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Mixed-ADC Massive MIMO

Ning Liang, Wenyi Zhang

arXiv:1504.03516v2cs.IT

TL;DR

The paper addresses how to reduce massive-MIMO receiver cost and power without sacrificing most conventional performance. It proposes a mixed-ADC architecture and analyzes achievable rates with GMI across fixed, fading, and multi-user channels. Numerical results show that relatively few high-resolution ADCs retain a large fraction of conventional capacity while substantially reducing energy consumption, including against antenna selection.

  • Problem

    Massive-MIMO hardware cost and circuit power scale with antenna count, while one-bit ADCs can incur large rate loss and complicate channel estimation.

  • Method

    The paper combines one-bit and high-resolution ADCs and uses generalized mutual information to analyze achievable rates across fixed SIMO, fading, and multi-user scenarios.

  • Results

    Relatively few high-resolution ADCs achieve a large fraction of conventional channel capacity or achievable rate, while mixed-ADC operation reduces energy consumption and outperforms equal-budget antenna selection.

  • Takeaways & Limitations

    Mixed-ADC receivers provide an attractive balance between spectral efficiency and energy efficiency in both single-user and multi-user massive MIMO.

  • Takeaways & Limitations

    The analysis focuses on narrow-band channels, leaving frequency-selective fading and related quantization interference for future work.

Abstract

from arXiv · show

Motivated by the demand for energy-efficient communication solutions in the next generation cellular network, a mixed-ADC architecture for massive multiple input multiple output (MIMO) systems is proposed, which differs from previous works in that herein one-bit analog-to-digital converters (ADCs) partially replace the conventionally assumed high-resolution ADCs. The information-theoretic tool of generalized mutual information (GMI) is exploited to analyze the achievable data rates of the proposed system architecture and an array of analytical results of engineering interest are obtained. For fixed single input multiple output (SIMO) channels, a closed-form expression of the GMI is derived, based on which the linear combiner is optimized. The analysis is then extended to ergodic fading channels, for which tight lower and upper bounds of the GMI are obtained. Impacts of dithering and imperfect channel state information (CSI) are also investigated, and it is shown that dithering can remarkably improve the system performance while imperfect CSI only introduces a marginal rate loss. Finally, the analytical framework is applied to the multi-user access scenario. Numerical results demonstrate that the mixed-ADC architecture with a relatively small number of high-resolution ADCs is able to achieve a large fraction of the channel capacity of conventional architecture, while reduce the energy consumption considerably even compared with antenna selection, for both single-user and multi-user scenarios.

I. INTRODUCTION

The paper proposes mixed-ADC massive MIMO to reduce hardware and energy costs while retaining much of conventional architectures’ performance, and evaluates it analytically across single-user and multi-user settings.

  • Massive MIMO improves spectral and radiated energy efficiency, but antenna-scaled hardware cost and circuit power motivate energy-efficient transceiver designs.
  • The mixed-ADC architecture partially replaces high-resolution ADCs with one-bit ADCs to reduce hardware cost and power consumption while retaining a large fraction of performance.
  • For fixed SIMO channels, the paper derives closed-form GMI expressions, optimizes the linear combiner, studies low- and high-SNR behavior, and evaluates dithering.
  • For ergodic fading channels, tight lower and upper GMI bounds are derived; a small number of high-resolution ADCs achieves a large portion of conventional channel capacity and outperforms equal-budget antenna selection.
  • In multi-user access, mixed-ADC systems with few high-resolution ADCs achieve a large fraction of conventional achievable rate and again outperform antenna selection with the same number of high-resolution ADCs.
  • At equal spectral-efficiency loss, both mixed-ADC and antenna-selection architectures reduce energy consumption, while mixed-ADC performs better, especially for multi-user systems.

III. GMI AND OPTIMAL COMBINING

The paper uses GMI to characterize achievable rates after quantization and linear combining, then optimizes the resulting correlation-based criterion for fixed SIMO channels.

