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Uplink Performance of Wideband Massive MIMO with One-Bit ADCs

Christopher Mollén, Junil Choi, Erik G. Larsson, Robert W. Heath

arXiv:1602.07364v2cs.IT

TL;DR

The paper studies how one-bit ADCs affect spectral efficiency in wideband massive MIMO, where ADC power consumption is significant. It derives achievable rates using low-complexity linear channel estimation and detection, finding that amplitude distortion vanishes with many channel taps and that OFDM matches single-carrier performance.

  • Problem

    The paper asks whether wideband massive MIMO can retain useful spectral efficiency with power-saving one-bit ADCs despite quantization distortion.

  • Method

    The paper analyzes single-carrier and OFDM transmission over frequency-selective Rayleigh channels using LMMSE channel estimation, linear combiners, and closed-form achievable-rate limits.

  • Results

    As channel taps grow, amplitude distortion vanishes, OFDM matches single-carrier performance, and quantized rates are approximately 60–70% of unquantized rates near 2 bpcu with an approximately 4 dB loss.

  • Takeaways & Limitations

    Wideband frequency selectivity makes low-complexity linear processing practical with one-bit ADCs, although matching unquantized performance at low-to-moderate SNR requires approximately three times more antennas.

Abstract

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Analog-to-digital converters (ADCs) stand for a significant part of the total power consumption in a massive MIMO base station. One-bit ADCs are one way to reduce power consumption. This paper presents an analysis of the spectral efficiency of single-carrier and OFDM transmission in massive MIMO systems that use one-bit ADCs. A closed-form achievable rate, i.e., a lower bound on capacity, is derived for a wideband system with a large number of channel taps that employs low-complexity linear channel estimation and symbol detection. Quantization results in two types of error in the symbol detection. The circularly symmetric error becomes Gaussian in massive MIMO and vanishes as the number of antennas grows. The amplitude distortion, which severely degrades the performance of OFDM, is caused by variations between symbol durations in received interference energy. As the number of channel taps grows, the amplitude distortion vanishes and OFDM has the same performance as single-carrier transmission. A main conclusion of this paper is that wideband massive MIMO systems work well with one-bit ADCs.

I. INTRODUCTION

The paper addresses wideband massive MIMO with one-bit ADCs using low-complexity linear estimation and detection, deriving achievable-rate results for single-carrier and OFDM transmission. It shows that frequency selectivity can make quantization distortion effectively additive and that OFDM approaches single-carrier performance as channel bandwidth increases.

  • Motivation: One-bit ADCs reduce conversion power and simplify the analog front end, while large antenna arrays can overcome their quantization-related performance loss.Automatic gain control becomes trivial because it only considers the input sign.
  • Contribution: The proposed system uses LMMSE channel estimation and low-complexity linear combiners for symbol detection, with achievable rates derived for single-carrier and OFDM transmission.The analysis assumes Rayleigh-fading channel taps with a general power delay profile and obtains closed-form limits for MRC and ZFC as the number of taps grows.
  • Related work: Previous linear detection work focused mainly on frequency-flat channels, leaving frequency-selective one-bit-quantized channels insufficiently studied.The paper targets this gap with wideband analysis and low-complexity processing.
  • Contribution: Approximately 4 dB separates the quantized and unquantized systems, while quantized performance reaches approximately 60–70% of unquantized performance near 2 bpcu.The loss can be reduced when longer pilot sequences are available.
  • Main result: As the number of channel taps grows, amplitude distortion vanishes, leaving circularly symmetric distortion and making OFDM performance equal to single-carrier performance.The amplitude component otherwise causes significant intersymbol interference in OFDM, whereas the circularly symmetric component becomes close to Gaussian in massive MIMO.
  • Novelty: Compared with earlier frequency-flat analyses, this paper additionally studies coherent amplitude distortion, channel-tap dependence, generic linear combiners, and pilot-length effects.It extends an earlier MRC-only conference derivation into a more general and complete treatment.

II. SYSTEM MODEL

The system is a synchronized massive MIMO uplink with M base-station antennas and K single-antenna users operating in either single-carrier or OFDM mode. It uses frequency-selective L-tap channels, one-bit in-phase and quadrature ADCs, and block-fading transmission with pilot and data periods.

