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Throughput Analysis of Massive MIMO Uplink with Low-Resolution ADCs

Sven Jacobsson, Giuseppe Durisi, Mikael Coldrey, Ulf Gustavsson, Christoph Studer

arXiv:1602.01139v3cs.IT

TL;DR

The paper addresses uplink throughput in MU massive MIMO with low-resolution ADCs when channel state information must be learned from quantized pilot observations. It proposes Bussgang-based channel estimation and rate approximations for finite-cardinality and Gaussian inputs. Results show that high-order constellations can work with 1-bit ADCs, while roughly 3-bit ADCs can approach infinite-resolution performance, including under power imbalance.

  • Problem

    Low-resolution ADCs introduce severe nonlinearity, making channel estimation and conventional signaling challenging when neither transmitter nor receiver has a priori CSI.

  • Method

    The paper develops a Bussgang-based channel estimator and achievable-rate approximations for finite-cardinality constellations and Gaussian inputs under training-based reception.

  • Results

    1-bit ADCs support reliable high-order-constellation communication with pilot-based estimation and MRC or ZF detection, while 3-bit ADCs can approach infinite-resolution performance.

  • Takeaways & Limitations

    Low-resolution massive-MIMO uplinks can retain high-order signaling and near infinite-resolution performance with only a few ADC bits, even with receive-power imbalances.

Abstract

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We investigate the uplink throughput achievable by a multiple-user (MU) massive multiple-input multiple-output (MIMO) system in which the base station is equipped with a large number of low-resolution analog-to-digital converters (ADCs). Our focus is on the case where neither the transmitter nor the receiver have any a priori channel state information. This implies that the fading realizations have to be learned through pilot transmission followed by channel estimation at the receiver, based on coarsely quantized observations. We propose a novel channel estimator, based on Bussgang's decomposition, and a novel approximation to the rate achievable with finite-resolution ADCs, both for the case of finite-cardinality constellations and of Gaussian inputs, that is accurate for a broad range of system parameters. Through numerical results, we illustrate that, for the 1-bit quantized case, pilot-based channel estimation together with maximal-ratio combing or zero-forcing detection enables reliable multi-user communication with high-order constellations in spite of the severe nonlinearity introduced by the ADCs. Furthermore, we show that the rate achievable in the infinite-resolution (no quantization) case can be approached using ADCs with only a few bits of resolution. We finally investigate the robustness of low-ADC-resolution MU-MIMO uplink against receive power imbalances between the different users, caused for example by imperfect power control.

I. INTRODUCTION

Massive MU-MIMO offers capacity and energy-efficiency gains through large antenna arrays, but low-resolution ADCs reduce hardware cost and power at the expense of signal quality. Prior work leaves open whether high-order signaling remains effective when channels must be learned from quantized observations without perfect CSI.

  • Motivation: Massive MIMO scales the base station antenna count to serve multiple users simultaneously, providing spatial-resolution, capacity, and array-gain benefits.The large array can also improve radiated energy efficiency.
  • Open problem: Prior analyses often model hardware distortion as independent additive Gaussian noise, while coarse quantization creates severe nonlinearity that can make conventional signaling and receivers suboptimal.Existing capacity results also commonly assume perfect CSI, leaving training-based high-order signaling insufficiently resolved.
  • Motivation: Low-resolution ADCs reduce base-station hardware and power costs, which is important because high-speed ADC power grows exponentially with resolution and linearly with sampling rate.Lower resolution can also relax RF-circuit quality requirements when quantization noise dominates other impairments.
  • Motivation: A 500-element array using paired 10-bit ADCs at 1 GS/s would generate 10 Tbit/s of data, exceeding common CPRI fronthaul capacity.This motivates low-resolution conversion in centralized baseband architectures.
  • Motivation: One-bit ADCs are especially attractive because comparators provide low hardware complexity and eliminate the need for automatic gain control.The paper treats separately quantized in-phase and quadrature components.

C. Contributions

The paper studies frequency-flat Rayleigh block-fading MU-MIMO uplinks without a priori CSI, where pilots and channel estimation must operate on quantized observations. It proposes Bussgang-based estimation and rate approximations, then evaluates their accuracy and the ADC resolution needed to approach infinite-resolution performance.