  • The framework models output quantization and linear combining through a memoryless nonlinear distortion function f(·) that maps (x, h, z) to the processed output.
  • GMI is a capacity lower bound characterizing rates achievable with Gaussian codebooks and nearest-neighbor decoding under the specified receiver processing.
  • The linear receiver is not necessarily optimal because a nonlinear receiver may outperform it, which the paper leaves for future work.
  • For given linear combiner w and ADC switch vector δ, Proposition 1 provides an explicit GMI expression after optimizing the nearest-neighbor decoding parameter.
  • The scaling parameter a adjusts power imbalance between the transmitted input and processed output and is selected to optimize decoding performance.
  • Because κ(w, δ) is the squared input-output correlation coefficient and GMI increases strictly with κ, the combiner and ADC switch are designed by maximizing κ.
  • Proposition 2 gives a closed-form expression for κ using the input-output correlation vector and processed-output covariance matrix, enabling optimization over w and δ.

B. Optimization of Linear Combiner

For fixed h and δ, the paper maximizes GMI by optimizing the linear combiner through an equivalent correlation objective. The resulting combiner is linear MMSE, and mixed ADCs outperform antenna selection with the same number of high-resolution ADCs.

  • Combiner optimization: For given h and δ, Proposition 3 gives the optimal linear combiner w.The resulting design is a linear MMSE combiner that minimizes the mean squared estimation error of x from r among linear combiners.
  • Comparison with antenna selection: The optimized mixed-ADC architecture achieves better performance than antenna selection with the same number of high-resolution ADCs.When high-resolution ADCs are assigned to the antennas with the strongest K link magnitude gains, the optimized GMI exceeds C(N, K, 0).
  • All-high-resolution special case: When K = N, the optimal combiner reduces to maximum ratio combining, and GMI equals the conventional architecture capacity C(N, N, 0).This is the all-high-resolution-ADC special case.

C. Asymptotic Behaviors of IGMI(wopt, δ)

The paper characterizes mixed-ADC GMI in low- and high-SNR regimes and studies dithering for high-SNR one-bit branches. One-bit ADCs contribute less as SNR grows, while dithering can mitigate this behavior.

  • Low-SNR behavior: At low SNR, one-bit quantization degrades part of the achievable rate by a factor of 2π relative to conventional high-resolution ADCs.The low-SNR expression also indicates that high-resolution ADCs should serve the antennas with the strongest K link magnitude gains.
  • High-SNR behavior: At high SNR, the high-resolution-ADC contribution grows linearly with Es, whereas the one-bit-ADC contribution approaches a positive constant.Thus high-resolution ADCs should again be switched to the antennas with the strongest K link magnitude gains.
  • High-SNR behavior: For pure one-bit quantization, GMI approaches a finite limit as Es increases.The paper attributes this saturation to the receiver’s inability to recover transmit-signal amplitude at sufficiently high Es.
  • Dithering: Dithering injects Gaussian noise before one-bit quantization when an antenna’s receive SNR exceeds threshold T, reducing its post-dither SNR to T.The optimal threshold Topt depends on K, N, and SNR and is found numerically.

IV. ERGODIC FADING CHANNELS

The fixed-channel GMI framework is extended to block-fading channels with instantaneous channel-dependent combiner and ADC-switch designs. Because the resulting optimization is difficult, the paper derives lower and upper GMI bounds.

  • Ergodic-fading formulation: For ergodic fading channels, the GMI optimization depends on the random channel realization and uses w and δ designed from instantaneous channel information.The analysis assumes a block-fading process across coherence intervals.
  • GMI bounds: The paper derives lower and upper bounds for IGMI in the ergodic fading scenario.The bounds are presented in Proposition 4 and are later evaluated numerically for tightness.
  • GMI bounds: The lower bound uses the fixed-channel optimal combiner as a feasible design, so it is not necessarily globally optimal for the fading-channel objective.The upper bound follows by exchanging the supremum and expectation operations.
  • GMI bounds: The resulting upper-bound argument follows directly from the fixed-SIMO expressions and established fixed-channel results.This connects the ergodic-fading analysis to the earlier fixed-channel framework.

