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Massive MIMO with 1-bit ADC

Chiara Risi, Daniel Persson, Erik G. Larsson

arXiv:1404.7736v1cs.IT

TL;DR

The paper asks whether massive MIMO can support uplink operation with 1-bit ADCs, which avoid AGC but can degrade performance. It develops LS-based channel estimation and MRC, ZF, and LS detection, plus an analytical MRC evaluation method. Numerical and analytical results show good mutual-information and SER performance, while the analytical expressions avoid symbol and channel-noise Monte Carlo simulation.

  • Problem

    1-bit ADCs avoid AGC and reduce hardware complexity and energy consumption, but their performance impact is severe; massive MIMO’s ability to average out ADC noise had not been investigated.

  • Method

    The paper discusses MAP and proposes sub-optimal LS channel estimation, designs MRC, ZF, and LS detection, and derives analytical MRC distributions and mutual-information and SER expressions.

  • Results

    Massive MIMO exhibits good performance with 1-bit receive quantization in mutual information and SER; analytical MRC results closely match Monte Carlo results.

  • Takeaways & Limitations

    Closed-form MRC analysis reduces computational complexity by avoiding Monte Carlo simulation of symbol and channel-noise vectors when computing mutual information.

Abstract

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We investigate massive multiple-input-multiple output (MIMO) uplink systems with 1-bit analog-to-digital converters (ADCs) on each receiver antenna. Receivers that rely on 1-bit ADC do not need energy-consuming interfaces such as automatic gain control (AGC). This decreases both ADC building and operational costs. Our design is based on maximal ratio combining (MRC), zero-forcing (ZF), and least squares (LS) detection, taking into account the effects of the 1-bit ADC on channel estimation. Through numerical results, we show good performance of the system in terms of mutual information and symbol error rate (SER). Furthermore, we provide an analytical approach to calculate the mutual information and SER of the MRC receiver. The analytical approach reduces complexity in the sense that a symbol and channel noise vectors Monte Carlo simulation is avoided.

I. INTRODUCTION

The paper motivates 1-bit ADCs as a lower-complexity, lower-energy option while asking whether massive MIMO can mitigate their performance impact. Existing massive MIMO receiver designs generally assume infinite-precision data, leaving this question open.

  • ADC motivation: Flash ADCs use automatic gain control to match received signals to comparator ranges, adding interface complexity.More ADC output bits also require more operational power.
  • 1-bit ADC motivation: 1-bit ADCs need no AGC and reduce hardware complexity and energy consumption, but generally cause severe performance degradation.Prior 1-bit ADC studies evaluate performance using bit-error rate and/or mutual information.
  • Massive MIMO gap: Massive MIMO uses M BS antennas and K users with M ≫ K ≫ 1, allowing many terminals to share the same time-frequency resource.Such systems are known to average out channel noise and fading.
  • Massive MIMO gap: The paper identifies an open question: whether massive MIMO can also average out ADC noise.Existing massive MIMO receiver designs assume infinitely precise received data.

A. Our contribution

The paper develops channel-estimation and detection methods for massive MIMO uplinks with 1-bit ADCs, then evaluates their information and error-rate performance analytically and numerically.

  • Scope: The study targets the uplink of a massive MIMO system employing 1-bit ADCs, distinguishing it from prior non-massive-MIMO treatments.The massive MIMO regime is defined by M ≫ K ≫ 1.
  • Channel estimation: MAP channel estimation has complexity exponential in K, motivating a sub-optimal LS channel-estimation approach for many-user settings.The paper explicitly quantifies the MAP complexity and proposes LS because that complexity is high.
  • Detection: MRC and ZF filters use the LS channel estimate, while the LS detection filter is computed directly from uplink training sequences.The LS detection filter does not rely on an intermediate channel estimate.
  • Analytical evaluation: The paper derives the MRC soft-estimate probability distribution and closed-form mutual-information expressions for hard and soft symbol estimates.These expressions characterize both discrete detected symbols and soft MRC outputs.
  • Evaluation: Numerical experiments evaluate mutual information and SER and compare numerical results with the closed-form expressions.The paper concludes that massive MIMO provides excellent SER and mutual-information performance across wide parameter ranges.