  • System model: The study considers a frequency-flat Rayleigh block-fading MU-MIMO uplink with N > K, where neither transmitter nor receiver knows the channel realization beforehand.The channel remains constant for T uses and changes independently between coherence intervals.
  • C. Contributions: A novel multi-bit channel estimator for nonuniform quantization regions uses Bussgang’s decomposition and recovers prior LMMSE estimators in the 1-bit case.The estimator is designed for channel learning from quantized observations.
  • C. Contributions: The paper provides an easy-to-evaluate finite-cardinality rate approximation that is explicit in pilot count and ADC resolution and accurate across a large SNR range.Accuracy is assessed against a numerically computed lower bound on achievable rates.
  • C. Contributions: A Bussgang-derived closed-form Gaussian-input rate approximation is accurate at low SNR for 1-bit ADCs but overestimates multiuser rates at high SNR.For 1-bit ADCs, the approximation recovers an earlier result.
  • C. Contributions: Numerical results suggest that about 3-bit ADCs can achieve performance close to infinite resolution across a large range of parameters, including substantial user power imbalances.The paper extends earlier analysis to ZF receivers, multi-bit quantization, and imperfect power control.

B. Quantization of a Complex-Valued Vector

The paper models separate in-phase and quadrature quantization with finite-resolution ADCs and uses Bussgang’s theorem to decompose Gaussian quantizer inputs into a linear term plus uncorrelated distortion.

  • ADC model: Each complex received sample is quantized componentwise by b-bit ADCs using thresholds and labels, producing r_t = Q_b(Hx_t + w_t).The real and imaginary parts are quantized separately, with labels selected according to their threshold intervals.
  • ADC model: The quantization labels are selected using a Gaussian approximation and rescaled so each quantized-output entry has variance Kρ + 1.This choice is suboptimal when low-SNR or large-user conditions do not support the Gaussian approximation.
  • Bussgang decomposition: Bussgang’s theorem decomposes a Gaussian-input quantizer output into a linear transformation of the input plus quantization distortion uncorrelated with that input.The decomposition applies to finite-resolution ADCs and supports both uniform and nonuniform quantizers.
  • Bussgang decomposition: The diagonal Bussgang gain matrix depends on the ADC labels, thresholds, and input covariance, while its infinite-resolution gain is 1.For 1-bit ADCs, the gain is G_1 = 2/π.

D. Channel Estimation

Channel estimation uses orthogonal pilot sequences and quantized observations to estimate the fading channel. The paper develops a Bussgang-based approximation with a simple closed form when the exact quantization-distortion covariance is unavailable.

  • Pilot-based estimation: The receiver reserves P pilot symbols per coherence interval, with K ≤ P ≤ T, and uses pairwise orthogonal pilot sequences across users.The pilots are transmitted to estimate the channel from quantized observations.
  • Estimator construction: The channel estimator generalizes the 1-bit LMMSE approach to multi-bit ADCs by applying Bussgang’s decomposition to the quantized pilot observations.The pilot and noise covariance matrices enter the estimator construction.
  • Estimator construction: The exact LMMSE estimator requires the quantization-distortion covariance, which is available in closed form for 1-bit ADCs but not generally for multi-bit ADCs.This unavailable covariance motivates the alternative estimator introduced next.
  • Approximation: The proposed approximation ignores temporal correlation by setting off-diagonal quantization-distortion covariance elements to zero.The assumption is accurate at low SNR or for many users and is exact when P = K.
  • Approximation: The simplified estimator coincides with the LMMSE estimator for P = K and with the classic MMSE estimator in the infinite-resolution case.For P = K, the associated MSE expressions are exact.

E. Data Detection

The paper considers computationally efficient linear reception for uplink data detection, using either maximal-ratio combining or zero-forcing based on the estimated channel.

  • Linear detection: The receiver forms a soft estimate for each user using a linear receive filter, choosing between maximal-ratio combining and zero-forcing.Linear processing is considered because it is less computationally demanding than nonlinear methods and performs well when antennas substantially outnumber users.
  • Linear detection: For zero-forcing, the kth user’s receive filter is the kth column of the pseudo-inverse of the estimated channel matrix.The pseudo-inverse is formed as ˆH(ˆH^HˆH)^−1.

F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?

With 1-bit ADCs, many receive antennas can make higher-order 16-QAM symbols distinguishable after MRC. Additive noise and independent fading phases support amplitude and phase discrimination, while high SNR preserves mainly phase information.