B. Training and Effect of Imperfect CSI

For imperfect CSI, channel estimation uses only the high-resolution ADCs in a round-robin schedule, after which the fixed-channel analysis is modified using the channel estimate and estimation error. Training consumes N/K symbol times and introduces a rate-loss factor.

  • Imperfect-CSI model: The imperfect-CSI analysis uses only high-resolution ADCs for channel estimation because coarsely quantized channel training is inefficient and difficult to analyze.The mixed-ADC receiver then designs w and δ from the channel estimate.
  • Training procedure: Round-robin training connects K high-resolution ADC pairs to successive groups of antennas, so the training phase lasts about N/K symbol times.The procedure estimates K channel coefficients per training symbol.
  • Imperfect-CSI model: The channel is decomposed into an estimated coefficient and an independent estimation error, with MSE_t = σ_t^2.The imperfect-CSI GMI analysis incorporates both the estimate and error into the received-signal model.
  • Training overhead: Training reduces the achievable rate through a leading factor involving T − N/K over T, where T is the coherence interval length.The rate-loss factor accounts for the symbols spent on channel training.

V. EXTENSION TO MULTI-USER SCENARIO

The multi-user extension derives user-specific GMI expressions and ergodic-channel bounds, then uses low-SNR behavior to motivate ADC switching while evaluating heuristic schemes at high SNR.

  • System model: The model serves M single-antenna users with perfect BS CSI and only K pairs of high-resolution ADCs.The remaining ADC pairs are one-bit ADCs.
  • Fixed multi-user channels: Proposition 6 gives the GMI of user j when signals from the other users are treated as noise.The expression uses the correlation vector between the quantized output and user j’s signal, together with the quantized-output covariance matrix.
  • ADC switching: At low SNR, the sum GMI suggests switching the K high-resolution ADC pairs to antennas with the largest aggregate channel gains.This follows the asymptotic behavior of the multi-user GMI as Es approaches zero.
  • High-SNR limitation: High-SNR GMI is analytically intractable, so no generally convincing ADC switching scheme is available for the multi-user scenario.The paper therefore evaluates two heuristic switching schemes numerically.
  • ADC switching: The numerical study compares random switching with norm-based switching based on the aggregate per-antenna channel-gain metric.The norm-based scheme implements the low-SNR recommendation, while the random scheme switches high-resolution ADCs randomly.

B. Ergodic Fading Channels

For ergodic fading channels, the paper derives tight GMI bounds, models receiver power, and evaluates spectral-efficiency and energy-efficiency trade-offs under perfect and imperfect CSI. Numerical results show that few high-resolution ADCs preserve much of conventional performance, while dithering can improve low-resolution operation.

  • Ergodic-channel analysis: Proposition 7 provides lower and upper bounds on the GMI for each user in ergodic fading channels.The bounds are numerically evaluated for tightness.
  • Power modeling: The receiver power models compare conventional architecture, antenna selection, and mixed-ADC architecture using circuit-power components.One-bit ADC power is neglected because it is considered marginal relative to other circuitry.
  • Numerical validation: The lower and upper GMI bounds virtually coincide for perfect CSI, making the lower bound sufficient for subsequent spectral-efficiency evaluation.This conclusion is based on the numerical results associated with Figure 3.
  • Energy-efficiency metrics: Normalized spectral efficiency and normalized energy consumption jointly characterize energy efficiency so that rate loss is considered alongside power reduction.The paper uses these metrics for single-user and multi-user comparisons.
  • Outage performance: At SNR = 0dB, K = 10 reaches 85% and K = 20 reaches 92% of conventional outage-capacity for Pout = 5%.The mixed-ADC architecture uses N = 100 antennas, and one-bit ADCs provide greater benefit at low to moderate SNR than at high SNR.

B. GMI for Ergodic Fading SIMO Channel

The ergodic-fading analysis derives tight GMI bounds and evaluates dithering, imperfect CSI, multi-user switching, and energy-efficiency tradeoffs. Numerical results show that a small number of high-resolution ADCs preserves much of conventional performance while reducing energy consumption.