II. MASSIVE MIMO UPLINK WITH 1-BIT ADC AND NO AGC

This section outlines the paper’s system model, proposed solutions, and performance analysis for the 1-bit-ADC massive MIMO uplink.

  • Section overview: Section II describes the channel model, detection filters, channel-estimation methods, and system-performance analysis.The system model, receiver methods, and performance procedures are organized into Sections II-A through II-C.
  • Section overview: Numerical and analytical procedures for estimating mutual information are discussed as part of the performance analysis.These procedures are included alongside the detection and channel-estimation methods.

A. System model

The system is a single-cell uplink with K single-antenna users transmitting QPSK symbols to an M-antenna base station through a noisy fading channel. A 1-bit quantizer acts separately on real and imaginary components before receive filtering and hard demodulation.

  • System configuration: The considered single-cell uplink has K single-antenna users and one base station equipped with M antennas.The received signal is represented in discrete-time complex baseband form.
  • Signal model: The channel matrix H has independent CN(0, 1) entries, while transmitted user symbols are independent QPSK variables with unit variance.The model also includes per-user transmit power and independent circularly symmetric complex Gaussian noise.
  • Quantization: The 1-bit quantizer applies the sign function separately to the real and imaginary parts of each complex received sample.For vectors, quantization is applied elementwise.
  • Detection pipeline: A receive filter A processes the quantized received vector to produce a soft transmitted-symbol estimate.QPSK demodulation then produces a hard estimate of the transmitted vector.

B. Receive filters

The paper develops receive filters for 1-bit-ADC massive MIMO using pilot-based channel information, including LS channel estimation, MRC, ZF, and a direct LS filter. MAP estimation is computationally exponential in the number of users, motivating the sub-optimal LS approach.

  • Channel estimation: Uplink pilots are used to estimate channel information and calculate the receive filter before data transmission.The coherence interval is divided into pilot and data phases, with 1-bit quantization applied during pilot reception.
  • Direct LS detection: A direct LS receive filter is derived from the uplink training sequences without relying on an intermediate channel estimate.Its complexity grows linearly with K and cubically with M, and it is intended for slowly varying channel conditions.
  • Channel estimation: MAP channel estimation requires a grid search over |A|^K possible user-channel vectors, giving exponential complexity in K.This makes MAP estimation less suitable for massive MIMO, where many users may be multiplexed.
  • Channel estimation: The LS estimator reduces channel-estimation complexity, growing linearly with M and cubically with K.The LS estimate requires the pilot-related matrix to be invertible and at least K training symbol transmissions.
  • Linear detection: MRC and ZF filters are constructed from the LS channel estimate for detecting transmitted symbols.These are conventional linear detectors applied after LS channel estimation.

C. Performance metrics

The paper evaluates system performance using mutual information per user and symbol error rate per user, calculating both metrics numerically and analytically.

  • Performance metrics: Performance is measured by mutual information per user and SER per user, with numerical and analytical calculation procedures provided.These are the two performance metrics considered in the section.

1) Mutual information:

For MRC, the paper models the soft symbol estimate analytically using a central-limit approximation, then derives transition probabilities and mutual information without repeated Monte Carlo sampling. The analytical probability density closely matches Monte Carlo results in the reported validation.