  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: Multiple receive antennas enable higher-order modulation with 1-bit ADCs, unlike the SISO case where QPSK is optimal.The section demonstrates this using single-user 16-QAM transmitted through a 1-bit-quantized receiver.
  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: P = 20 pilot symbols are used to obtain the channel estimates for the single-user MRC outputs shown in Fig. 2.The figure varies the number of receive antennas N and SNR ρ for 16-QAM inputs.
  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: Averaging the four possible 1-bit quantizer outputs through MRC produces different mean values for inner and outer constellation points.The resulting averaged value is smaller for inner points than for outer points, allowing amplitude information to be conveyed.
  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: At 20 dB SNR, negligible additive noise makes the MRC output lie approximately on a circle, so amplitude cannot convey information while phase remains detectable.The high-SNR analysis treats the received signal and MRC output through approximations involving the fading phases.
  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: For 0 < θ < π/2, the quantized signal component has phase 0 with probability 1 − 2θ/π and phase π/2 with probability 2θ/π, yielding mean phase θ.Independent fading phases make the phase of the MRC estimate converge to θ as N grows through the central limit theorem.
  • F. High-Order Modulation Formats with 1-bit ADCs: Why Does it Work?: N = 200 antennas are sufficient to distinguish the phase of 16-QAM points at 20 dB SNR.The conclusion depends on independent fading coefficients so that the central limit theorem applies.
  • III. ACHIEVABLE RATE ANALYSIS: The paper characterizes rates for finite-cardinality constellations and gives a Bussgang-based Gaussian-input approximation that is accurate at low SNR.The rate analysis complements the high-order-modulation discussion by addressing achievable throughput under low-resolution quantization.
  • A. Sum-Rate Lower-Bound for Finite-Cardinality Inputs: The finite-cardinality rate lower bound applies to arbitrary input distributions and ADC resolutions, with tighter numerical evaluation as the grid spacing decreases.The conditional distributions are estimated by Monte Carlo sampling and mapped onto a rectangular complex-plane grid.

B. Sum-Rate Approximation for Finite-Cardinality Inputs

The paper develops tractable achievable-rate approximations for finite-cardinality and Gaussian inputs under quantized massive-MIMO detection. Bussgang decomposition models quantization distortion, while Gaussian approximations trade exactness for efficient evaluation under stated regimes.

  • B. Sum-Rate Approximation for Finite-Cardinality Inputs: The finite-cardinality approximation assumes the real and imaginary parts of the soft estimate are conditionally jointly Gaussian.Their conditional mean and covariance determine the approximation.
  • B. Sum-Rate Approximation for Finite-Cardinality Inputs: The approximation is evaluated through Gaussian-mixture entropies, with MRC and ZF using different procedures for selecting the conditional covariance.The resulting approximation is reported accurate for all system parameters considered.
  • C. Sum-Rate Approximation for Gaussian Inputs: For Gaussian inputs, the paper extends Bussgang-based rate analysis beyond the 1-bit case to multi-bit ADCs.The quantizer input is modeled as a Gaussian vector under covariance assumptions that are accurate at low SNR or with many UEs.
  • C. Sum-Rate Approximation for Gaussian Inputs: Bussgang’s theorem decomposes the received signal into a linear component and quantization distortion d.Substitution into the detector defines effective noise containing channel-estimation error, additive noise, and quantization distortion.
  • C. Sum-Rate Approximation for Gaussian Inputs: The Gaussian rate expressions include pilot overhead through the factor (T − P)/T and recover known imperfect-CSI rates when G∞ = 1.For 1-bit ADCs, the ZF expression also recovers a previously reported approximation.
  • C. Sum-Rate Approximation for Gaussian Inputs: The Gaussian-input approximations are accurate in the low-SNR regime despite the assumptions used to derive them.This accuracy is assessed numerically in the paper’s subsequent evaluation.

IV. NUMERICAL RESULTS

The numerical study uses coordinated, round-robin pilot transmission in the multi-user uplink. Idle users allow each pilot transmission to use K times the data-symbol power while satisfying the average-power constraint.