  • Ergodic-fading GMI bounds: The lower and upper GMI bounds virtually coincide, supporting use of the lower bound for subsequent spectral-efficiency evaluation.This tightness is reported for the ergodic-fading analysis and remains negligible with imperfect CSI.
  • Asymptotic behavior: With merely one pair of high-resolution ADCs, the GMI increases linearly with 10 log10(SNR) at high SNR, whereas pure one-bit quantization suffers significant high-SNR rate loss.At low SNR, the one-bit case closely approaches configurations with K > 0.
  • Dithering: Gaussian dithering can substantially improve spectral efficiency, especially when K = 0, but its benefit for K > 0 declines as K or SNR increases.The high-resolution ADC contribution increasingly reduces the relative benefit of dithering.
  • Multi-user access: At SNR = 0 dB and N = 100, norm-based switching with K = 10 and K = 20 achieves 77% and 81% of the conventional per-user rate, respectively.The mixed-ADC architecture also achieves noticeably higher spectral efficiency than antenna selection.
  • Energy efficiency: If 10% spectral-efficiency degradation is allowed in the single-user case, antenna selection reduces energy consumption by more than 60%, while mixed ADCs provide about 10% further reduction at low to moderate SNR.In the high-SNR regime, antenna selection may be more energy efficient because one-bit ADCs become less beneficial.
  • Energy efficiency: In the multi-user case, mixed ADCs always outperform antenna selection across the considered SNR range, with a larger gap as the number of users increases.At K ≈20, a 20% spectral-efficiency loss corresponds to a 70% energy-consumption reduction for the stated parameters.

APPENDIX

The appendix develops lemmas and correlation identities used to evaluate the mixed-ADC receiver's correlation vector and covariance matrix. These quantities support the subsequent closed-form GMI derivation.

  • Auxiliary lemmas: Lemma 1 relates expectations involving jointly Gaussian real variables and their signs through their correlation coefficient.The lemma is introduced for evaluating sign-dependent correlation terms.
  • Auxiliary lemmas: Lemma 2 establishes symmetry between E[S* · sgn(S + T)] and E[S · sgn*(S + T)] for independent complex Gaussian variables.Its proof decomposes the complex expectation into real and imaginary components.
  • Correlation evaluation: The appendix evaluates the correlation vector R_rx and covariance matrix R_rr, including diagonal terms, using the introduced lemmas and sign correlations.These calculations are then combined to obtain the covariance entries and conclude the proof.

B. Asymptotic behavior of IGMI(wopt, δ) in low SNR regime

The low-SNR analysis examines the asymptotic behavior of the optimized GMI using limiting expressions for the correlation and covariance quantities. The resulting expansion yields the low-SNR asymptote.

  • Low-SNR limit: The analysis takes the limit of κ(w_opt, δ) as E_s → 0 using limits of R_rx/E_s and R_rr.Continuity of matrix inversion is used when the limiting covariance matrix is nonsingular.
  • Low-SNR limit: The logarithmic expansion log(1 + x/(1 − x)) = x + o(x) as x → 0 is used to obtain equation (22).This converts the limiting effective expression into the stated low-SNR asymptotic form.

C. Asymptotic behavior of IGMI(wopt, δ) in high SNR regime

The high-SNR analysis partitions the receiver quantities into high-resolution and one-bit ADC components and derives the corresponding effective SNR. The resulting expression characterizes optimized GMI behavior at large signal energy.

  • High-SNR setup: The channel vector is rearranged so that coefficients associated with high-resolution ADCs occupy the first K positions.This ordering supports the block-matrix treatment of the covariance matrix.
  • High-SNR setup: As E_s tends to infinity, the covariance matrix and its inverse are represented in partitioned blocks involving A, B, and U.The square blocks A and B are invertible, while U is rectangular.
  • High-SNR derivation: The Sherman-Morrison formula and the inverse of a partitioned matrix are applied to simplify the high-SNR covariance expressions.These matrix identities lead to the final effective-SNR calculation.
  • High-SNR result: The high-SNR derivation concludes with an effective-SNR expression involving π − [4 + O(1/E_s)]q^H B^-1q.This expression is obtained after simplifying κ(w_opt, δ).
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