  • System model: The base station has M antennas serving K single-antenna users, with complex-baseband signals sampled uniformly at the Nyquist rate under perfect synchronization.The matched receive filter must therefore be analog because no intermediate oversampling step is used.
  • Signal and quantization model: The received signal contains user transmissions weighted by transmit powers Pk, thermal noise, and one-bit quantization of separately sampled in-phase and quadrature components.Thermal noise is IID over symbol duration and antenna, and each user symbol has zero mean and unit power.
  • Channel model: The channel between each user and antenna has L impulse-response taps combining small-scale fading with large-scale fading.The small-scale fading is estimated, while its mean, variance, and power-delay-profile structure are known a priori or modeled.
  • Channel model: Wideband examples use tens of channel taps, including L = 15 for a 15 MHz, 1 µs-delay channel and L = 75 for a 5 µs-delay urban channel.These examples motivate the frequency-selective regime analyzed in the paper.
  • Signal and quantization model: The quantizer uses a zero threshold and scales its output so that |qm[n]| = 1.The paper assumes nonzero thresholds would provide no significant improvement in the high-interference multiuser setting.
  • Transmission modes: Transmission supports single-carrier with frequency-domain equalization and OFDM, using blocks of pilot and data symbols with a cyclic prefix of length L−1.The cyclic prefix enables a frequency-domain input-output relation after prefix removal.

III. QUANTIZATION

The quantization analysis models one-bit quantization through an optimally scaled distortion that is uncorrelated with the received signal. In wideband channels, averaging across many taps makes received power nearly constant and reduces the relative distortion variance.

  • Quantization model: The quantization distortion is defined through a scaling factor ρ chosen by the Wiener solution to minimize error variance.The resulting distortion is uncorrelated with the received signal by the orthogonality principle, although its distribution depends nonlinearly on the received-signal distribution.
  • Wideband averaging: When the number of channel taps L is large, the expected received power at each symbol duration approaches its average received power.The convergence follows from an averaging effect over channel taps and can be accurate even when L is shorter than the block length N.
  • Wideband averaging: A large number of users K can provide a similar received-power averaging effect when no user dominates the received power.Power control makes this condition plausible when the users’ received powers have similar magnitudes.
  • Distortion characterization: Under IID Rayleigh fading, the analysis gives the one-bit quantizer’s scaling factor and distortion variance in closed form.The limiting variance coincides with the standard one-bit quantization mean-squared error when received power is constant and ρ^2 = 2π¯Prx.
  • Distortion characterization: The relative quantization distortion variance approaches a wideband limit as L grows, and finite-wideband distortion variance is smaller than the narrowband limit.This follows from the concavity of the square-root function in the derived expression.
  • Reference case: With no quantization, the quantization error and relative distortion are both zero, so the derived expressions also cover the unquantized system.This provides the comparison baseline for later rate analysis.

IV. CHANNEL ESTIMATION

The paper develops low-complexity channel estimation for one-bit wideband massive MIMO under block fading, using known pilot phases and LMMSE estimation. Quantized estimation quality can approach the unquantized case, but finite coherence time limits feasible pilot lengths.

  • The uplink is split into disjoint pilot and data blocks sharing the same block-fading channel within one coherence time.
  • Known phase-rotated pilots produce noisy, periodically sampled observations of each user’s frequency-domain channel.The sampling period is tied to the number of users, and the channel impulse response can be recovered when the sampling condition is met.
  • The channel estimate uses an LMMSE estimator, with estimation error uncorrelated to the estimate; its estimate variance defines channel estimation quality.
  • The difference in channel estimation quality is less than 2 dB when quantized and unquantized systems use the same pilot excess factor under a uniform power-delay profile.
  • The quantized system’s quality improves with pilot excess factor, but practical coherence-time constraints limit feasible values to approximately µ ≤ 830/K in the considered outdoor channel.
  • Nonconstant pilot phases distribute training energy across more low-SNR observations, which improves quantized channel estimation relative to concentrating energy in a few high-SNR observations.The asymptotic limits require received power to become constant as L grows, excluding intervals with no received signal.

V. UPLINK DATA TRANSMISSION

The uplink data-transmission analysis studies practical linear detection for one-bit massive MIMO, including quantization-induced symbol-estimation errors and OFDM effects. It derives an achievable rate to evaluate system performance.

