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Channel Estimation and Performance Analysis of One-Bit Massive MIMO Systems

Yongzhi Li, Cheng Tao, Gonzalo Seco-Granados, Amine Mezghani, A. Lee Swindlehurst, Liu Liu

arXiv:1609.07427v3cs.IT

TL;DR

One-bit ADCs offer a low-power alternative for massive MIMO, but reliable channel estimation and performance analysis remain necessary under coarse quantization. The paper uses Bussgang-based BLMMSE estimation and derives rate approximations for MRC and ZF, finding improved estimation, resource-allocation gains, and a quantified antenna requirement for MRC.

  • Problem

    The paper addresses channel estimation and uplink performance for massive MIMO with one-bit ADCs, where accurate BS-side CSI is indispensable but difficult to obtain under one-bit output quantization.

  • Method

    The paper uses separate Bussgang decompositions for pilot and data phases to formulate a BLMMSE channel estimator, then derives low-SNR rate approximations and resource-allocation methods.

  • Results

    The BLMMSE estimator outperforms previously proposed methods, while optimal resource allocation improves performance and MRC requires approximately 2.2-2.3 times more antennas than a full-precision system for similar spectral efficiency.

  • Takeaways & Limitations

    One-bit massive MIMO can achieve similar power efficiency to conventional massive MIMO, but system design must account for quantization, training, data power, and receiver choice.

Abstract

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This paper considers channel estimation and system performance for the uplink of a single-cell massive multiple-input multiple-output (MIMO) system. Each receive antenna of the base station (BS) is assumed to be equipped with a pair of one-bit analog-to-digital converters (ADCs) to quantize the real and imaginary part of the received signal. We first propose an approach for channel estimation that is applicable for both flat and frequency-selective fading, based on the Bussgang decomposition that reformulates the nonlinear quantizer as a linear functionwith identical first- and second-order statistics. The resulting channel estimator outperforms previously proposed approaches across all SNRs. We then derive closed-form expressions for the achievable rate in flat fading channels assuming low SNR and a large number of users for the maximal ratio and zero forcing receivers that takes channel estimation error due to both noise and one-bit quantization into account. The closed-form expressions in turn allow us to obtain insight into important system design issues such as optimal resource allocation, maximal sum spectral efficiency, overall energy efficiency, and number of antennas. Numerical results are presented to verify our analytical results and demonstrate the benefit of optimizing system performance accordingly.

I. INTRODUCTION

The paper motivates one-bit ADCs for massive MIMO as a low-power alternative while addressing channel estimation and uplink performance under coarse quantization. It proposes Bussgang-based estimation and analytical resource and antenna design results for one-bit systems.

  • Massive MIMO uses large BS antenna arrays to serve multiple users and improve spectral and energy efficiency with CSI and simple MRC or ZF processing.
  • High-resolution ADCs can consume several watts each, making their power and cost prohibitive in large antenna arrays.
  • One-bit ADCs use simple comparators with negligible power, but their benefits depend on reliable BS-side CSI and are strongest in low-SNR operation.
  • The BLMMSE estimator applies separate Bussgang decompositions to pilot and data phases and supports flat and frequency-selective channels.
  • The paper derives low-SNR, large-user closed-form uplink-rate approximations for MRC and ZF using estimated CSI, accounting for noise and quantization effects.
  • Optimal resource allocation can change the preferred training length and improve performance, while MRC one-bit systems require approximately 2.2-2.3 times more antennas than conventional systems.

A. Bussgang-Based Channel Estimator

The paper develops a Bussgang-based linear channel estimator for one-bit massive MIMO that captures quantization statistics and supports analytical performance evaluation.

  • Bussgang decomposition replaces the nonlinear one-bit quantizer with a statistically equivalent linear operator and quantizer-noise term.The operator is chosen so the equivalent quantizer noise is uncorrelated with the received signal.
  • The estimator is initially derived for an identity channel covariance and can be extended to a generic channel covariance matrix.The derivation states that the generic covariance case is readily modified from the simplified model.
  • The BLMMSE estimator accounts for correlations among quantizer-noise elements through the arcsine law, making it more general than earlier estimators.The paper states that the estimator reduces to the earlier form when τ = K.
  • At high training power, the BLMMSE channel estimate exhibits an error floor caused by one-bit quantization.The cited result identifies this floor as training power increases toward infinity.

