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Capacity Analysis of One-Bit Quantized MIMO Systems with Transmitter Channel State Information

Jianhua Mo, Robert W. Heath

arXiv:1410.7353v3cs.IT

TL;DR

The paper addresses capacity characterization for wideband MIMO systems using one-bit ADCs, where high-resolution conversion is power-intensive. It analyzes one-bit-quantized channels with CSI at both ends, deriving exact MISO capacity, capacity bounds for SIMO and MIMO channels, and input-design methods. The results include tight finite-SNR bounds under full row rank and show that sparse mmWave capacity is limited by propagation paths.

  • Problem

    High-speed, high-resolution ADCs are costly and power-hungry in wideband systems, motivating capacity analysis of MIMO channels with one-bit ADCs.

  • Method

    The paper derives analytical capacities and bounds across SISO, MISO, SIMO, MIMO, and sparse mmWave channels, and uses convex optimization to design transmitter constellations.

  • Results

    The paper obtains closed-form MISO capacity, infinite-SNR SIMO and MIMO results, and a finite-SNR MIMO upper bound that is tight when the channel has full row rank.

  • Takeaways & Limitations

    One-bit quantized systems can be analyzed and approached with designed discrete inputs, while sparse mmWave capacity is limited by the number of propagation paths.

  • Takeaways & Limitations

    The analysis assumes complete and perfect CSI at both the transmitter and receiver.

Abstract

from arXiv · show

With bandwidths on the order of a gigahertz in emerging wireless systems, high-resolution analog-to-digital convertors (ADCs) become a power consumption bottleneck. One solution is to employ low resolution one-bit ADCs. In this paper, we analyze the flat fading multiple-input multiple-output (MIMO) channel with one-bit ADCs. Channel state information is assumed to be known at both the transmitter and receiver. For the multiple-input single-output channel, we derive the exact channel capacity. For the single-input multiple-output and MIMO channel, the capacity at infinite signal-to-noise ratio (SNR) is found. We also derive upper bound at finite SNR, which is tight when the channel has full row rank. In addition, we propose an efficient method to design the input symbols to approach the capacity achieving solution. We incorporate millimeter wave channel characteristics and find the bounds on the infinite SNR capacity. The results show how the number of paths and number of receive antennas impact the capacity.

I. INTRODUCTION

The paper studies one-bit-quantized MIMO channels with CSI at both transmitter and receiver, motivated by the power cost of high-speed, high-resolution ADCs in wideband systems. It characterizes capacities and bounds across channel configurations and SNR regimes, and develops input-design methods for approaching capacity.

  • Motivation: High-speed, high-resolution ADCs become costly and power-hungry as bandwidth and sampling rate increase.Flash ADC power grows exponentially with resolution because an ideal b-bit ADC uses 2^b−1 comparators.
  • Research gap: Prior quantized-MIMO capacity analyses without transmitter CSI optimized neither the input distribution nor the constellation, limiting their applicability beyond achievable-rate characterization.Earlier work commonly assumed BPSK, QAM, or independent QPSK signaling.
  • Motivation: One-bit ADCs reduce receiver power and circuit complexity because they can be implemented as simple comparators and may eliminate automatic gain control.The paper identifies this architecture as especially attractive for wideband systems.
  • Contributions: The paper derives the full-SNR MISO capacity, infinite-SNR SIMO capacity, and infinite- and finite-SNR MIMO bounds for one-bit quantization.The finite-SNR upper bound is tight for row-full-rank channels, while the MIMO infinite-SNR analysis uses combinatorial geometry.
  • Contributions: A convex-optimization-based method designs transmitter input alphabets to approach the capacity-achieving solution.The method is motivated by the difficulty of directly optimizing discrete input distributions for one-bit channels.
  • mmWave extension: For sparse mmWave channels, infinite-SNR capacity is mainly limited by the number of propagation paths, with the single-path case admitting a capacity-achieving strategy.The paper also investigates how receive-antenna count affects the resulting bounds.

III. SISO AND MISO CHANNEL CAPACITIES WITH ONE-BIT QUANTIZATION

This section derives exact one-bit-quantized capacities for SISO and MISO channels with transmitter and receiver CSI. The optimal strategy rotates the signal to align with the channel and uses QPSK signaling, while one-bit quantization causes a 1.96 dB low-SNR power loss.

