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Quantized Massive MU-MIMO-OFDM Uplink

Christoph Studer, Giuseppe Durisi

arXiv:1509.07928v2cs.IT

TL;DR

The paper studies the underexplored problem of coarse-quantized massive MU-MIMO uplinks over frequency-selective wideband channels. It develops MAP channel-estimation and data-detection algorithms for OFDM and finds that four-bit ADCs can achieve near-optimal performance in sufficiently large arrays without added baseband complexity.

  • Problem

    Frequency-selective wideband quantized massive MU-MIMO-OFDM uplinks are less understood than frequency-flat systems, despite their relevance to broadband communication and ADC power, cost, and data-rate constraints.

  • Method

    The paper develops quantized MAP channel-estimation and data-detection algorithms, including convex optimization methods and low-complexity mismatched models for massive MU-MIMO-OFDM.

  • Results

    Four-bit ADCs achieve near-optimal performance when the BS-antenna-to-user ratio is eight or higher, with no additional baseband-processing complexity compared to infinite precision.

  • Takeaways & Limitations

    Coarse quantization can support low-cost and low-power massive MU-MIMO-OFDM implementations while reducing ADC output rates from 10 bit to 4 bit by 2.5×.

Abstract

from arXiv · show

Coarse quantization at the base station (BS) of a massive multi-user (MU) multiple-input multiple-output (MIMO) wireless system promises significant power and cost savings. Coarse quantization also enables significant reductions of the raw analog-to-digital converter (ADC) data that must be transferred from a spatially-separated antenna array to the baseband processing unit. The theoretical limits as well as practical transceiver algorithms for such quantized MU-MIMO systems operating over frequency-flat, narrowband channels have been studied extensively. However, the practically relevant scenario where such communication systems operate over frequency-selective, wideband channels is less well understood. This paper investigates the uplink performance of a quantized massive MU-MIMO system that deploys orthogonal frequency-division multiplexing (OFDM) for wideband communication. We propose new algorithms for quantized maximum a-posteriori (MAP) channel estimation and data detection, and we study the associated performance/quantization trade-offs. Our results demonstrate that coarse quantization (e.g., four to six bits, depending on the ratio between the number of BS antennas and the number of users) in massive MU-MIMO-OFDM systems entails virtually no performance loss compared to the infinite-precision case at no additional cost in terms of baseband processing complexity.

I. INTRODUCTION

The paper addresses coarse-quantized massive MU-MIMO uplinks over frequency-selective wideband channels, extending prior work focused mainly on frequency-flat channels. It develops algorithms intended to preserve reliable transmission while reducing ADC power, cost, and data rates.

  • Paper scope: The paper investigates quantized massive MU-MIMO-OFDM uplinks and develops channel-estimation and data-detection methods for reliable wideband transmission.The considered architecture has U single-antenna users and a base station with B ≫ U antennas and coarse ADCs.
  • Motivation: Coarse ADC quantization reduces massive-MIMO hardware power and cost because ADC power consumption scales roughly exponentially with quantization bits.Lower precision also reduces the raw ADC data transferred between spatially separated antenna arrays and baseband processing.
  • Prior work: Prior massive-MIMO results indicate resilience to coarse-quantization noise, but they primarily concern frequency-flat fading channels and restricted receiver or modulation assumptions.Examples include 1-bit ADC systems using QPSK, maximum-ratio combining or zero forcing, and least-squares channel estimation.
  • Research gap: Frequency-selective channels are more representative of modern broadband wireless systems than the frequency-flat channels emphasized in earlier massive-MIMO quantization studies.OFDM can orthogonalize frequency-selective channels, but this relies on infinite-precision time-domain ADCs.
  • Technical challenge: Low-precision ADCs make channel estimation and data detection challenging because quantization introduces a strong nonlinear distortion.The distortion is signal-dependent, so additive Gaussian impairment models are a poor match for coarse quantization.

C. Contributions

The paper contributes exact or efficient optimization-based algorithms for quantized channel estimation and data detection, together with models and system assumptions for massive MU-MIMO-OFDM.

  • Algorithmic contributions: MAP channel estimation becomes a convex problem under suitable fading-channel and noise assumptions and can be solved exactly with efficient numerical methods.The paper uses computationally efficient methods for the resulting convex optimization problem.
  • Algorithmic contributions: The paper formulates MAP data detection with quantization and develops a low-complexity MMSE data-detection algorithm.The detector is designed for the quantized massive MU-MIMO-OFDM setting.
  • Quantization models: Two mismatched quantization models trade error-rate performance against computational complexity and are evaluated through numerical simulations.These models support simpler receiver algorithms than the exact quantization model.
  • System model: The quantizer maps complex RF-chain samples independently through their real and imaginary components into finite quantization labels.With Q labels per real dimension, Qb = log2 |A| bits are used per real dimension and 2Qb bits per complex sample.
  • System model: The likelihood of a quantization label is computed using independent real and imaginary thermal-noise components and independently operated scalar quantizers.The thermal noise is modeled as circularly symmetric complex Gaussian noise added before quantization.

