unitary factors and a diagonal matrix of singular values for digital design.
- SVD-based baseband digital precoders and combiners are employed to further enhance spectral efficiency.
- The baseband precoder and combiner are finally normalized.
IV. ONE-BIT RESOLUTION ANALOG PRECODER AND COMBINER DESIGN
The one-bit design replaces exhaustive binary beamformer search with a candidate-set procedure based on a rank-1 channel approximation, while jointly selecting precoder-combiner pairs to improve performance at quadratic complexity.
- One-bit phase shifters reduce power consumption and hardware complexity, motivating a simpler binary analog beamformer design.
- Exhaustive search for the one-bit optimization has exponential complexity O(2^NtNr), making it unsuitable for large antenna arrays.
- The method approximates the interference-included equivalent channel with its strongest singular component and decouples precoder and combiner design.
- For a given auxiliary phase, the conditionally optimal binary precoder is determined by the signs of cosine phase differences.
- The algorithm constructs Nt candidate precoders and corresponding combiners, reducing each candidate search to linear complexity O(Nt).
- Joint selection over candidate precoder and combiner sets may improve on the rank-1 solution with quadratic complexity O(NtNr).
V. HYBRID PRECODER AND COMBINER DESIGN FOR MULTIUSER MMWAVE MIMO SYSTEMS
The proposed low-resolution hybrid beamforming design is extended to a multiuser mmWave MIMO uplink system, with Algorithm 2 providing the one-bit analog beamformer procedure.
- The low-resolution hybrid precoder and combiner designs are extended from the preceding systems to a multiuser mmWave MIMO uplink.
- Algorithm 2 computes the dominant singular vectors of Ql before constructing binary beamformer candidates.
- The candidate combiner set Wl is constructed by applying the analogous procedure used for the precoder set Fl.
A. System Model and Problem Formulation
The system model considers a multiuser mmWave MIMO uplink in which users transmit through low-resolution analog precoders and the base station combines signals with analog and digital processing to maximize uplink sum-rate.
- The base station has Nr antennas and NRF RF chains and simultaneously serves K mobile users, each with Nt antennas and one RF chain.
- Each user employs a constant-magnitude, low-resolution analog precoder to transmit one data stream.
- The received signal includes each user's uplink channel and complex Gaussian noise before base-station combining.
- The base station applies a low-resolution analog combining matrix followed by a baseband digital combiner for each user.
- The design objective is to jointly optimize low-resolution analog precoders, combiners, and digital combiners to maximize uplink sum-rate.
B. Low-Resolution Hybrid Precoder and Combiner Design
The multiuser design successively constructs low-resolution analog beamformer pairs to enhance desired-channel gain and suppress inter-user interference, then computes digital combiners from the resulting effective channels.
- The first user's analog precoder and combiner maximize its channel gain under finite phase-shifter constraints.
- For subsequent users, the algorithm removes previously selected combiner components before designing beamformers that suppress interference.Previously determined analog combiners are orthonormalized using Gram-Schmidt before modifying the current user's channel.
- The analog beamformer pair for each remaining user is obtained by solving an optimization on the modified channel.
- After all analog beamformers are selected, effective baseband channels are formed and digital combiners are obtained for the users.
- The multiuser procedure is summarized as an algorithm that takes channel and beamformer inputs and outputs digital precoders and combiners.
VI. SIMULATION RESULTS
The simulations evaluate the proposed low-resolution designs over sparse mmWave channels using spectral-efficiency plots that vary SNR, antenna count, iteration count, and phase-shifter resolution.
- The simulations use a geometric mmWave channel model with L propagation paths and uniform linear arrays.The channel environment includes L = 6 scatterers.
- Figure 3 measures spectral efficiency versus SNR, while Figure 4 measures spectral efficiency versus the number of antennas at fixed SNR.
- Figure 5 measures spectral efficiency versus the number of iterations, and Figure 6 measures it versus phase-shifter resolution.
A. Simulation Results of a Point-to-Point mmWave System
The proposed low-resolution hybrid beamforming algorithms outperform existing designs across point-to-point and multiuser settings, with particularly strong performance using one-bit phase shifters. Increasing resolution improves spectral efficiency, but low resolutions can approach ideal benchmarks.
- Point-to-point evaluation: The point-to-point evaluation compares 2-bit and 1-bit proposed algorithms against low-resolution, fully digital, and infinite-resolution hybrid beamforming benchmarks.The setup uses 64-antenna transmitter and receiver arrays, six RF chains, and six data streams over 10^6 channel realizations.
- Point-to-point evaluation: The proposed algorithms outperform competing low-resolution designs, especially with 1-bit phase shifters, while the 2-bit design approaches fully digital and infinite-resolution performance.Similar conclusions hold when spectral efficiency is evaluated versus the number of antennas.
- Resolution and convergence: B = 3 bits is sufficient to closely approach the ideal unquantized case, while higher resolutions provide only marginal additional spectral efficiency at greater cost and complexity.The proposed algorithms outperform the other two low-resolution methods across all phase-shifter resolutions.
- One-bit validation: The one-bit algorithm matches optimal exhaustive search in the examined Nt = 8, Nr = 8, Ns = 1 setting, indicating optimal or near-optimal performance there.The comparison uses a simple configuration because exhaustive search has exponential complexity.
- Multiuser evaluation: In the multiuser uplink, the proposed 2-bit design outperforms codebook-based alternatives, while the 1-bit design remains comparable or competitive as the number of users increases.With K > 5, the 1-bit method remains competitive with TS-HB and MMSE-HB; the 2-bit method always outperforms the compared algorithms in the reported user sweep.
- Overall findings: The study concludes that the proposed algorithms are effective for low-resolution hybrid beamforming, particularly when one-bit phase shifters are used.The conclusion covers both point-to-point and multiuser MIMO systems.
APPENDIX A PROOF OF PROPOSITION 1
The appendix reformulates the optimization problem and derives an equivalent expression by rearranging terms. The proof then uses phase alignment to maximize the resulting amplitude.
- Equivalent formulation: Optimization problem (19) is equivalently reformulated through algebraic transformations involving Nt and Nr.The supplied proof passages present the reformulation steps without the complete displayed expressions.
- Summation rearrangement: Because one term is independent of summation index j, it can be moved outside the first summation to obtain the next expression.This rearrangement simplifies the structure used in the proof.
- Phase alignment: Aligning the phases of the first and second terms maximizes their combined amplitude, completing the proof of Proposition 1.The phase-alignment argument is the final step establishing the claimed equivalence.