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Compressed Sensing Based Multi-User Millimeter Wave Systems: How Many Measurements Are Needed?
Ahmed Alkhateeb, Geert Leus, Robert W. Heath
TL;DR
The paper tackles the difficulty of acquiring channel knowledge for directional mmWave beamforming with large arrays and low pre-beamforming SNR. It proposes compressed sensing channel estimation with conjugate analog beamforming for multi-user downlink systems, then relates measurement count to achievable rate. The proposed operation approaches perfect-channel performance with substantially lower training overhead, while measurement count must be selected to balance estimation accuracy and overhead.
Problem
Large mmWave antenna arrays, low pre-beamforming SNR, and hardware constraints make complete channel state information difficult to obtain for beamformer design.
Method
The paper proposes multi-user downlink operation combining compressed sensing channel estimation, simultaneous channel estimation across mobile stations, and conjugate analog beamforming.
Results
The proposed operation needs ∼280–330 measurements to approach the achievable rate with perfect channel knowledge, versus ∼2048 symbols for traditional lower-frequency solutions.
Takeaways & Limitations
Measurement count should be optimized because achievable rate depends on the trade-off between channel-estimate accuracy and training overhead, especially for fast channels.
Takeaways & Limitations
The analysis seeks a sufficient, not necessarily minimum, number of measurements and relies on assumptions including U ≤ NRF and a single-path channel model.
Abstract
from arXiv · showhide
Millimeter wave (mmWave) systems will likely employ directional beamforming with large antenna arrays at both the transmitters and receivers. Acquiring channel knowledge to design these beamformers, however, is challenging due to the large antenna arrays and small signal-to-noise ratio before beamforming. In this paper, we propose and evaluate a downlink system operation for multi-user mmWave systems based on compressed sensing channel estimation and conjugate analog beamforming. Adopting the achievable sum-rate as a performance metric, we show how many compressed sensing measurements are needed to approach the perfect channel knowledge performance. The results illustrate that the proposed algorithm requires an order of magnitude less training overhead compared with traditional lower-frequency solutions, while employing mmWave-suitable hardware. They also show that the number of measurements need to be optimized to handle the trade-off between the channel estimate quality and the training overhead.
1. INTRODUCTION
The paper addresses the difficulty of obtaining channel state information for directional mmWave beamforming and proposes compressed sensing for low-overhead multi-user operation.
- 1. INTRODUCTION: Large antenna arrays, low pre-beamforming SNR, and hardware constraints make complete channel state information difficult to obtain in mmWave systems.The paper motivates low-complexity channel estimation for practical mmWave operation.
- 1. INTRODUCTION: Analog beamforming avoids some hardware limitations but prior beam-training methods support only single-stream transmission and have user-scaled training overhead.The cited disadvantages motivate a different multi-user channel-estimation strategy.
- 1. INTRODUCTION: The proposed operation combines compressed sensing channel estimation with conjugate analog beamforming for multi-user mmWave systems.Its training overhead does not scale with the number of users.
- 1. INTRODUCTION: The paper characterizes a lower bound on achievable rate as a function of compressed sensing measurements and evaluates the operation through simulations.The evaluation uses achievable rate to relate measurement count and channel-estimation performance.
2. SYSTEM MODEL
The modeled system serves multiple mobile stations with single-stream analog beamforming over a sparse, single-path mmWave channel. It uses RF precoding and combining under assumptions on RF-chain capacity, array geometry, and channel structure.
- 2. SYSTEM MODEL: The base station has NBS antennas and NRF RF chains, serves U mobile stations, and uses one stream per station with U ≤ NRF.The system assumes the number of simultaneously served users does not exceed the available RF chains.
- 2. SYSTEM MODEL: The base station uses U of its NRF RF chains and an NBS×U RF precoder to transmit the U×1 symbol vector.The transmitted signal is x = Fs, with total average transmit power PT.
- 2. SYSTEM MODEL: The RF precoder is implemented with quantized analog phase shifters, whose entries have magnitude 1/√NBS for power normalization.Each entry is represented using a quantized phase angle.
