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Hybrid Architectures with Few-Bit ADC Receivers: Achievable Rates and Energy-Rate Tradeoffs

Jianhua Mo, Ahmed Alkhateeb, Shadi Abu-Surra, Robert W. Heath

arXiv:1605.00668v2cs.IT

TL;DR

Wireless systems with large antenna arrays need lower-power receiver designs, but prior work largely considers either few RF chains with full-resolution ADCs or many RF chains with low-resolution ADCs. The paper proposes a generalized hybrid architecture with few-bit ADCs, derives achievable rates for channel-inversion and SVD-based transmission, and evaluates rate-power trade-offs. Its simulations show comparable low-to-medium-SNR rates and that coarse quantization usually maximizes energy efficiency.

  • Problem

    Prior receiver designs largely treat few RF chains with full-resolution ADCs and low-resolution ADCs with one RF chain per antenna as separate extremes.

  • Method

    The paper proposes a generalized hybrid architecture with few-bit ADC receivers, derives achievable rates for channel-inversion and SVD-based transmission, and analyzes rate-power trade-offs.

  • Results

    Simulations show comparable performance to fully digital or infinite-bit hybrid receivers at low-to-medium SNRs, while coarse quantization normally achieves maximum energy efficiency.

  • Takeaways & Limitations

    Hybrid combining with coarse quantization provides a better energy-rate trade-off than hybrid combining with full-resolution ADCs or 1-bit ADC combining.

Abstract

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Hybrid analog/digital architectures and receivers with low-resolution analog-to-digital converters (ADCs) are two low power solutions for wireless systems with large antenna arrays, such as millimeter wave and massive MIMO systems. Most prior work represents two extreme cases in which either a small number of RF chains with full-resolution ADCs, or low resolution ADC with a number of RF chains equal to the number of antennas is assumed. In this paper, a generalized hybrid architecture with a small number of RF chains and finite number of ADC bits is proposed. For this architecture, achievable rates with channel inversion and SVD based transmission methods are derived. Results show that the achievable rate is comparable to that obtained by full-precision ADC receivers at low and medium SNRs. A trade-off between the achievable rate and power consumption for different numbers of bits and RF chains is devised. This enables us to draw some conclusions on the number of ADC bits needed to maximize the system energy efficiency. Numerical simulations show that coarse ADC quantization is optimal under various system configurations. This means that hybrid combining with coarse quantization achieves better energy-rate trade-off compared to both hybrid combining with full-resolutions ADCs and 1-bit ADC combining.

I. INTRODUCTION

Large-array wireless systems need lower-power receiver architectures because fully digital designs are difficult to realize, while prior hybrid and few-bit ADC designs occupy two extremes. This paper generalizes the architecture by combining few-bit ADCs with a limited number of RF chains, derives achievable rates, and evaluates energy-rate trade-offs.

  • Motivation: Massive MIMO and mmWave systems use large antenna arrays to support many users, high received power, and ultra-high data rates.These benefits come with substantial mixed-signal hardware cost and power consumption in fully digital implementations.
  • Motivation: Fully digital receivers are difficult to realize because assigning one RF chain per antenna increases mixed-signal hardware cost and power consumption.Reducing RF chains can save power, but phase-shifter consumption may offset those savings.
  • Research gap: Prior approaches represent two extremes: hybrid architectures use few RF chains with high-resolution ADCs, whereas 1-bit receivers use RF chains equal to the antenna count.The latter assumption can leave hardware costs high and forgo possible analog beamforming gains.
  • Contribution: The proposed generalized hybrid architecture combines a limited number of RF chains with finite-resolution ADCs and supports hybrid precoding and combining.The paper develops channel-inversion and SVD-based transmission methods with derived achievable rates.
  • Results: At low-to-medium SNRs, the proposed few-bit architecture can achieve performance comparable to fully digital or infinite-bit hybrid receivers.The comparison is especially relevant to mmWave communications, where low-to-medium SNR operation is important.

II. SYSTEM MODEL

The proposed MIMO receiver combines hybrid analog/digital processing with a limited number of few-bit ADC RF chains. The model retains full-resolution DACs and assumes perfect channel knowledge.