B. Extension to Frequency Selective Fading with OFDM

The channel-estimation method extends to frequency-selective fading with OFDM by modeling each subcarrier’s transmission through a finite-tap circulant channel after cyclic-prefix removal.

  • OFDM signal model: The OFDM extension uses Nc subcarriers, a unitary IFFT, and a cyclic prefix satisfying L − 1 ≤ Ncp ≤ Nc.L denotes the number of channel taps.
  • OFDM signal model: After cyclic-prefix removal, each BS antenna receives a time-domain convolution involving a circulant channel matrix and additive white Gaussian noise.The circulant matrix is determined by the channel’s L taps.
  • OFDM signal model: The circulant-convolution representation supports equivalent frequency-domain formulations because circulant convolution is commutative.The finite number of channel taps determines the structure of the channel representation.
  • OFDM signal model: Stacking the received signals across antennas and users produces a time-domain model containing all channel taps between the BS antennas and users.The stacked channel vector includes the M K L channel-tap coefficients.

L hTD + nTD, (20)

For one-bit OFDM reception, Bussgang decomposition enables channel estimation from the quantized time-domain signal, while the resulting quantization-noise covariance is generally non-diagonal.

  • Quantized OFDM model: One-bit quantization is applied to the stacked time-domain received signal, yielding a quantized signal for wideband channel estimation.The wideband model is formulated directly in the time domain after stacking.
  • System implication: Because of one-bit quantization, OFDM cannot decompose the wideband channel into many parallel narrowband channels as in conventional systems.The limitation follows from the time-domain quantized model.
  • Bussgang-based estimation: Bussgang decomposition reformulates the nonlinear quantizer as a linear term plus quantizer noise chosen uncorrelated with the quantizer input.This decomposition provides the basis for the wideband BLMMSE estimator.
  • Bussgang-based estimation: The wideband BLMMSE estimator uses the channel covariance and obtains the quantized-signal covariance through the arcsine law.The quantizer-noise covariance is then derived from the resulting covariance expressions.
  • Bussgang-based estimation: Unlike narrower prior derivations, the general estimator accounts for non-diagonal quantizer-noise covariance caused by the arcsine law.The observation applies to arbitrary linear modulation schemes, not only OFDM.

C. Low SNR Approximate BLMMSE Channel Estimate Covariance

At low SNR or with many users, the quantizer-noise covariance becomes approximately diagonal, yielding a tractable BLMMSE channel-estimate covariance and supporting rate analysis with linear receivers.

  • Approximation regime: The arcsine operation makes a general closed-form BLMMSE mean-squared-error expression difficult, motivating a low-SNR covariance approximation.Massive-MIMO array gain motivates focusing on relatively low-SNR operation.
  • Approximation regime: At low SNR or for large numbers of users, the quantizer-input covariance is diagonally dominant, enabling an arcsine-law approximation.The approximation relies on small non-diagonal covariance elements.
  • Approximation result: At low SNR, quantizer noise is approximated as uncorrelated with variance 1 − 2/π.This approximation follows from the small off-diagonal terms in the covariance matrices.
  • Approximation result: The resulting BLMMSE channel-estimate covariance has uncorrelated elements, providing the approximation used for uplink achievable-rate evaluation.The covariance result is subsequently applied to linear receiver analysis.
  • Data detection model: During data transmission, the BS models user interference, channel-estimation error, AWGN noise, and quantizer noise separately.These four terms are included in the received-signal performance analysis.
  • Data detection model: The Bussgang decomposition is applied separately for each channel realization, ensuring quantizer noise is uncorrelated with the desired signal.This differs from a model that uses the same decomposition across channel realizations.
  • Data detection model: The BS can reconstruct the quantizer-input covariance from quantizer-output measurements and approximate the gain matrix using channel hardening without perfect CSI.The gain approximation is based on large-user massive-MIMO conditions and i.i.d. unit-variance channels.
  • Data detection model: The BLMMSE channel estimate is used to construct an MRC or ZF linear receiver that separates the quantized signal into user streams.Detection multiplies the quantized signal by the receiver matrix.