B. First Mismatched Quantization Model

The first mismatched quantization model replaces exact quantization likelihoods with a simpler Gaussian error model, enabling lower-complexity estimation and detection with limited performance loss.

  • Computational trade-off: The mismatched likelihood is used instead of the exact quantization likelihood to design simpler channel-estimation and data-detection algorithms.The approximation targets lower computational complexity and often faster algorithms.
  • Model construction: Each quantization label is assigned a complex value y(q) whose real and imaginary parts lie within the corresponding quantization bin.The assigned value is defined using the centroid of the quantization region under the unquantized noisy-signal density.
  • Model assumptions: The model assumes quantization error is independent of both the unquantized signal and thermal noise, and approximates that error as circularly symmetric complex Gaussian.Its variance depends on the quantization label.
  • Performance trade-off: Despite its approximations, the first mismatched model can closely approach the error-rate performance of algorithms based on the exact quantization model.The paper notes that the density used for the centroid can be approximated uniformly over the quantization region or taken from the known signal distribution.

III. QUANTIZED SINGLE-INPUT SINGLE-OUTPUT OFDM

This section models quantized SISO-OFDM transmission over frequency-selective channels and develops convex MAP/ML channel-estimation formulations despite quantization nonlinearity.

  • System model: Quantized SISO-OFDM transmits frequency-domain symbols through a frequency-selective channel, then quantizes the received time-domain signal.The channel matrix is circulant, and the noise is i.i.d. circularly symmetric complex Gaussian.
  • System model: The quantizer prevents direct conversion of the time-domain input-output relation into a diagonal frequency-domain relation using a receiver DFT.This nonlinearity couples the otherwise frequency-domain-orthogonal representation.
  • MAP channel estimation: A single OFDM training symbol is assumed for pilot-based MAP channel estimation.The channel typically has fewer degrees of freedom than the number of OFDM tones, supporting accurate estimation with one training symbol.
  • MAP channel estimation: The negative log-likelihood of the quantized observations is smooth and convex in the complex received variable.The proof separates real and imaginary components and establishes convexity of both terms.
  • MAP channel estimation: MAP channel estimation is convex for log-concave channel priors, including the Gaussian prior associated with i.i.d. Rayleigh fading.With no prior knowledge, the formulation becomes convex ML channel estimation instead.

C. MAP and MMSE Data Detection

This section formulates quantized MAP data detection using an estimated channel, then explains why exact finite-alphabet detection is computationally infeasible for practical OFDM sizes.

  • Detection formulation: Quantized data detection uses either the MAP or ML channel estimate obtained from the convex channel-estimation problem.The resulting detector operates on quantized measurements through the estimated channel model.
  • Detection formulation: MAP detection incorporates prior information about the transmitted data vectors, whereas ML detection assumes all transmit vectors are equally likely.The ML formulation uses p(s) = |O|^-W for every s in the finite alphabet.
  • Hard-output MAP detection: Exact detection is at least as hard as conventional unquantized MIMO detection with W spatial streams because the effective channel matrix is generally non-diagonal.The detector is mismatched because the channel is estimated rather than perfectly known.
  • Hard-output MAP detection: For practical OFDM systems with hundreds or thousands of subcarriers, exhaustive MAP detection is infeasible and requires approximate algorithms.The exact-search complexity grows exponentially in W.

2) Hard-Output and Soft-Output MMSE Detection:

This section introduces a low-complexity quantized MMSE detector by relaxing the finite alphabet and using a Gaussian approximation, with hard- and soft-output options.

  • MMSE detection: Relaxing the finite-alphabet constraint makes quantized data detection convex and efficiently solvable.The detector approximates transmitted symbols as i.i.d. circularly symmetric complex Gaussian variables with variance E_s.
  • Hard-output and soft-output MMSE detection: The resulting quantized MMSE estimate can be mapped element-wise to the nearest constellation point for hard-output detection.This produces a hard-output MMSE estimate.
  • Hard-output and soft-output MMSE detection: The same estimate can generate soft information as max-log log-likelihood ratios for coded detection.The relevant constellation subsets depend on the mapping between bits and symbols.
  • Hard-output and soft-output MMSE detection: The soft-output calculation approximates every post-equalization SINR ρ_w as 1 because the convex solution generally does not provide exact SINR values.Prior work is cited as showing near-optimal performance for this approximation in massive-MIMO max-log decoding.
  • Mismatched quantization model: Under the first mismatched quantization model, DFT-domain quantization noise is generally correlated, complicating channel estimation and data detection.The model assigns quantization-noise variances according to the received quantization labels.