  • SISO channel: The SISO capacity optimization is difficult because mutual information contains multiple integrals and sums, but the single-receiver case admits an exact solution.The analysis proceeds from SISO to MISO capacity-achieving strategies.
  • SISO channel: Rotated QPSK signaling with uniform probabilities achieves the one-bit-quantized SISO capacity.The rotation compensates for the scalar channel phase and decouples the real and imaginary components.
  • MISO channel: The one-bit-quantized MISO capacity is achieved by maximal-ratio-transmission beamforming and QPSK signaling.The beamformer transforms the MISO channel into an equivalent SISO channel with gain ||h||.
  • MISO channel: 2 bps/Hz is the high-SNR limit of the one-bit-quantized MISO capacity.This is the same limiting value identified for the corresponding one-bit-quantized SISO capacity.
  • MISO channel: 1.96 dB is the low-SNR power loss caused by one-bit quantization in the MISO channel with transmitter CSI.The loss is expressed as 10 log10(π/2).
  • MISO channel: Independent QPSK signaling across transmitter antennas incurs a 1/Nt power loss relative to the optimal beamforming strategy.The paper attributes the gap to the array gain provided by beamforming.

IV. SIMO AND MIMO CHANNEL CAPACITIES AT INFINITE SNR WITH ONE-BIT QUANTIZATION

At infinite SNR, one-bit SIMO capacity is characterized through distinguishable quantization regions, while MIMO capacity follows from the geometry of channel-induced partitions. The SIMO capacity lies between log2(4Nr) and log2(4Nr + 1) and approaches the upper bound for sufficiently many receive antennas.

  • SIMO Channel with One-Bit Quantization: The SIMO analysis partitions transmit symbols into zero, threshold-phase, and non-threshold-phase categories with distinct quantization transition probabilities.Non-threshold-phase symbols produce deterministic outputs, while the zero symbol produces uniformly distributed outputs.
  • SIMO Channel with One-Bit Quantization: At most 4Nr+1 input symbols are needed for the SIMO capacity-achieving distribution, giving the upper bound log2(4Nr + 1).The optimal distribution assigns zero probability to symbols on region boundaries.
  • SIMO Channel with One-Bit Quantization: For Nr = 3, the optimal SIMO constellation contains 12 nonzero symbols and the zero symbol.The nonzero symbols occupy the 12 regions formed by the channel-rotated quantization thresholds.
  • SIMO Channel with One-Bit Quantization: The SIMO capacity is at least log2(4Nr), achieved by transmitting the 4Nr distinguishable symbols with equal probability.This construction excludes the zero symbol.
  • SIMO Channel with One-Bit Quantization: When Nr ≥6, the high-SNR SIMO capacity is very close to log2(4Nr + 1).The capacity and both bounds are compared numerically in Fig. 3.

B. MIMO Channel Capacity with One-Bit Quantization

At infinite SNR, one-bit MIMO capacity is linked to the number of regions induced by channel hyperplanes and their dual subspace intersections. The resulting bounds depend on channel rank and general position, with exact capacity in the full-rank equality case.

  • MIMO Channel Capacity with One-Bit Quantization: Under general position, the transmitter approaches infinite-SNR capacity by sending the zero symbol and one symbol from each region when Nt < Nr.When Nt ≥Nr, all 2^2Nr quantization outputs can be used with equal probabilities, without transmitting zero.
  • MIMO Channel Capacity with One-Bit Quantization: Hyperplanes induced by the equivalent real channel divide signal space into regions corresponding to distinguishable quantization outputs.The dual geometric formulation counts orthants intersected by the channel-column subspace.
  • MIMO Channel Capacity with One-Bit Quantization: The geometric capacity characterization relies on the condition of general position, which may not hold for practical channel matrices.Continuous channel coefficients satisfy this condition with probability one.
  • MIMO Channel Capacity with One-Bit Quantization: The infinite-SNR MIMO capacity satisfies 2^rank(H) ≤ C1bit,MIMO ≤ log2(K(Nr, rank(H)) + 1) when Nr > rank(H).When Nr = rank(H), the capacity equals 2Nr.
  • MIMO Channel Capacity with One-Bit Quantization: The high-SNR capacity is not generally invariant to exchanging transmit and receive dimensions, unlike unquantized MIMO with CSIT.K(Nr, Nt) is generally asymmetric, increases with Nr, and saturates at 2Nr when Nt ≥Nr.