B. Uplink Achievable Rate Approximation at Low SNR

The paper develops a closed-form low-SNR approximation of the achievable uplink rate for one-bit massive MIMO with MRC and ZF processing. The approach models correlated quantization noise and uses a Gaussian worst-case assumption to obtain tractable rate expressions.

  • Rate approximation: A closed-form approximation is derived for the achievable rate of one-bit massive MIMO uplinks at low SNR, addressing the limited insight provided by prior mutual-information expressions.The approximation is developed for both MRC and ZF receivers.
  • Quantization-noise modeling: The covariance matrix of the data-phase quantization noise is generally non-diagonal, with correlations computed using the arcsine law.At low SNR or for large K with i.i.d. channels, a diagonal approximation can be used when the received-signal covariance is diagonally dominant.
  • Rate approximation: A Gaussian worst-case assumption with the same covariance matrix yields a lower bound on the achievable rate despite the quantization noise being non-Gaussian.The detected signal is rewritten using a known mean gain and effective noise before deriving the bound.
  • Approximation accuracy: The effective-noise Gaussian approximation is expected to become asymptotically tight as the number of antennas M grows, by the central limit theorem.The paper states that later numerical results quantify the gap between the approximation and the ergodic-rate lower bound.
  • Receiver-specific results: Theorems provide low-SNR achievable-rate approximations for MRC and ZF receivers using CSI estimated by the BLMMSE channel estimator.The results specifically concern the kth user in a one-bit massive MIMO uplink.

V. ONE-BIT MASSIVE MIMO SYSTEM DESIGN

This section uses the achievable-rate approximation to study how antenna count and transmit-power scaling affect one-bit massive MIMO system design. With fixed channel-estimation accuracy, user transmit power can decrease as 1/M while maintaining asymptotic spectral efficiency.

  • Design framework: The system-design analysis evaluates training length, training power, data power, and BS antenna count using sum spectral efficiency as the performance metric.The coherence interval has length T, and both MRC and ZF receivers are considered.
  • Power efficiency: With fixed training power and channel-estimation accuracy, each user’s transmit power can decrease proportionally to 1/M while maintaining a given sum spectral efficiency for both MRC and ZF.The asymptotic performance of MRC and ZF is the same in this case.

2) Case II:

The paper examines joint resource allocation and antenna requirements for one-bit massive MIMO. Equal scaling of training and data power is less aggressive than fixed-accuracy scaling, while optimized resources reduce the antenna overhead relative to earlier fixed-resource analyses.

  • Case II: joint power scaling: When training and data powers both scale as 1/M^c, constant performance is obtained with c = 1/2, so power cannot decrease as aggressively as when channel-estimation accuracy is fixed.The asymptotic performance for MRC and ZF is again the same, but has a different value.
  • Case II: asymptotic interpretation: The resulting spectral-efficiency expressions are equivalent to K interference-free SISO channels with effective transmit powers 2σ^2E_u/π and 4τE_u^2/π^2, respectively.The paper concludes that spectral efficiency increases proportionally to the number of users K despite one-bit ADCs.
  • Resource allocation: Unlike conventional MIMO, the optimal training length in one-bit massive MIMO is not always τ = K and depends on parameters including T and total energy budget P.The optimum is evaluated numerically because no closed-form expression for τ* is obtained.
  • Antenna requirements: With optimized training length and training and data powers, fewer than 2.5 times more antennas are needed for MRC, and for ZF at low SNR, to match conventional massive MIMO performance.This improves on the 2.5-times antenna requirement reported for the special case τ = K and ρ_d = ρ_p.

VI. NUMERICAL RESULTS

Numerical experiments compare BLMMSE with prior channel estimators and examine the effect of correlated quantization noise. BLMMSE performs best overall, and modeling spatial correlation becomes increasingly important as SNR or channel correlation increases.

  • A. Channel Estimation Performance: Figure 2 compares estimator MSE versus SNR for M = 16, K = 4, and τ = 20, including LS, nML, and other one-bit channel-estimation approaches.The LS and nML estimators are taken from prior work.
  • A. Channel Estimation Performance: As SNR increases, ignoring correlation among quantization-noise elements causes a small performance loss, supporting the use of the arcsine-law covariance model.The gap is larger when the quantization noise is spatially correlated.
  • A. Channel Estimation Performance: BLMMSE outperforms the previously proposed channel estimators across the evaluated SNR range.At low SNR, BLMMSE matches the method using uncorrelated quantization noise; a gap emerges as SNR increases.
  • A. Channel Estimation Performance: For a spatially correlated channel with a non-diagonal channel covariance matrix, the MSE gap exceeds 1 dB, showing the impact of spatially correlated quantization noise.The example uses a typical urban channel with a 10° angle spread.

