Source-linked AI summary
One-Bit Massive MIMO: Channel Estimation and High-Order Modulations
Sven Jacobsson, Giuseppe Durisi, Mikael Coldrey, Ulf Gustavsson, Christoph Studer
TL;DR
The paper asks how much information can be transmitted over fading channels when one-bit ADCs are used and channel realizations are unknown in advance. It analyzes LS channel estimation with joint pilot-data processing in SISO and LS estimation with MRC in massive-MIMO uplinks. It finds capacity achievement in SISO and support for multiuser operation and high-order constellations in massive MIMO despite one-bit quantization nonlinearity.
Problem
The paper studies achievable rates when one-bit ADCs quantize fading-channel outputs and neither transmitter nor receiver has a priori channel-state information.
Method
The paper analyzes LS channel estimation with joint pilot-data processing for SISO and LS estimation with MRC for multiuser massive-MIMO uplinks.
Results
LS with joint pilot-data processing is capacity achieving in SISO, while LS with MRC supports multiuser operation and high-order constellations in one-bit massive MIMO.
Takeaways & Limitations
Massive antenna arrays can support high-order modulation under one-bit quantization, with spatial oversampling serving as an alternative to temporal oversampling.
Abstract
from arXiv · showhide
We investigate the information-theoretic throughout achievable on a fading communication link when the receiver is equipped with one-bit analog-to-digital converters (ADCs). The analysis is conducted for the setting where neither the transmitter nor the receiver have a priori information on the realization of the fading channels. This means that channel-state information needs to be acquired at the receiver on the basis of the one-bit quantized channel outputs. We show that least-squares (LS) channel estimation combined with joint pilot and data processing is capacity achieving in the single-user, single-receive-antenna case. We also investigate the achievable uplink throughput in a massive multiple-input multiple-output system where each element of the antenna array at the receiver base-station feeds a one-bit ADC. We show that LS channel estimation and maximum-ratio combining are sufficient to support both multiuser operation and the use of high-order constellations. This holds in spite of the severe nonlinearity introduced by the one-bit ADCs.
I. INTRODUCTION
The paper studies one-bit ADC receivers for fading channels without a priori CSI, addressing low-complexity operation in wideband and massive-MIMO settings. It shows that joint LS estimation supports capacity in SISO, while LS estimation with MRC supports multiuser massive MIMO and high-order constellations.
- Motivation: One-bit ADCs reduce hardware complexity because zero-threshold quantization requires no automatic gain controller.Low-resolution ADCs are also motivated by the power cost of high-speed conversion and the many RF chains in massive MIMO base stations.
- Scope: The study analyzes Rayleigh block-fading MIMO channels with one-bit ADCs when neither transmitter nor receiver has a priori CSI, focusing on Nyquist-rate sampling.The receiver must therefore learn the fading channel from quantized outputs.
- Contributions: For SISO, jointly processing pilot and data symbols with LS channel estimation is capacity achieving, eliminating the gap to capacity despite LS being simpler than MAP estimation.This contrasts with the generally smaller, rather than zero, gap associated with joint pilot-data processing in infinite-precision systems.
- Contributions: For massive-MIMO uplinks, LS channel estimation with MRC supports multiuser operation and high-order constellations such as 16-QAM.The result addresses the severe nonlinearity introduced by one-bit quantization.
- Contributions: 16-QAM rates exceed previously reported QPSK rates at SNR values as low as −15 dB when the base station has at least 100 antennas.The paper interprets this as spatial oversampling potentially replacing temporal oversampling.
II. SYSTEM MODEL
The system is a single-cell uplink with K single-antenna users and a base station having N much larger than K antennas, operating over independent Rayleigh block-fading channels. The received signal is quantized separately in its real and imaginary components by one-bit ADCs, with no a priori channel knowledge at either side.
- Network model: The base station serves K single-antenna users with an antenna array of N ≫ K elements.The model is a single-cell uplink.
- Channel model: Each subchannel follows independent Rayleigh block fading, remaining constant for T channel uses before evolving independently across coherence blocks.T is the channel coherence time.
- Signal model: The pre-quantization received signal is modeled across all base-station antennas within one coherence block, with H connecting users to antennas and W representing AWGN.The channel and noise entries are modeled as independent circularly symmetric complex Gaussian variables.
- Quantization: Each antenna separately quantizes the real and imaginary received components using one-bit zero-threshold comparators.The quantizer maps each complex sample to one of four outputs determined by the signs of its real and imaginary parts.
- Information setting: The capacity analysis assumes neither users nor the base station know the channel realization a priori and codes across many coherence blocks.The sum-rate capacity is optimized over input distributions satisfying an average-power constraint.
- Capacity formulation: The SNR is denoted by ρ, while the resulting sum-rate capacity generally lacks a closed-form expression even with infinite-precision quantization.The noise variance is normalized to one.
III. SISO CASE
In the SISO one-bit quantized fading channel, pilot-based LS estimation gives a capacity lower bound, while joint pilot-and-data processing makes LS estimation capacity achieving.