V. BOUNDS OF MIMO CHANNEL CAPACITY WITH ONE-BIT QUANTIZATION AT FINITE SNR

The finite-SNR analysis introduces an upper bound and two lower bounds for one-bit MIMO capacity. The quantity log2 K(Nr, Nt) is reported as close to the high-SNR capacity and varies systematically with antenna dimensions.

  • V. BOUNDS OF MIMO CHANNEL CAPACITY WITH ONE-BIT QUANTIZATION AT FINITE SNR: Finite-SNR capacity analysis proposes a new upper bound together with two lower bounds.The section uses these bounds to study one-bit MIMO capacity beyond the infinite-SNR regime.
  • V. BOUNDS OF MIMO CHANNEL CAPACITY WITH ONE-BIT QUANTIZATION AT FINITE SNR: log2 K(Nr, Nt) is close to the high-SNR capacity of the MIMO channel with Nt transmit and Nr receive antennas.Figures 4 and 5 examine this quantity across receive- and transmit-antenna dimensions.

A. Upper Bound of MIMO Channel Capacity at finite SNR

The finite-SNR upper-bound derivation minimizes conditional output entropy under a transmit-power constraint. The bound is achieved for equal nonzero singular values but can be loose when the channel lacks full row rank.

  • A. Upper Bound of MIMO Channel Capacity at finite SNR: The entropy-based upper bound follows from the limited number of one-bit quantization outputs and the largest singular value σmax of H.The output-count argument uses at most 2^2Nr quantization outputs.
  • A. Upper Bound of MIMO Channel Capacity at finite SNR: The upper-bound derivation minimizes H(r|x) subject to the transmit-power constraint.The conditional entropy is analyzed using independent Gaussian noise components and the convexity of the binary entropy expression.
  • A. Upper Bound of MIMO Channel Capacity at finite SNR: The upper bound is achieved when H has Nr equal singular values, equivalently HH* = σ^2I, using simple channel inversion.This condition corresponds to a full-row-rank channel with equal singular values.
  • A. Upper Bound of MIMO Channel Capacity at finite SNR: When rank(H) < Nr, the upper bound can be loose because it approaches 2Nr at high SNR while SIMO capacity is around log2(4Nr).The gap is explicit for Nr ≥2.
  • A. Upper Bound of MIMO Channel Capacity at finite SNR: At low SNR, the derived expression includes ln 2 + o(Pt), and one-bit quantization incurs the reported low-SNR loss.The supplied derivation states the low-SNR asymptotic result and its quantization penalty.

B. Lower Bounds on MIMO Channel Capacity at Finite SNR

The section develops achievable-rate lower bounds for one-bit quantized MIMO channels using channel inversion and related signaling strategies, comparing them with QPSK and capacity bounds. Channel inversion is especially effective for well-conditioned or structurally symmetric channels, but its gain and tightness depend on channel properties.

  • Channel inversion: Channel inversion precoding with independent QPSK signaling provides a simple achievable-rate strategy when HH∗ is invertible.The transmitted QPSK vector has identity covariance, and the precoder is scaled to satisfy the transmission-power constraint.
  • Tightness conditions: Channel inversion is capacity-achieving when H has Nr identical singular values.In that case, the channel decomposes into Nr parallel one-bit quantized SISO channels with equal channel gain.
  • Tightness conditions: The power loss of channel inversion relative to the optimum is bounded, and the strategy is nearly optimum when H has a small condition number.The bound is expressed using the eigenvalues of HH∗ and the associated inverse-channel power requirement.
  • Low-SNR comparison: At low SNR, independent QPSK signaling achieves a rate proportional to (2/π) tr(HH∗)Pt/(Nt ln 2).This result is attributed to prior analysis for independent QPSK signaling with one-bit ADCs.
  • Low-SNR comparison: When Nt ≫ Nr, channel inversion benefits from array gain that increases with the number of transmitter antennas.With relatively small transmitter arrays, channel inversion does not provide gain over simple QPSK signaling.