  • IV. NUMERICAL RESULTS: Users coordinate pilot transmission so that only one UE transmits pilots at a time in a round-robin schedule.The remaining UEs remain idle during each pilot transmission.
  • IV. NUMERICAL RESULTS: Time-interleaved pilots give the pilot covariance P P_t = Pρ I_K.This setup is part of the numerical evaluation of the channel-estimation and detection schemes.
  • IV. NUMERICAL RESULTS: Each user can transmit pilots at K times the data-symbol power while still meeting the average-power constraint.The increase follows from the users’ idle time during the round-robin pilot schedule.

A. Channel Estimation

The proposed rate approximations closely track achievable rates across broad conditions, while low-resolution ADCs support high-order constellations and approach infinite-resolution performance with a few bits.

  • Channel-estimation approximation: The MSE approximation is exact when P = K and accurate at low SNR when P = 3K, with accuracy improving as ADC resolution increases.The approximation relies on negligible off-diagonal covariance terms, which diminish with higher ADC resolution.
  • Single-user performance: Despite 1-bit ADCs, 16-QAM and 64-QAM outperform QPSK at SNR values as low as ρ = −15 dB with MRC.The comparison uses N = 200, T = 1142, and numerically optimized pilot length P.
  • Rate approximations: The finite-cardinality approximation closely tracks simulated rates for all SNR values, whereas the Gaussian approximation is accurate mainly at low SNR.The finite-cardinality approximation also predicts the SNR beyond which rates with finite constellations stop increasing.
  • Quantization impact: With QPSK, the rate difference between 1-bit and infinite-resolution ADCs is marginal, while higher-order constellations experience more pronounced rate loss.This comparison highlights the stronger sensitivity of higher-order signaling to coarse quantization.
  • Multi-user performance: For K = 10 users, rates with 16-QAM and 64-QAM saturate at the same high-SNR level under both MRC and ZF.The observed saturation indicates effective distortion and interference limitation.
  • ADC resolution: At ρ = −10 dB with 64-QAM and ZF, 2-bit ADCs achieve 90% of the infinite-resolution rate, compared with 71% for 1-bit ADCs.Increasing resolution beyond 3 bits appears unnecessary for the illustrated system parameters.

C. Impact of Large-Scale Fading and Imperfect Power Control

The study evaluates received-power imbalance under imperfect power control and finds that low-resolution ADC systems retain substantial throughput, with performance improving as ADC resolution increases.

  • Motivation and setup: Large received-power differences can make low-power signals indistinguishable from high-power interferers when ADC resolution is too low.The analysis models imperfect power control through user-distance spreads in a single-cell urban-macro scenario.
  • Evaluation metric: The 10% worst throughput is evaluated for a user 185 meters from the BS as a function of the distance spread Δd.The study uses 16-QAM, known received signal powers, and 10^3 random interfering-user drops per Δd value.
  • Results: With uncoordinated transmission and ZF, 1-bit ADCs attain 57% of the perfect-power-control rate, while 3-bit ADCs attain 79%.The results show that low-resolution ADCs can retain high rates even without power control.
  • Conclusion: The paper concludes that increasing ADC resolution to about 3 bits provides near infinite-resolution performance and robustness to user received-power differences.The stated robustness includes differences caused by large-scale fading or imperfect power control.
  • Scope boundary: The analysis is limited to a frequency-flat Rayleigh block-fading channel; an OFDM extension for frequency-selective channels remains under investigation.The paper also emphasizes comparing ADC architectures using total RF and baseband power consumption.

APPENDIX B DERIVATION OF (32)

The derivation constructs a Gaussian approximation for the detected symbol conditioned on the channel estimate, using approximate moments and receiver-specific covariance evaluation.

  • Gaussian approximation: The derivation approximates the conditional distribution of the real and imaginary detected-symbol components as a bivariate Gaussian.The approximation uses mean μ(x_k, Ĥ) and a 2 × 2 covariance matrix Σ(x_k, Ĥ).
  • Rate expression: The achievable-rate approximation includes the rate loss caused by transmitting P pilot symbols for channel estimation.The conditional output distribution given the channel estimate is treated as a Gaussian mixture after conditioning on constellation symbols.
  • Moment construction: The conditional mean and covariance are derived from the received-signal components under assumptions about antenna-wise conditional uncorrelatedness and Gaussianized interference.For the single-user case, the interference approximation is exact because no interference is present.
  • ZF implementation: For ZF, the covariance is obtained through Monte Carlo simulations because direct covariance computation does not provide a satisfactory approximation.Estimating the covariance requires substantially fewer noise and interference realizations than estimating the full probability mass functions.
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