  • The architecture combines hybrid analog/digital precoding with pairs of few-bit ADCs to reduce hardware cost and power consumption.
  • The system supports Nt and Nr antennas, limited transmit and receive RF chains, and Ns communicated data streams.
  • The transmitter uses full-precision DACs, whereas the receiver uses few-bit ADCs with 1–4 bits.
  • Compared with a fully digital receiver, the proposed architecture uses fewer high-resolution ADC pairs.
  • The analysis assumes high-resolution DACs and perfect channel knowledge; jointly analyzing low-resolution ADCs and DACs is left for future work.

RFHFRFFBBs + W∗

The paper formulates capacity optimization for the hybrid quantized channel and develops two transmission strategies to make the problem tractable. The formulation relies on semi-unitary analog processing and perfect channel knowledge.

  • The received signal results from analog combining, component-wise quantization of real and imaginary parts, and digital combining.
  • The baseband combiner is ignored because any invertible combiner does not affect channel capacity.
  • Capacity optimization maximizes mutual information over the signal distribution and quantization function, including ADC thresholds.
  • The analog precoder and combiner are assumed semi-unitary, which keeps the effective noise white Gaussian under the stated approximation.
  • The problem is difficult because quantization is nonlinear, depends on the input distribution, and is coupled with non-convex equality constraints.
  • Two transmission strategies are developed, including their precoding, signal distribution, and quantization designs, with achievable rates analyzed.

IV. UPPER BOUND OF THE ACHIEVABLE RATE

The paper derives upper bounds for achievable rates with one-bit quantization and introduces channel-inversion transmission, whose streams can be analyzed independently. The bounds expose finite-quantization limitations and special cases of optimality.

  • One-bit achievable-rate upper bounds increase linearly with power at low SNR but saturate at high SNR because quantization outputs are finite.
  • Upper bounds for multi-bit quantization remain unknown and are left for future work.
  • The upper bounds depend on analog precoder and combiner choices, with a looser alternative independent of analog precoding.
  • The one-bit bounds use the maximum singular value of H and can be achieved when the effective channel G is full rank with the specified receive-chain condition.
  • Infinite-bit upper bounds grow without bound as power increases, unlike the finite one-bit bounds.
  • Channel-inversion transmission removes interference among streams, quantizes each stream separately, and therefore permits an exact closed-form achievable rate.

RFHFRFFBBs + W∗

The paper analyzes effective-channel design, alternating-projection analog precoding, and one-bit channel-inversion performance. The algorithm converges rapidly, while rate and power behavior depend on channel conditioning and RF-chain allocation.

  • Maximizing SNR is equivalent to maximizing the harmonic mean of the squared singular values of the effective channel G.
  • Choosing singular vectors associated with the largest Ns singular values of H maximizes this objective but may not satisfy constant-norm constraints.
  • Alternating projection iteratively enforces constant-norm and semi-unitary constraints for the analog precoder and combiner.
  • The algorithm converges within less than 100 iterations to a normalized distance below 10^-5.
  • DFT-matrix columns satisfy constant-norm and orthogonality constraints, but searching their best combination becomes more complex than alternating projection for large antenna arrays.
  • For one-bit quantization, channel-inversion power loss is at most 10 log10 Ns dB, and the loss is small for a well-conditioned effective channel.
  • With one receive RF chain, the proposed transmission method achieves the channel capacity.
  • Turning off some receive RF chains decreases power consumption when the effective channel lacks an inverse.

C. Rate Analysis with Few-Bit Quantization

The paper models few-bit quantization using discrete signaling and derives a channel-inversion transmission method with analog and digital precoding, signaling, and quantization steps.

  • Signaling and Quantization: Quantizer thresholds are selected at the midpoints of input mass-point locations.The text notes that this combination is close to an iterative optimum and is optimal for one-bit quantization.
  • Signaling and Quantization: The transmitter uses 2^b-QAM signaling, while the receiver applies uniform quantization.This signaling and quantization combination is assumed in the proposed method.
  • Channel Inversion Based Transmission Method: The proposed channel-inversion method combines analog and digital precoding design with discrete signaling and receiver quantization.The method is summarized as Transmission Method 2.
  • Few-Bit Quantization Model: For multi-bit quantization, the analysis uses transition probabilities between input symbols and quantizer outputs.The transition probability matrix is extended from the two-bit example to higher-resolution ADCs.

12 SNRCI

The analysis derives achievable-rate expressions and bounds for channel-inversion and SVD-based transmission with few-bit ADCs, while identifying high-SNR modeling and optimization limitations.