  • The analysis studies one block of Nd uplink data symbols and applies practical linear symbol detection based on the estimated channel.
  • The paper analyzes the distribution of quantization-induced symbol-estimation error and its effect on OFDM before deriving an achievable rate.

A. Receive Combining

Receive combining uses FIR filters derived from the estimated channel to recover time- or frequency-domain transmit signals. MRC, ZFC, and RZFC are considered, with different practical trade-offs.

  • The base station combines received signals with an FIR filter whose transfer function and impulse response produce estimates of transmitted signals.
  • Combiner weights are formed from the estimated channel matrix using MRC, ZFC, or RZFC with selectable regularization.
  • RZFC is preferred in practice for superior performance, while MRC and ZFC are included for mathematical tractability.
  • MRC concentrates combining energy in exactly L taps and permits much of its processing to be distributed locally across antennas.

B. Quantization Error and its Effect on Single-Carrier and OFDM Transmission

Quantization error separates into amplitude distortion and circularly symmetric distortion; as channel taps grow, amplitude distortion vanishes, making OFDM match single-carrier performance.

  • Quantization error has amplitude and circularly symmetric components, with amplitude distortion harming OFDM more than single-carrier transmission.The circularly symmetric component is distinct from the amplitude-related error that produces stronger OFDM degradation.
  • The quantization-distortion decomposition models one component through an LMMSE estimate of the distortion conditioned on the transmitted signal and channel taps.The residual term has the smallest variance and is uncorrelated with the channel.
  • As L → ∞, the amplitude-distortion term vanishes because τ[n] → 0, while the growing number of error terms yields a Gaussian limit.The rate of disappearance depends on how closely received interference power remains constant across symbol durations.
  • In wideband systems, negligible amplitude distortion makes OFDM perform as well as single-carrier transmission without additional signal processing.The resulting quantization error can be treated as additional AWGN, supporting parallel frequency-flat channel analysis and arbitrary QAM constellations.
  • Figure 3 shows that narrowband amplitude distortion creates oblong symbol-estimate clouds and OFDM intersymbol interference, whereas wideband estimates are comparable across waveforms.The comparison uses 128 antennas, no thermal noise, perfect channel state information, and zero-forcing combining.

C. Achievable Rate

The paper derives achievable-rate expressions for one-bit-ADC wideband massive MIMO and analyzes their asymptotic behavior for linear receivers, pilot design, and transmission mode. Quantization is noncoherent noise that vanishes with many antennas, while ZFC can retain high-SNR rate ceilings from residual distortion.

  • The achievable rate is derived in closed form as the number of channel taps L grows, closely approximating systems with practically large L.The analysis targets the uplink with one-bit ADCs and linear processing.
  • Single-carrier transmission has the same achievable rate as OFDM because the Fourier transform is unitary.
  • Quantization distortion is noncoherent noise: its variance does not scale with antenna number M, while the desired-signal gain does.Consequently, the distortion disappears as M grows, allowing arbitrarily high rates in the large-antenna limit.
  • With equal channel-estimation quality, one-bit quantization causes a 2/π ≈ −2 dB SINR loss; with unit pilot excess factors and equal powers, the loss increases to −4 dB.The −4 dB result assumes µq = µ0 = 1, equal receive powers, and a uniform power delay profile.
  • Quantized ZFC has a residual-interference rate ceiling at high SNR, even with perfect channel state information, whereas its low-SNR loss can match the 2/π factor.The residual interference is attributed to distortion of the received signals.
  • Increasing quantized pilot length improves the rate ratio, with the largest improvement occurring from µq = 1 to µq = 2 before saturation in most systems.In the studied systems, quantized MRC reaches approximately 60–70% of the unquantized rate; quantized ZFC ranges from around 60% to 40% at low SNR as users increase from 5 to 30.

VI. NUMERICAL EXAMPLES

The numerical results show that the large-tap closed-form limit accurately approximates achievable rates and that one-bit systems can approach unquantized performance with additional antennas. Frequency selectivity mitigates amplitude distortion, while weak users experience substantially larger performance gaps.

