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Near Maximum-Likelihood Detector and Channel Estimator for Uplink Multiuser Massive MIMO Systems with One-Bit ADCs
Junil Choi, Jianhua Mo, Robert W. Heath
TL;DR
One-bit ADCs reduce the power burden of uplink massive MIMO but make detection and channel estimation challenging. The paper reformulates ML detection and channel estimation as convex optimization, proposes one- and two-stage nML detectors, and reports strong performance with practical implementation. Its channel estimator, however, tends to overestimate the channel norm.
Problem
High-resolution ADCs are power-inefficient for massive MIMO, while one-bit ADCs impose severe quantization thresholds that complicate uplink detection and channel estimation.
Method
The paper reformulates exhaustive-search ML detection as ML estimation and convex optimization, then develops one- and two-stage nML detectors plus a structurally matched one-bit ML channel estimator.
Results
The two-stage nML detector achieves almost the same performance as the original ML detector, while both nML detectors outperform the ZF-type detector across reported SNR and antenna regimes.
Takeaways & Limitations
The unified detector and channel-estimation framework supports a practical low-power uplink massive MIMO solution using one-bit ADC hardware, including multiple users and higher-order constellations.
Takeaways & Limitations
The proposed one-bit channel estimator tends to overestimate the channel norm.
Abstract
from arXiv · showhide
In massive multiple-input multiple-output (MIMO) systems, it may not be power efficient to have a high-resolution analog-to-digital converter (ADC) for each antenna element. In this paper, a near maximum likelihood (nML) detector for uplink multiuser massive MIMO systems is proposed where each antenna is connected to a pair of one-bit ADCs, i.e., one for each real and imaginary component of the baseband signal. The exhaustive search over all the possible transmitted vectors required in the original maximum likelihood (ML) detection problem is relaxed to formulate an ML estimation problem. Then, the ML estimation problem is converted into a convex optimization problem which can be efficiently solved. Using the solution, the base station can perform simple symbol-by-symbol detection for the transmitted signals from multiple users. To further improve detection performance, we also develop a two-stage nML detector that exploits the structures of both the original ML and the proposed (one-stage) nML detectors. Numerical results show that the proposed nML detectors are efficient enough to simultaneously support multiple uplink users adopting higher-order constellations, e.g., 16 quadrature amplitude modulation. Since our detectors exploit the channel state information as part of the detection, an ML channel estimation technique with one-bit ADCs that shares the same structure with our proposed nML detector is also developed. The proposed detectors and channel estimator provide a complete low power solution for the uplink of a massive MIMO system.
I. INTRODUCTION
The paper develops practical detection and channel-estimation methods for uplink multiuser massive MIMO with one-bit ADCs, addressing hardware constraints while retaining tractable processing. Its nML detectors use convex optimization, and the two-stage variant improves performance toward original ML detection.
- Motivation: One-bit ADCs can reduce massive-MIMO hardware power consumption, but their severe quantization threshold makes uplink multiuser detection challenging.ADC power consumption grows exponentially with resolution, motivating one-bit conversion for cost-efficient and green implementations.
- Problem: The original ML detector is impractical for many users because exhaustive-search complexity grows exponentially with the number of uplink users.Prior low-resolution-ADC work included linear, message-passing, and other detectors, but the paper targets general multiuser transmission with one-bit ADCs.
- Channel Estimation: The proposed ML channel estimator shares the detector structure, estimates channel direction and norm, and avoids assuming channel sparsity.The shared structure permits both detection and channel estimation to use the same algorithmic framework.
- Detection Method: The proposed nML detector relaxes ML detection into an ML estimation problem, converts it into a convex optimization problem, and then performs symbol-by-symbol detection.This reformulation enables efficient standard convex-optimization techniques while supporting arbitrary constellation sizes.
- Detection Method: The two-stage nML detector combines the original ML structure with the reduced candidate set produced by the one-stage nML detector.Numerical results report significantly improved detection in the high-SNR regime and performance similar to the original ML detector.
- Analysis: The paper derives an exact condition-specific probability for distinct transmit vectors to produce the same quantized output, with that probability tending to zero as antennas increase.The analysis relates this probability to the numbers of antennas and users.
A. Massive MIMO Received Signal Model
The paper models uplink multiuser massive MIMO with one-bit quantization of each received signal’s real and imaginary components. It specifies the multiuser signal, noise, CSI, constellation, norm, and single-cell assumptions used for detection.
- Each cell has a base station with Nr receive antennas and K single-antenna users transmitting independent symbols simultaneously.