  • Discrete estimate: Mutual information for the discrete QPSK estimate is computed from transition probabilities p(bx_k|x_k,H) and the uniformly distributed QPSK input.The transition probabilities can be evaluated numerically or through complementary error functions once the soft-estimate density is known.
  • Soft-estimate model: The soft MRC estimate is rewritten as a sum of antenna-dependent terms, enabling a central-limit approximation under independence and finite-variance conditions.The approximation relies on the stated independence assumption for the contributing terms.
  • Discrete estimate: The conditional density f(ex_k|x_k,H) is used to calculate QPSK transition probabilities and the mutual information between transmitted and estimated symbols.The four QPSK constellation points are used when forming the conditional probabilities.
  • Validation: Figure 1 compares analytical and Monte Carlo conditional distributions of the real and imaginary parts of ex_k across a 3 × 2 grid of channel realizations.The first and second columns show the real-part and imaginary-part distributions, respectively; rows represent different random channels.
  • Validation: The derived probability density closely matches the Monte Carlo distribution, with similar results reported across other SNR values and tested channel realizations.The reported validation uses a 400-antenna base station serving K = 20 users at −20 dB SNR.
  • Computational approach: Using the analytical density avoids Monte Carlo simulation over many transmit-vector and channel-noise realizations when evaluating the mutual information.The approach replaces repeated sampling with the distribution in (12).

2) SER:

The paper defines SER as the probability that the estimated QPSK symbol differs from the transmitted symbol. For MRC, SER can be calculated analytically from transition probabilities rather than solely through Monte Carlo simulation.

  • SER definition: SER is defined as p(bx_k ≠ x_k), the probability that the QPSK symbol estimate differs from the transmitted symbol.The average SER is expressed over the channel and transmitted-symbol realizations.
  • Analytical evaluation: For MRC, analytical SER uses transition probabilities p(bx_k|x_k,H) calculated with the same procedure used for analytical mutual information.Numerical SER instead estimates the error probability through Monte Carlo realizations of x, n, and H.

III. SIMULATION RESULTS

The simulations evaluate mutual information and SER for MRC, ZF, and LS receivers under varying antenna counts, training lengths, CSI assumptions, and quantizer modeling. Massive MIMO generally supports strong performance, while analytical MRC results closely match Monte Carlo evaluations.

  • Simulation setup: The simulations use Monte Carlo channel, symbol, and noise realizations to evaluate mutual information and SER for QPSK systems.Mutual information is computed per user, while SER is also evaluated per user.
  • Training length and receiver comparison: With M = 400 and K = 20, LS requires around N = 50M training time slots to match full-CSI ZF, whereas MRC and ZF converge with N = 5M.The LS filter is subsequently omitted because it needs more training data and higher computational complexity than ZF and MRC channel estimation.
  • Training length and receiver comparison: With M = 20 and K = 20, LS performs better than ZF and MRC when N = 50M, but all methods perform worse than in the massive MIMO setting.This contrasts with the M = 400, K = 20 case, where LS requires substantially more training to perform well.
  • Training length and receiver comparison: For M = 400 and K = 20, MRC and ZF become equivalent in mutual information for SNR greater than −5 dB, with channel-estimation error suppressed even for short training sequences.In the same SNR regime, the maximal possible QPSK capacity of 2 bits is obtained.
  • Quantizer effects: Assuming perfect CSI and neglecting quantizer effects gives an upper performance bound, while all four considered cases reach 2 bits of QPSK mutual information above −10 dB.The comparison uses ZF with M = 400 and K = 20 and a training sequence of length N = 50K.

IV. CONCLUSIONS

The paper evaluates 1-bit-ADC massive MIMO using MRC, ZF, and LS receivers, and develops analytical performance results for MRC. Analytical results closely match Monte Carlo simulations, while massive MIMO maintains good performance despite 1-bit quantization.

  • Mutual information and SER are evaluated for MRC, ZF, and LS receive filters.
  • ZF generally performs better than MRC and LS filters.
  • At sufficiently high SNR, all filters reach the maximum possible QPSK capacity for quantized and unquantized cases, regardless of training-sequence length.
  • Analytical MRC results closely match symbol-and-noise-vector Monte Carlo simulation results.
  • The analytical MRC treatment reduces complexity by avoiding symbol and channel noise vector Monte Carlo simulation.
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