- Capacity characterization: The SISO capacity expression includes an SNR threshold ρc defined through an optimization problem.The threshold is characterized by maximizing over nonnegative ρc.
- Pilot-based estimation: Pilot symbols and their one-bit quantized outputs are used to estimate the fading channel before data detection.The receiver reserves P of T channel uses for pilots, then estimates the channel from the pilot pair.
- Numerical comparison: At ρ = 10 dB, the pilot-based LS-estimation lower bound is essentially a constant gap below capacity across the considered T values, except at T = 2.For this SNR, capacity equals the QPSK rate for all T; the gap to perfect receiver-CSI capacity decreases as T grows.
- Pilot-based estimation: The pilot-based LS-estimation lower bound coincides with the QPSK-achievable rate when T = 2.This equality is stated in Lemma 1 for the SISO case.
- Joint pilot-and-data processing: LS channel estimation combined with joint pilot-and-data processing achieves the channel capacity.The result removes the need for more sophisticated channel-estimation techniques in this setting.
IV. MASSIVE MIMO CASE
The massive MIMO uplink uses pilot-based LS channel estimation and MRC to evaluate multiuser operation with one-bit ADCs. Despite quantization nonlinearity, optimized 16-QAM can outperform QPSK, although performance depends on SNR, coherence time, antennas, and interference.
- Massive MIMO receiver: Pilot-based LS estimation and MRC are used to separate multiuser streams while limiting receiver complexity.The analysis excludes joint pilot-data processing because it is computationally demanding for massive MIMO.
- High-order modulation: Independent fading phases and additive noise allow MRC outputs from one-bit samples to retain a higher-cardinality alphabet capable of representing 16-QAM.Each antenna output has four possible quantized values, but MRC averages them into a scalar alphabet whose cardinality grows with pilots and antennas.
- High-order modulation: At high SNR, negligible noise causes the 16-QAM MRC output to approach a circle, so transmitted amplitude no longer conveys information.With fully correlated fading and negligible noise, the constellation collapses toward a noisy QPSK diagram.
- Achievable rates: 16-QAM outperforms QPSK at SNR values as low as −15 dB and reaches 4 bits per channel use for large SNR in the single-user case.The comparison uses N = 400, T = 1000, and pilot counts optimized for each SNR.
- Multiuser effects: With 16-QAM, the rate difference between single-user and 20-user operation is more pronounced, suggesting that quantization partly destroys inter-user channel orthogonality and creates interference limitation.For QPSK, the corresponding single-user versus multiuser rate difference is marginal.
- Coherence time: For K = 20 users, achievable rates approach perfect-CSI rates more slowly as coherence time increases, and rates are zero when T ≤20.At least 20 pilot symbols are required for orthogonal pilot sequences when K = 20.
- Antenna scaling: With ρ = −10 dB and T = 1000, 16-QAM outperforms QPSK even with substantially fewer than 400 receive antennas, whereas QPSK saturates rapidly as antennas increase.Pilot counts are optimized separately for each antenna-array size.
V. CONCLUSIONS
The paper analyzes one-bit quantized Rayleigh block-fading receivers without a priori channel knowledge. It establishes capacity achievement for LS estimation with joint pilot-data processing in SISO and shows that LS estimation with MRC supports high-order constellations in massive MIMO.
- For SISO channels, LS estimation with joint pilot-data processing is capacity achieving.
- In one-bit massive MIMO, high-order constellations can achieve higher rates than QPSK despite quantizer nonlinearity and multiuser interference.
- The conclusion also notes that zero-forcing gives similar results and that optimized constellations or higher-resolution ADCs remain open directions.
APPENDIX A PROOF OF (12)
The appendix derives the SISO achievable-rate expression by reducing the QPSK analysis to a real BPSK channel and evaluating sign-mismatch probabilities. Beta-function identities then yield equation (12).
- QPSK inputs are reduced to a real-valued BPSK analysis because symmetry makes QPSK rates twice the BPSK rates.
- The proof organizes channel-output probabilities by the number of sign mismatches between pilot/data input vectors and quantized outputs.
- Substitution into the mutual-information expression, together with a beta-function identity, produces equation (12).
APPENDIX B PROOF OF THEOREM 2
The proof constructs an achievable-rate scheme for joint pilot-data processing using progressively updated LS channel estimates, then establishes its optimality by induction over coherence time.
- Achievable-rate construction: The scheme uses the first symbol in each coherence block as a pilot and decodes each later symbol using an LS estimate from preceding observations.The estimate for symbol n uses one pilot and n − 2 earlier data symbols.
- Achievable-rate construction: The resulting achievable-rate expression is evaluated using the channel estimate based on the past n − 1 symbols.The notation ˆh(n −1) records how many input symbols contribute to the estimate.
- Inductive proof: The proof begins at coherence time T = 2, where the joint pilot-data expression coincides with the right-hand side of an earlier rate expression by Lemma 1.This establishes the induction base case.
- Inductive proof: For the induction step, assuming the result at coherence time T, the argument replaces the relevant rate and mutual-information terms using the induction hypothesis and the earlier expression.The remaining equality follows from a binomial identity and algebraic manipulation.