B. Validation of Achievable Rate Results

The simulations validate the achievable-rate analysis and examine spectral- and energy-efficiency trade-offs under antenna scaling and resource allocation. Optimized one-bit systems retain substantial performance but require additional antennas relative to conventional systems.

  • Achievable-rate validation: The closed-form MRC and ZF rate approximations closely track ergodic sum spectral efficiency across SNRs and antenna counts.Fig. 4 compares the approximations with ergodic rates for M = {32, 64, 128} and K = τ = 8.
  • Power efficiency: Under antenna scaling, sum spectral efficiency converges to a constant for both MRC and ZF receivers when data or both training and data power decrease with M.With ρd = Eu/M in Case I, and ρp = ρd = Eu/M in Case II, the constant in Case II is reached only for very large M.
  • Resource allocation: A sum spectral efficiency of 15 bits/s/Hz requires 1.9 times less bit energy with optimal allocation than the benchmark for M = 128, for both MRC and ZF.Doubling antennas from 128 to 256 further reduces bit energy by about 2.2 at the same spectral efficiency under optimal allocation.
  • Resource allocation: Optimal training length is not always equal to the number of users, and ZF requires a higher-quality channel estimate than MRC to reduce interuser interference.The proposed allocation adjusts training length, training power, and data-transmission power.
  • One-bit versus conventional systems: At M = 400 and ρ = −10dB, one-bit MRC and ZF achieve 23.2 and 24.6 bits/s/Hz, respectively, or 73.68% and 69.76% of conventional performance.These comparisons optimize training length, training power, and data power for each system.
  • One-bit versus conventional systems: With optimal resource allocation, one-bit MRC needs approximately 2.2-2.3 times more antennas for equivalent performance, while ZF needs increasingly more antennas as transmit power rises.Without optimal allocation, the antenna ratio is 2.5 for MRC and at low SNR for ZF.

APPENDIX A

The appendix derives the BLMMSE channel estimator and the rate-analysis ingredients for one-bit quantized observations. It uses Bussgang-based uncorrelated quantization noise and Gaussian approximations to obtain tractable receiver expressions.

  • Channel-estimation derivation: The conditional quantization-noise covariance is incorporated into the BLMMSE estimate rather than modeled only as independent additive noise.The appendix removes the fixed quantization noise from the conditional expectation and identifies the resulting estimate as linear MMSE.
  • Channel-estimation derivation: The Bussgang choice makes quantization noise uncorrelated with the pilot observation and therefore with the channel.This property supports the subsequent linear MMSE estimation analysis.
  • Achievable-rate derivation: For rate analysis, uncorrelated interference and quantization noise are treated as independent Gaussian noise under a worst-case assumption.This yields the variance of the effective noise used in the achievable-rate lower bound.
  • Achievable-rate derivation: The BLMMSE estimate is approximated as Gaussian because quantization noise makes it non-Gaussian, enabling the subsequent MRC moment calculations.The approximation invokes Cramér’s central limit theorem.

APPENDIX D

The appendix reformulates sum spectral efficiency in terms of training allocation variables and shows how the optimal training length and power split are obtained. It concludes that one-bit systems need not use a pilot length equal to the user count.

  • Optimization formulation: Sum spectral efficiency is rewritten as a function of the power-allocation variable γ and training length τ for MRC and ZF.The receiver-specific expression includes coefficients such as a3 for the ZF case.
  • Optimization formulation: The optimal solution distributes total power between training and data through γ* and τ*, defining corresponding training and data powers.The construction satisfies γ*P = τ*ρp* and (1 − γ*)P = (T − τ*)ρd*.
  • Training-length conclusion: Because the spectral-efficiency objective is not monotonic in τ at fixed γ*, the optimal training length is not always equal to K.The benchmark choice τ = K therefore cannot generally be assumed optimal in one-bit systems.
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