2) Additive Quantization Noise Model (AQNM):

The AQNM approach models quantization error as Gaussian noise to obtain a lower bound on achievable rate. For one-bit quantization, this bound is useful at low SNR but is loose at high SNR.

  • AQNM formulation: The AQNM lower bound treats quantization error as Gaussian distributed noise and uses a distortion factor ρ.The normalized channel matrix and eigenvalues of its Gram matrix enter the resulting bound.
  • SNR regimes: The AQNM lower bound is quite tight at low SNR when additive white Gaussian noise dominates.Its low-SNR behavior reflects the regime in which thermal noise is stronger than quantization noise.
  • SNR regimes: At high SNR, the AQNM lower bound is loose because quantization noise dominates.When HH∗ is invertible, channel inversion instead approaches an achievable rate of 2Nr at high SNR.
  • AQNM formulation: Because each row of the normalized channel has unit norm, concavity of log2(1 + x) yields a simplified rate bound.For one-bit quantization, the distortion factor is ρ = π−2.

VI. NUMERICAL INPUT OPTIMIZATION METHODS FOR THE MIMO CHANNEL

The paper proposes a convex-optimization procedure for constructing input symbols associated with feasible one-bit quantizer outputs, then optimizes their probabilities. The approach approaches infinite-SNR capacity under general-position conditions, while low-rank mmWave channels require specialized treatment.

  • Optimization method: For each possible quantization output r, the method searches for an input x satisfying r = sgn(Hx).This avoids requiring HH∗ to be invertible, unlike channel inversion.
  • Optimization method: The optimization maximizes the minimum distance between Hx and the one-bit ADC threshold at zero.A nonnegative distance guarantees that the resulting quantizer output equals the target vector r.
  • Optimization method: Each fixed-sign optimization problem is convex and can be solved with a software solver such as CVX.The compact formulation preserves linear inequality constraints for fixed quantization signs.
  • Complexity reduction: The exhaustive procedure considers 2^2Nr possible quantization outputs, but feasibility checks can reduce the number of optimization problems solved.Fourier–Motzkin elimination can test the associated systems of linear inequalities before optimization.
  • Capacity approach: At high SNR, the convex-optimization method approaches infinite-SNR capacity when the channel satisfies the general-position condition.Under that condition, M = K(Nr, Nt), and the lower bound converges to M.
  • mmWave channels: For mmWave channels, sparse scattering makes H low rank, so the general-position condition cannot be applied directly.The channel is modeled with L paths and L < min{Nt, Nr}; a reduced-dimensional equivalent channel restores the relevant structure for the bound.
  • mmWave channels: Multipath improves infinite-SNR capacity in mmWave MIMO systems, while one-path channels admit a capacity-achieving beamforming strategy.For one path, matched-filter beamforming reduces the channel to an equivalent SIMO channel, whose symbols are designed by a cutting-plane method.

VIII. SIMULATION RESULTS

The simulations evaluate capacity bounds, optimized input distributions, and achievable rates across SIMO, MISO, MIMO, and mmWave settings. Results illustrate finite-SNR effects, discrete optimal constellations, and the practical role of antenna array gain.

  • Simulation scope: The simulations compare capacities and achievable rates for SIMO, MISO, MIMO, and mmWave channels across SNR regimes.They also evaluate the proposed convex-optimization method and the mmWave channel model.
  • SIMO results: For h = [e^jπ/8, e^−jπ/8]^T and Pt = 10, the optimal SIMO input distribution achieves about 2.52 bps/Hz.The displayed distribution contains rotated 8-PSK symbols and the zero symbol.
  • MISO results: One-bit MISO capacity approaches 2 bps/Hz at high SNR.For ||h|| = 16, the capacity is close to the upper bound when SNR exceeds −15 dB.
  • MISO results: The MISO transmitter antenna array provides power gain, while one-bit quantization causes about 2 dB power loss at low and medium SNR.The simulations report that array gain can make the high-SNR regime appear at relatively low practical SNR.
  • SIMO method: The SIMO input distributions are obtained by optimizing probabilities over a fine discrete grid using an iterative cutting-plane method.The algorithm converges in several tens of iterations, with each iteration solving a convex problem.
  • SIMO results: For h = [1, 2e^−jπ/3]^T and Pt = 20 dB, the achievable SIMO rate is 3.0050 bps/Hz.The paper notes that optimal constellations for general channels need not be regular.

