Source-linked AI summary
Analysis of One-Bit Quantized Precoding for the Multiuser Massive MIMO Downlink
Amodh Kant Saxena, Inbar Fijalkow, A. Lee Swindlehurst
TL;DR
The paper studies how one-bit DACs affect linear precoding in the massive MIMO downlink, where reducing RF complexity and power consumption is important. Using Bussgang-based analysis, it derives SQINR and SER results for general linear precoders and asymptotic results for quantized ZF. Quantized ZF depends mainly on M/K, can outperform ML encoding at low-to-moderate SNRs, and a modified Bussgang-based precoder achieves lower SER at high SNR.
Problem
The paper examines how massive-MIMO downlink linear precoding can use one-bit DACs to reduce power consumption and circuit complexity despite quantization distortion.
Method
The paper applies the Bussgang theorem to derive SQINR and SER expressions for general quantized linear precoders, then analyzes quantized ZF asymptotically and designs a modified precoder.
Results
Quantized ZF performance depends primarily on M/K, outperforms ML encoding at low-to-moderate SNRs in the studied scenario, and the modified precoder achieves lower SER at high SNR.
Takeaways & Limitations
Simple quantized ZF is an attractive downlink approach for low-to-medium-SNR massive-MIMO scenarios, while the modified precoder improves high-SNR performance.
Abstract
from arXiv · showhide
We present a mathematical analysis of linear precoders for downlink massive MIMO multiuser systems that employ one-bit digital-to-analog converters at the basestation in order to reduce complexity and mitigate power usage. The analysis is based on the Bussgang theorem, and applies generally to any linear precoding scheme. We examine in detail the special case of the quantized zero-forcing (ZF) precoder, and derive a simple asymptotic expression for the resulting symbol error rate at each terminal. Our analysis illustrates that the performance of the quantized ZF precoder depends primarily on the ratio of the number of antennas to the number of users, and our simulations show that it can outperform the much more complicated maximum likelihood encoder for low-to-moderate signal to noise ratios, where massive MIMO systems are presumed to operate. We also use the Bussgang theorem to derive a new linear precoder optimized for the case of one-bit quantization, and illustrate its improved performance.
I. INTRODUCTION
Massive MIMO offers capacity and robustness gains through many-antenna spatial multiplexing, but RF power consumption and hardware costs motivate one-bit DACs and simpler downlink precoding. This paper analyzes quantized linear precoding using Bussgang-based SQINR and SER expressions, focusing on quantized ZF and its dependence on the antenna-to-user ratio.
- Massive MIMO uses many BS antennas to provide spatial multiplexing, more degrees of freedom per user, and improved robustness and data rates.
- As antennas increase, fixed circuit consumption and RF-chain inefficiency raise power use and hardware costs despite beamforming gains.
- One-bit DACs reduce power consumption and eliminate highly linear amplifiers and back-off operation, while introducing distortion that signal processing must mitigate.
- The paper addresses limited downlink evidence on low-resolution DACs by analyzing one-bit DAC effects on linear precoding for massive MIMO.
- Using Bussgang modeling, the paper derives closed-form SQINR and SER expressions for general linear precoders and an asymptotic quantized-ZF expression.
- Quantized ZF performance depends primarily on M/K and can outperform maximum-likelihood encoding at low-to-moderate SNRs, where massive MIMO is expected to operate.
C. One-bit Quantized Linear Precoding
The paper models one-bit quantization of a standard linear precoder using the Bussgang theorem, representing quantization through a statistically equivalent linear component and uncorrelated noise. This enables analysis of the resulting signal and quantization-noise covariances for arbitrary full-column-rank precoders.
- Quantized linear precoding: One-bit DACs quantize the output xP = Ps of a standard linear precoder before transmission.The quantizer constrains each transmitted component through the signs of its real and imaginary parts.
- Bussgang model: The Bussgang decomposition represents quantization with a linear term F xP and an uncorrelated distortion q.F is chosen so that the cross-correlation between xP and q is zero, providing second-order statistical equivalence.
- Covariance analysis: Under the large-K Gaussian approximation, the quantized signal’s inter-correlation is reduced by the factor sqrt(2/π) relative to normalized unquantized correlations.This relation is used to derive covariance expressions for the quantized data and quantization noise.
- Bussgang model: The matrix determining F is rank deficient, but the analysis states that a unique F is unnecessary because the available expression is sufficient.This constrains direct recovery of F while preserving the quantities needed for the subsequent performance analysis.
- Covariance analysis: The covariance of the one-bit quantization noise is obtained using the arcsin law for hard-limiting quantizers.The analysis uses these covariance expressions to approximate the resulting signal-to-quantization-and-interference-plus-noise ratio.
B. Impact on the Signal of Interest
The section introduces the noiseless received signal and examines its cross-correlation with the desired symbol vector. This correlation provides the basis for characterizing how quantized precoding affects the signal of interest.
- Signal of interest: The noiseless received signal vector is denoted by ˜s.It is used as the signal component before incorporating receiver noise.
- Signal of interest: The cross-correlation between ˜s and the desired symbol vector s is derived to quantify the retained signal component.This relation supports the subsequent characterization of quantization’s effect on the desired signal.
HFP ,
For an arbitrary full-rank precoder, the paper defines the effective channel contribution as G = HFP and uses the Bussgang representation to characterize the quantized received signal and its SQINR. The quantization effect on the desired signal is captured by a diagonal scaling matrix.
- Effective channel: The effective channel associated with the Bussgang linear component is defined as G = HFP.This matrix links the channel H, Bussgang factor F, and precoder P.