2) MAP and MMSE Data Detection:

This section develops MAP and MMSE detection under mismatched quantization models, contrasting a potentially prohibitive correlated-noise formulation with a tone-wise low-complexity alternative.

  • MAP and MMSE data detection: MAP detection under the first mismatched model uses the MAP channel estimate but has prohibitive complexity because the effective covariance matrix is generally non-diagonal.The same covariance structure also complicates the corresponding estimation problem.
  • MAP and MMSE data detection: Relaxing the alphabet constraint yields a mismatched quantized MMSE detector formulated as a W-dimensional least-squares problem.Its solution supports both hard-output and soft-output detection.
  • Second mismatched quantization model: Mismatch 2 replaces independent, non-identically distributed quantization errors with i.i.d. Gaussian errors having the average quantization-noise variance.The average variance is γ = (1/W) Σ_w γ²(q_w).
  • Second mismatched quantization model: Mismatch 2 keeps total noise uncorrelated after the DFT, so estimation and detection decouple into W independent one-dimensional problems.This permits low-complexity tone-wise frequency-domain processing using standard OFDM techniques.

IV. QUANTIZED MASSIVE MU-MIMO-OFDM UPLINK

The section formulates a coded massive MU-MIMO-OFDM uplink with quantized time-domain antenna outputs and frequency-selective channels. A mapping between per-frequency and per-antenna orientations enables the quantized system input-output model.

  • System model: The system uses U independent single-antenna users, B ≥ U BS antennas, and W-tone OFDM symbols over training and data phases.Training comprises U OFDM symbols, followed by D data symbols, with T = U + D total OFDM symbols.
  • System model: Known QPSK training symbols support channel estimation, while coded data symbols use data tones and BPSK symbols occupy pilot tones.Guard tones remain unused during training and separate data transmission from pilot signaling.
  • Quantized reception: Each BS antenna quantizes its noisy time-domain received vector before forwarding the quantized matrix to channel estimation and data detection.The receiver estimates frequency-domain channel matrices for all OFDM tones from these quantized observations.
  • System model: The quantized MU-MIMO-OFDM input-output relation is Qb = Q(FHZb + Nb), with frequency-domain symbol matrices as inputs and quantized time-domain matrices as outputs.The mapping T and the single-antenna OFDM relation produce this antenna-indexed model.
  • Data orientation: Mapping T preserves the frequency-tones × BS-antennas × OFDM-symbols data cube while changing its association to per-frequency or per-antenna matrices.For one OFDM symbol, the same received signal can be represented as W vectors of length B or B vectors of length W.
  • Channel model: Frequency-domain channel matrices are Fourier transforms of time-domain impulse-response matrices whose nonzero support is limited to L taps, with L ≤ P.The cyclic-prefix length P is assumed to cover the channel impulse response.

B. MAP Channel Estimation

MAP channel estimation is posed as a convex optimization problem over all OFDM tones using quantized measurements, pilot matrices, and a bounded delay-spread constraint. The formulation assumes no prior channel distribution is available at the receiver.

  • MAP formulation: Q-CHE estimates all frequency-domain channel matrices from quantized measurements and known orthogonal pilot matrices.The optimization minimizes the negative log-likelihood of the quantized data under the channel model.
  • Assumptions: The receiver uses only the assumption that the channel impulse response does not exceed the cyclic-prefix length P.Accurate prior channel statistics are assumed unavailable, particularly in settings with differing user path losses.
  • Optimization constraints: Q-CHE jointly estimates the channel over all W OFDM tones while enforcing at most P nonzero time-domain taps.Its constraints transform frequency-domain channel representations to antenna and time-domain forms and impose the delay-spread bound.
  • Solution: Because Q-CHE is convex, it can be solved exactly and efficiently using first-order methods.The affine constraints encode representation changes and the finite channel impulse-response support.

C. MMSE Data Detection

Quantized MMSE detection jointly estimates data across subcarriers and antennas, while mismatched quantization models provide lower-complexity alternatives. The second mismatched model restores independent per-subcarrier detection.