  • Rate convergence: The large-tap limit R′k closely approximates the achievable rate Rk, including at L = 15 taps for 5 users and even L = 1 for 30 users.The approximation is accurate across the studied antenna and user configurations.
  • Rate convergence: Amplitude distortion lowers performance for small L with 5 users, but disappears as the number of taps increases and the rate approaches R′k.The improvement saturates once amplitude distortion becomes negligible.
  • Antenna scaling: 2.5 times more antennas are needed at high SNR, and 2.6 times more at low SNR, for quantized systems to match unquantized MRC performance.At high SNR, this corresponds to approximately a 4 dB antenna-related gap.
  • Transmission schemes: At high SNR, the single-carrier and OFDM curves coincide for both MRC and ZFC under the same pilot budget.The figure uses βkPk/N0 = 10 dB and Np = KL pilot symbols.
  • Antenna scaling: Weak users require 10.4 times more antennas with ZFC and 10.6 times with MRC, compared with 2.6 and 2.5 times under equal SNR.The larger gap is attributed to degraded channel estimation when pilot orthogonality is lost through quantization.
  • SNR dependence: At −5 dB SNR, quantized MRC achieves approximately 70% of unquantized performance, while ZFC achieves 60%, with about 2 bpcu for MRC.The quantized rate is limited by a rate ceiling, which can be compensated by increasing antennas.

VII. CONCLUSION

The paper derives and evaluates an achievable-rate lower bound for wideband massive MIMO with one-bit ADCs and low-complexity estimation and detection. It concludes that frequency selectivity reduces quantization-related distortion, making both single-carrier and OFDM transmission practical, though oversampling may improve performance.

  • VII. CONCLUSION: The derived achievable rate is a capacity lower bound for one-bit massive MIMO with estimated CSI and frequency-selective Rayleigh-fading channels.The rate converges to a closed-form limit as the number of channel taps grows.
  • VII. CONCLUSION: Frequency-selective channels spread received interference evenly over time, making quantization error additive and circularly symmetric.This supports low-complexity receive combining and channel estimation for multiuser detection.
  • VII. CONCLUSION: One-bit ADCs require approximately three times more antennas at low to moderate SNR to match unquantized performance with the proposed estimator.The alternative is reduced rate performance.
  • VII. CONCLUSION: Amplitude distortion becomes negligible in wideband systems, so OFDM is affected by one-bit quantization in the same way as single-carrier transmission.The remaining quantization error is circularly symmetric Gaussian in massive MIMO.
  • VII. CONCLUSION: Oversampling the received signal may achieve better performance than the paper’s achievable-rate bound.The authors identify oversampling receivers as a direction for future research.

APPENDIX B PROOF OF LEMMA 3

The proof establishes the asymptotic behavior of the relevant correlation by using Gaussian conditioning and the scaling expression, showing that it vanishes as the number of channel taps grows.

  • APPENDIX B PROOF OF LEMMA 3: The variables hmk[ℓ], ym[n], and em[n] form a Markov chain because they are functions of one another.This relationship supports the conditional-expectation steps in the proof.
  • APPENDIX B PROOF OF LEMMA 3: The conditional mean of a Gaussian variable given a Gaussian-noisy observation is identified with its LMMSE estimate.This fact is used to evaluate the conditional expectation in (92).
  • APPENDIX B PROOF OF LEMMA 3: As L →∞, Prx[n] converges almost surely to its limiting value and the correlation goes to zero.The scaling expression for ρ is used in this asymptotic step.

APPENDIX C PROOF OF THEOREM 1

The proof decomposes the estimated signal into uncorrelated components and evaluates the resulting expectations to obtain the asymptotic rate expression as the number of channel taps tends to infinity.

  • APPENDIX C PROOF OF THEOREM 1: The estimated signal is represented as a sum of distinct terms before their pairwise correlations are analyzed.The decomposition is the starting point for proving the rate result.
  • APPENDIX C PROOF OF THEOREM 1: Each term in the decomposition is uncorrelated with the others, including the quantization error and transmit signal after their correlation is shown to vanish.The proof uses Gaussian transmit signals and a Markov-chain argument.
  • APPENDIX C PROOF OF THEOREM 1: Evaluating the expectations in the achievable-rate expression and taking L →∞ yields the limiting rate R′k.Corollary 1 is used in the final asymptotic step.
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