- 2. SYSTEM MODEL: Each mobile station receives through an NMS×NBS channel matrix Hu with Gaussian noise and applies an RF combiner wu to produce a scalar.The model uses a narrowband block-fading channel.
- 2. SYSTEM MODEL: The channel model assumes limited scattering and represents each mmWave channel with a single geometric path having complex gain αu and AoA/AoD array responses.Uniform arrays are used in the simulations, although the results are stated to generalize to arbitrary antenna arrays.
3. PROPOSED DOWNLINK SYSTEM OPERATION
The proposed multi-user mmWave downlink estimates channels with compressed sensing and then uses conjugate analog beamforming and combining. Random training beams exploit channel sparsity, allowing all mobile stations to estimate their channels simultaneously without user-scaled training overhead.
- The operation has two phases: compressed sensing based downlink channel estimation followed by conjugate analog beamforming and combining.
- Random beamforming and projections estimate sparse mmWave channels with relatively low training overhead.
- All mobile stations can estimate their channels simultaneously, so the proposed training overhead does not scale with the number of users.
- The received training matrix uses MBS base-station beamforming vectors and MMS measurement vectors, collected into P and Q.
- Assuming grid-quantized AoDs and AoAs, the channel is represented by a sparse vector whose nonzero path gain identifies the corresponding departure and arrival directions.
- Sparse recovery algorithms such as LASSO and OMP recover the support of zu, which determines the estimated AoA and AoD; the simulations use OMP for low complexity.
- Each mobile station forms its analog combiner from its estimated AoA and feeds its estimated AoD index back to the base station, which designs the beamforming matrix accordingly.
4. ACHIEVABLE AND EFFECTIVE RATES
The analysis relates achievable rate to compressed sensing support-recovery probability and accounts for channel-training overhead through an effective-rate formulation. Under tractable assumptions, it motivates choosing both the measurement design and measurement count to balance estimation accuracy against overhead.
- The evaluation studies how achievable-rate performance relates to the number of compressed sensing measurements.
- Under constant path gains, uniform arrays, and grid sizes GBS = NBS and GMS = NMS, the dictionary matrices become DFT matrices.
- The average achievable rate is lower bounded using the probability of correct support recovery of the sparse channel vector.
- The effective-rate analysis captures a trade-off between accurate channel estimates, represented by small ϵ, and the larger training overhead from more measurements.
- Mϵ denotes the number of measurements needed to guarantee support recovery with probability at least 1 −ϵ.
5. SIMULATION RESULTS
Simulations evaluate the proposed multi-user mmWave operation under a 64-antenna BS, four 32-antenna MSs, and LOS single-path channels. They show that roughly 280–330 compressed sensing measurements approach perfect-channel performance, while effective achievable rate requires balancing estimation accuracy against training overhead.
- Simulation setup: The simulation uses a 64-antenna BS, four 32-antenna MSs, 28 GHz operation, 50 MHz bandwidth, and a 500 m BS–MS distance.All channels are LOS and single-path; the average transmit power is 37 dBm and Q = 4.
- Training overhead: ∼280 − 330 measurements are needed to approach the achievable rate with perfect channel knowledge.This is compared with ∼ NBSNMS = 2048 symbols required by traditional lower-frequency solutions.
- Training overhead: The proposed training overhead does not depend on the number of users, unlike adaptive compressed sensing whose overhead scales with the number of users.For four users, adaptive compressed sensing would require 200 −400 measurements based on ∼50 −100 measurements per user.
- Effective achievable rate: Effective achievable rate must balance channel-estimate accuracy against training overhead, especially for fast-fading channels.Figure 3 reports effective achievable rate for different channel fading coherence values.
6. CONCLUSION
The paper proposes and evaluates a low-complexity compressed sensing channel-estimation operation for multi-user mmWave systems. Results indicate that relatively small training overhead can achieve good performance, but measurements must be selected carefully to maximize effective system sum-rate.
- 6. CONCLUSION: The proposed low-complexity operation achieves good performance with relatively small training overhead relative to the channel matrix dimensions.The operation is based on compressed sensing channel estimation.
- 6. CONCLUSION: The number of measurements should be selected carefully to maximize the effective system sum-rate.The conclusion identifies measurement selection as a key performance trade-off.