C. MIMO Channel with One-Bit Quantization

The paper evaluates achievable rates for one-bit-quantized MIMO systems across SNR regimes and mmWave channels, comparing constellation-design methods with theoretical bounds and unquantized baselines. Convex optimization approaches the infinite-SNR capacity, while channel rank, SNR, and path count shape performance.

  • Simulation setup: The simulations generate independent CN(0, 1) channel coefficients and evaluate achievable rates for several MIMO dimensions.The considered systems include 2 × 2, 2 × 4, and 3 × 2 configurations.
  • Input design: The input alphabet contains 2^2Nr symbols, constructed using channel inversion or convex optimization, with either equal or Blahut–Arimoto-optimized probabilities.The optimized-probability variant is denoted “BA” in the figures.
  • Medium and high SNR: Convex optimization converges to 4 bps/Hz for the 2 × 2 and 2 × 4 cases and approximately 5.7 bps/Hz for the 3 × 2 case.These limits are the corresponding infinite-SNR upper bounds, verifying that the method can approach the capacity.
  • Comparisons: The quantized systems incur less than 5 dB power loss relative to unquantized systems at medium SNR.Independent QPSK signaling is inferior to the two proposed methods at high SNR, while probability optimization improves rates at low and medium SNR but not at high SNR.
  • mmWave channels: In the mmWave 4 × 256 system, high-SNR rates converge to 4, 7, and 7.9 bps/Hz for L = 1, 2, and 3 paths, respectively.The convergence occurs only at very high SNR, above 60 dB in the figure, and the paper concludes that capacity is limited by the number of paths.
  • Conclusions: The conclusions relate infinite-SNR MIMO capacity to a combinatorial geometry problem and state that channel inversion is near-optimal for full-row-rank channels with small condition number.The paper also concludes that treating quantization error as Gaussian noise is unsuitable at high SNR.
  • Scope and future work: The analysis assumes complete and perfect CSI at both transmitter and receiver, with imperfect-CSI robustness identified as future work.The paper also points to compressed channel estimation and CSI feedback for sparse mmWave channels.

APPENDIX A BOUNDS OF THE ACHIEVABLE RATE OF CONVEX

The appendix derives a lower bound for the achievable rate of the convex-optimization strategy by analyzing quantization vectors, candidate symbols, and their induced entropies. It uses concavity to reduce the relevant minimization to extreme probability points.

  • Bound construction: The appendix considers M convex-optimization problems with nonzero objective values and denotes their quantization vectors, optimal solutions, and objective values.These quantities are used to construct the rate lower bound.
  • Entropy evaluation: The candidate symbols are assumed to be transmitted with equal probability 1/M when evaluating the conditional entropy.The derivation then relates the resulting conditional entropy to mutual information.
  • Entropy evaluation: The conditional-entropy calculation uses independence across receive antennas and independence of the Gaussian noises’ in-phase and quadrature components.The noise components have variance 1/2.
  • Extreme-point analysis: Concavity of the entropy function implies that the minimum occurs at extreme points of the probability region.The appendix identifies two kinds of extreme points for the subsequent bound analysis.
  • Extreme-point analysis: The first extreme-point family assigns q/M to each of the M designated symbols and zero probability to the remaining symbols, with one designated symbol receiving the residual mass.This family is specified by the probability cases in the appendix.
  • Extreme-point analysis: The second extreme-point family assigns q/M to the first M symbols, 1 − q to one additional symbol, and zero to the other remaining symbols.The selected additional symbol is indexed within the candidate-symbol range.
  • Bound scaling: The appendix states that d_min = α(H)P_t, where α(H) is a channel-dependent constant.This relation supplies the channel-dependent scaling used in the bound.
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