- The received signal combines users’ channel-weighted symbols with complex AWGN at each base station.
- The detector assumes M-ary constellation symbols with zero mean, unit average power, and a normalized transmitted-vector norm constraint.
- The initial detector analysis assumes perfect local CSI, neglects pilot contamination, and focuses first on a single cell without inter-cell cooperation.
- Each received signal is quantized by separate one-bit ADCs for its real and imaginary parts, producing four possible complex outputs.
III. POSSIBLE DETECTORS USING ONE-BIT ADCS
The paper reformulates one-bit-ADC detection in the real domain, defining sign-refined channel quantities and the ML objective over the discrete real-valued constellation space.
- The detector discussion recasts distributed-reception detectors for uplink multiuser massive MIMO and examines their characteristics and limitations.
- Writing the model in real-vector form exploits the independent real and imaginary Gaussian-noise components for one-bit ADC analysis.
- The quantized received vector is represented in the real domain, and its signs are used to construct a sign-refined channel matrix for each receive antenna.
- The constellation size M determines the real-symbol search set used by the detector.
- The ML detector is defined using the sign-refined observations and the Cartesian-product real constellation, with the likelihood term involving the Gaussian cumulative distribution function.
B. ZF-Type Detector Reformulation
The ZF-type detector provides a lower-complexity alternative to exhaustive ML detection by estimating, normalizing, and then slicing the transmitted vector symbol by symbol.
- Brute-force ML detection has complexity M^K, which grows exponentially with the number of users.
- The ZF-type detector first computes a zero-forcing estimate and then normalizes it before symbol-by-symbol detection.
- Normalization is unimportant for PSK constellations but crucial for QAM constellations.
- ZF-type detectors reach a higher error-rate floor than ML detectors as SNR increases, although both floors are caused by one-bit ADCs.
IV. NEAR ML DETECTOR IMPLEMENTATION
The nML detector relaxes the discrete ML search into a convex ML-estimation problem, solves it iteratively, and uses the estimate for symbol slicing. Its suboptimality is principally bounded by the relaxation, with performance improving as receive antennas increase.
- The proposed nML approach converts the original ML detection problem into convex optimization by relaxing constraints on the transmitted vector.
- The relaxed ML estimator uses the constraint ∥´xR∥2 ≤ K instead of the discrete constellation constraint, making the problem efficiently solvable.
- As Nr tends to infinity for arbitrary ρ > 0, the ML detector based on the relaxed formulation achieves the correct decision in probability.
- The iterative solver uses projected-gradient-style updates after initialization related to a maximum-ratio estimate, with one-bit quantization reflected in the sign-refined channel quantities.
- The nML estimate is normalized and then mapped to transmitted symbols through symbol-by-symbol detection in the real-valued representation.
- The algorithm’s complexity is dominated by iterations, with convergence requiring fewer than 20 iterations in an example, and each iteration uses limited matrix-vector operations.
- The nML detector is suboptimal to original ML because of its relaxed estimation space, but the suboptimality does not come into play when Nr is large.
B. Analyses in Asymptotic Regimes
The analysis examines when distinct transmit vectors produce distinct one-bit quantized signals and how the relaxed nML formulation relates to the original detector. In a special case, the probability of identical quantized outputs vanishes as the number of receive antennas grows.
- The relaxed nML formulation has the same high-SNR performance as the original formulation with a relaxed norm constraint.
- Different transmit vectors must produce different quantized received signals to avoid detection errors.
- For the analyzed special case, Pr(ŷ1 = ŷ2) → 0 as Nr → ∞.
- At arbitrary SNR, the special-case transmit vectors yield different quantized signals as the number of receive antennas tends to infinity.
C. Two-Stage nML Detector
The two-stage nML detector narrows the candidate search using the one-stage nML output, then applies a second-stage detection procedure. This improves high-SNR detection while retaining substantially lower search complexity than exhaustive ML detection.
- The two-stage detector constructs a reduced candidate set from the one-stage nML estimate and detected symbols.
- The second stage tests transmit vectors close to the one-stage nML estimate.
- As c → ∞, the two-stage nML detector becomes the original ML detector.
- With a proper c, the method improves detection especially at high SNR with marginal additional computational complexity.
D. Extension to Multicell Setting
The paper adapts the detectors to multicell interference and develops one-bit channel estimators using training observations. The channel-estimation optimization is convex but requires a norm constraint to address norm overestimation.
- In multicell operation, inter-cell interference is modeled as additional AWGN using its estimated long-term statistics.