- Signal scaling: For any full-rank precoder, one-bit quantization affects the signal of interest through a diagonal matrix.The section uses this result to characterize the resulting SQINR.
- Received-signal model: The quantized received signal is represented through H(FPs + q), with the received quantization disturbance denoted by d = Hq.The covariance of d is then used in the performance analysis.
- Performance measure: The resulting SQINR expression applies to an arbitrary one-bit quantized linear precoder.The paper then uses this quantity to approximate each user’s decoding error probability under Gray-mapped QPSK.
IV. ASYMPTOTIC PERFORMANCE OF THE ONE-BIT QUANTIZED ZERO-FORCING PRECODER
The paper specializes the general analysis to quantized ZF precoding and adopts a large-system regime in which antennas and users grow while their ratio remains finite. Wishart-matrix asymptotics make the inverse-channel behavior increasingly deterministic.
- ZF specialization: The analysis specializes the general SQINR result to the zero-forcing precoder.This provides a focused setting for studying one-bit quantization under ZF transmission.
- Asymptotic regime: The asymptotic regime lets both the number of antennas M and users K grow large while maintaining a finite M-to-K ratio.The resulting expressions are intended to provide additional insight into quantized ZF performance.
- Channel assumptions: The channel model uses diagonal user gains and an i.i.d. circularly symmetric Gaussian matrix with independent real and imaginary components.These assumptions support the random-matrix analysis of the ZF precoder.
- Random-matrix analysis: The relevant channel Gram matrix is modeled as a complex Wishart matrix, enabling analysis of the inverse matrix required by ZF.The variance of inverse-matrix elements is characterized using Wishart-distribution properties.
- Asymptotic result: As M becomes asymptotically large, the variance of every element of the inverse matrix goes to zero, yielding a deterministic limiting value.This deterministic behavior is used to analyze the one-bit quantized ZF precoder.
- Asymptotic result: In the asymptotic case, the rows of H become quasi-orthogonal, simplifying the diagonal terms relevant to ZF analysis.The paper applies these approximations to the performance of the one-bit quantized ZF precoder.
B. Asymptotic Received Downlink Signal
For equal path losses, the asymptotic received signal becomes interference-free across users, with its useful gain increasing with the antenna-to-user ratio M/K. Simulations closely match the asymptotic scaling formula when M/K > 10.
- For large M and K, the received user signals become uncorrelated across users, so multi-user interference disappears.
- The useful received signal has gain proportional to √K, and larger M/K pushes it deeper into the correct decision region.
- The simulated average scaling factor agrees very well with the asymptotic formula for M/K > 10.
C. Asymptotic SQINR and Probability of Error
The quantized ZF analysis shows that SQINR and decoding-error probability depend critically on M/K. At the typical loading factor M/K ≃10, the simulated noiseless SER is below 10^-4.
- SQINR increases approximately linearly with the antenna-to-user ratio M/K.
- At the typical loading factor M/K ≃10, the noiseless SER is below 10^-4.
- The simulations match the asymptotic SER analysis well across the examined M/K values.
D. Simulations
Simulations validate the asymptotic SER analysis for equal and unequal channel gains and compare quantized ZF with ML encoding. Quantized ZF is significantly better over a broad low- to medium-SNR range, while ML is superior at high SNR.
- With unequal channel gains, the SER approaches an error floor of the order of 10^-4 when M ∼10K at high SNR.
- The unequal-gain simulations agree very well with the asymptotic analysis.
- ML encoding is superior at high SNR, but quantized ZF performs significantly better across a broad low- to medium-SNR range.
- As M increases from 20 to 300 for K = 4, received symbols move farther from decision boundaries in the noiseless simulations.
V. BUSSGANG ADAPTED ONE-BIT ZF PRECODER
The Bussgang-adapted precoder uses the quantized-ZF analysis to enforce a diagonal cross-covariance structure for arbitrary M and K. This improves the precoder design beyond the asymptotic ZF property.
- The adapted design enforces a diagonal cross-covariance structure for any values of M and K.
A. Principle
The Bussgang-adapted precoder is constructed by scaling the ZF precoder so that the quantized system preserves the desired signal structure. This cancels multi-user interference and supports SQINR and SER analysis for QPSK signaling.
- The precoder takes the form H†D, where H† is the ZF pseudoinverse and D is a positive diagonal scaling matrix.
- Λ does not affect the signal of interest, so it is set to the identity matrix for simplicity.
- D is selected so that the diagonal covariance condition diag(TD^2T^H) = I_M holds for T = H†.
- The resulting received signal has all multi-user interference canceled.
- The analysis then obtains the kth user's SQINR and QPSK SER from the received-signal representation.
B. Proposed Algorithm
The proposed algorithm compares quantized ZF with a Bussgang-adapted auxiliary precoder and selects the lower-error option. Across the reported K = 3 and K = 10 cases, it achieves a lower error floor than ZF, with the improvement increasing slightly as M grows.
- Proposed Algorithm: The algorithm first tests quantized ZF and, when needed, compares it with an auxiliary precoder H†D using noiseless decoded errors.
- K = 3 users: For K = 3 users, the Bussgang-adapted approach is compared with quantized ZF across transmit SNR and varying M.
- Proposed Algorithm: The precoding matrix producing fewer noiseless transmission errors is selected for the transmitted vector.
- Results: The new algorithm achieves a lower error floor than ZF in all reported cases, and its improvement increases slightly with M at fixed K.
- Results: The analysis derives closed-form SQINR and SER expressions for linear precoders and finds asymptotic scaling proportional to M/K.
- K = 10 users: For K = 10 users, the same SER comparison is made across transmit SNR and varying M.