  • Exact quantization: Q-DET solves a convex optimization problem for each OFDM symbol to estimate transmitted data symbols across users, subcarriers, and BS antennas.The estimates support either hard decisions or soft-information computation.
  • Exact quantization: Q-DET constrains pilot-tone symbols to their known values while transforming per-frequency variables into per-BS-antenna representations.Its objective is the MIMO generalization of the SISO-OFDM MMSE objective.
  • Mismatched models: The mismatched model replaces the exact conditional probabilities in Q-CHE and Q-DET with mismatched probabilities without changing problem dimensionality.The resulting algorithms offer improved numerical stability and often faster convergence.
  • Second mismatched model: The second mismatched quantization model decouples channel estimation into U × B independent W-dimensional subproblems after receiver-side DFT processing.Although the resulting channel-estimation problem has a closed-form solution, a first-order method is used to reduce complexity and memory requirements.
  • Second mismatched model: The second mismatched model enables independent MMSE detection per subcarrier, requiring |Ωdata| independent B × U problems.This matches the per-subcarrier structure of conventional unquantized MIMO-OFDM detection.
  • Optimization: The Q-CHE, Q-DET, and MQ-CHE problems are large-dimensional convex programs solved using low-complexity algorithms.The paper summarizes first-order methods for these optimization problems.

A. Forward-Backward Splitting

Forward-backward splitting solves the paper’s large convex estimation and detection problems by combining gradient steps for smooth objectives with proximal steps for constraints. The implementation exploits efficient transforms and step-size selection.

  • FBS framework: FBS is a first-order method for convex objectives combining a smooth differentiable function h with a potentially nonsmooth convex function g.Its applicability depends on efficient evaluation of the proximal operator associated with g.
  • FBS framework: Each FBS iteration computes a proximal operator, the gradient of h, matrix-vector products with A and A^H, and a positive step-size multiplication.Iterations continue until convergence.
  • Application to receiver algorithms: The Q-CHE, Q-DET, and MQ-CHE objectives are smooth and convex, while their affine constraints are represented through the characteristic function g.This places all three problems in the standard FBS form.
  • Proximal operations: For Q-CHE, the proximal step transforms channel matrices to the time domain, zeros taps beyond P, and transforms them back.This enforces the finite delay-spread constraint during optimization.
  • Evaluation: The numerical study evaluates the proposed quantized massive MU-MIMO-OFDM uplink algorithms using a selected set of system parameters.The paper notes that the simulations are limited by space constraints.

A. Simulation Parameters

The simulations evaluate quantized massive MU-MIMO-OFDM uplinks using an SNR operating point and compare receiver architectures across antenna ratios and quantization levels. Results show that increasing the BS-antenna-to-user ratio enables near-optimal performance with four-bit ADCs and no additional baseband complexity.

  • Simulation setup: The simulations use an IEEE 802.11n-like 40 MHz system with 128 OFDM subcarriers, coded transmission, and 16-QAM data symbols.A rate-5/6 convolutional code, random interleaving, and max-log soft-input Viterbi decoding are used.
  • Performance metric: Performance is measured by the minimum average receive SNR required to achieve 1% packet-error rate.This metric is evaluated for the Quantizer, Mismatch 1, and Mismatch 2 receiver architectures.
  • BS-antenna-to-user ratio 2: With 16 BS antennas and 8 users, Quantizer outperforms both mismatched receivers for 3 ≤ Qb ≤ 5, while Qb < 3 cannot reach 1% PER.For Qb ≥ 6, Mismatch 2 outperforms both Quantizer and Mismatch 1 because it computes post-equalization SINR values for soft-output detection.
  • BS-antenna-to-user ratio 4: With 32 BS antennas and 8 users, Quantizer and Mismatch 2 achieve similar SNR operating points when Qb ≥ 5, while Quantizer reaches 1% PER with Qb = 2.Mismatch 2 is preferred in the higher-bit regime because it does not increase complexity relative to conventional infinite-precision algorithms.
  • Higher antenna ratios: At BS-antenna-to-user ratios of 8 and 16, Mismatch 2 matches Quantizer, and Qb = 4 produces only a 0.25 dB SNR gap to infinite-precision ADCs.The gap to the infinite-precision SIMO system with channel estimation is less than 1 dB.
  • Conclusions and open issues: Four-bit ADCs achieve near-optimal performance at BS-antenna-to-user ratios of eight or higher without additional baseband-processing complexity, while reducing ADC output rates 2.5× versus 10-bit precision.The paper identifies synchronization, noise-variance estimation, SINR extraction, SC-FDMA extension, and post-ADC filtering as future-work issues.
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