- The proposed detectors adapt to multicell systems by replacing ρ with the effective multicell parameter ρi,MC.
- Channel estimation uses the same detector structure after reversing the roles of the channel and transmitted signal.
- The estimator uses T known training observations, producing 2T one-bit outputs per receive antenna.
- The unconstrained ML channel-estimation problem is solvable by standard convex optimization because Φ(·) is log-concave.
- A norm constraint is imposed because the unconstrained estimator tends to overestimate the channel norm.
VI. SIMULATION RESULTS
Monte Carlo simulations evaluate the proposed detectors in single-cell and multicell scenarios, using fixed algorithm parameters and comparing detection performance across system settings.
- The simulations use ε = 10^-3 and κ = 0.01 for one-stage nML, and c = 1.3 for two-stage nML.
- The evaluation first compares detector performance in a single-cell scenario and then includes the multicell scenario.
A. Single Cell Scenario
The single-cell evaluation compares nML detectors with ML, ZF-type, and GAMP detectors for detection, and evaluates channel estimators using MSE and normalized MSE. Results show that the two-stage nML detector closely matches ML in a small QPSK setting, remains competitive with GAMP for higher-order constellations, and that the proposed ML channel estimator performs best among the compared estimators.
- Detection performance: The two-stage nML detector achieves almost the same SER as the original ML detector, while both one- and two-stage nML detectors outperform the ZF-type detector.The comparison uses K = 4, M = 4 (QPSK), and Nr = 32; the original ML detector uses exhaustive search, whereas the one-stage nML detector uses convex optimization.
- Detection performance: For K = 8 and M = 8 (8PSK), the two-stage nML and GAMP detectors are comparable and outperform the one-stage nML and ZF-type detectors at the same Nr.The ZF-type detector exhibits an error-rate floor, while the other detectors do not reach such a floor until 10^-5 SER.
- Detection performance: With M = 16 (16QAM), the two-stage nML and GAMP detectors remain comparable, while increasing Nr to 196 mitigates error-rate floors for all detectors.The result is presented as evidence of the benefit of massive MIMO for using one-bit ADCs.
- Detection performance: The proposed detectors provide much better SER performance than the ZF-type detector with the same number of receive antennas, supporting more users under the same system setup.The paper qualifies this practical preference by assuming the base station has sufficient computational power.
- Complexity: Algorithm 1 requires less than 20 iterations on average, and its iteration count decreases as the number of receive antennas increases.The additional comparison size in the two-stage nML detector is described as marginal, indicating improved efficiency with large Nr.
B. Multicell Scenario
The multicell study compares nML and ZF-type detectors under coordinated and uncoordinated user placement. The one-stage nML detector performs better in both scenarios, with scheduling or power control mitigating near-far effects.
- B. Multicell Scenario: The study compares coded BER versus distance for nML and ZF-type detectors under coordinated and uncoordinated user dropping.The multicell parameters are given in Table II, and the comparison is plotted in Fig. 8.
- B. Multicell Scenario: The one-stage nML detector outperforms the ZF-type detector in both user dropping scenarios.
- B. Multicell Scenario: As distance d increases, both detectors’ BER worsens because the received signal power decreases.
- B. Multicell Scenario: For large d, coordinated user placement yields much better BER because similar received signal powers avoid the near-far effect.The coordinated scenario places center-cell users within (d−20m, d+20m), while the uncoordinated scenario randomly drops other users.
- VII. CONCLUSION: The conclusion reports that the proposed detectors outperform ZF-type detection in practical channel-coded and multicell settings.It also identifies future extensions to frequency-selective channels and reduced training overhead for channel estimation.
APPENDIX A PROOF OF LEMMA 1
The appendix proof analyzes high-SNR behavior under the stated assumptions and reduces equality of two detected outputs to independent sign equations.
- APPENDIX A PROOF OF LEMMA 1: In the high-SNR regime, Φ(t) is increasing and bounded above by 1 for the indexed variables in the proof.
- APPENDIX A PROOF OF LEMMA 1: The proof invokes an arbitrary parameter satisfying 0 < α < 1 when analyzing the norm square of ˇx(2).
- APPENDIX A PROOF OF LEMMA 1: The norm constraint forces R,ML to equal K in the proof’s argument.
- APPENDIX A PROOF OF LEMMA 1: Under the assumptions on H and x_k, the condition ˆy1 = ˆy2 decomposes into 2N_r independent sign equations.The equations take the form sgn(u − v) = sgn(u + v).