  • Channel-Inversion Analysis: The channel-inversion rate analysis derives a Fano-based lower bound from the error probability of 2^b-PAM signaling.The bound applies to the sum rate of 2N_s sub-channels.
  • Channel-Inversion Analysis: As SNR_CI increases, the symbol error probability decreases to zero and the lower bound approaches 2N_sb bps/Hz.For one-bit quantization, the expression reduces to the corresponding one-bit result.
  • Transmission Methods: The channel-inversion precoder generally performs well at high SNR but poorly at low SNR.This motivates analysis of an SVD-based alternative at low SNR.
  • AQNM Analysis: The AQNM is accurate enough at low SNR for a lower bound but is not accurate at high SNR.The discrepancy is attributed to Gaussian input assumptions, worst-case Gaussian quantization noise, and MSE-oriented Max-Lloyd quantization.
  • SVD-Based Analysis: At low SNR, eigenmode beamforming maximizes the analyzed rate expression, and achieves the one-bit quantized-channel upper bound.The beamforming direction is the right singular vector associated with the largest singular value of the effective channel.
  • Limitations: For higher SNR, the optimal digital and analog precoders are unknown, so the paper uses suboptimal design procedures.The achievable-rate expression before quantization is also unknown in the relevant analysis.
  • SVD-Based Transmission Method: The SVD-based design uses alternating projection for analog precoding and SVD with waterfilling for digital precoding.Gaussian signaling and Max-Lloyd quantization are used in this design.

VII. SIMULATION RESULTS

The simulations evaluate the proposed methods in a measured-inspired mmWave MIMO setting with large antenna arrays and limited RF-chain counts.

  • Simulation Setup: The evaluation uses a mmWave MIMO channel with large antenna arrays and limited transmit and receive RF chains.The simulations average results over 100 channel realizations.
  • Channel Model: The channel model reflects the assumption that mmWave propagation consists mainly of line-of-sight and a few non-line-of-sight clusters.The text contrasts this sparse clustering with lower-frequency channels.
  • Channel Model: The simulated channel contains 4 clusters, each with 5 rays, and an angle spread of 7.5 degrees.These parameters are selected according to 28 GHz urban-macro NLOS measurement results.

A. Achievable Rates

The simulations compare channel inversion and SVD across SNR, ADC resolution, and RF-chain count. Few-bit quantization causes little loss at low and medium SNR, while the preferred transmission method depends on resolution and SNR.

  • SNR Comparison: At high SNR with one transmit RF chain, channel inversion achieves 2^b bps/Hz while SVD saturates below that rate.At low SNR, the two methods have close performance.
  • SNR Comparison: 1.46, 3.09, and 4.86 bps/Hz are reported for b = 1, 2, and 3, respectively.These values are reported for the evaluated channel-inversion case.
  • SNR Comparison: With 4 receive RF chains, channel inversion exceeds SVD at high SNR but performs much worse at low SNR.The low-SNR degradation is linked to the small fourth singular value in the four-cluster channel.
  • Rate Bounds: The channel-inversion lower bound is tight for one-bit ADCs and becomes tight at high SNR for other resolutions.The channel-inversion method reaches the quantized-channel upper bound at sufficiently high SNR.
  • Hybrid and Fully-Digital Architectures: The hybrid-versus-digital rate gap separates RF-chain loss from ADC-resolution loss, with the latter below 3 dB for Hybrid-3bit below 10 dB SNR.The hybrid architecture rate is the maximum of the channel-inversion and SVD methods.
  • Hybrid and Fully-Digital Architectures: A one-RF-chain hybrid receiver is much worse than the fully-digital architecture because it supports only single-stream transmission.With 4 receive RF chains, the gap between the architectures is small in the evaluated setting.
  • ADC Resolution: At SNR = −10 dB, 5-bit SVD quantization matches the infinite-bit hybrid performance, while at SNR = 10 dB, 7-bit quantization does so.High-resolution ADCs provide little gain over few-bit ADCs at low SNR; channel inversion is better at low resolution, whereas SVD is better at high resolution.
  • RF-Chain Scaling: SVD rates increase with the number of receive RF chains, while turning off RF chains can reduce power consumption and support fewer streams.The channel-inversion method achieves the largest rate in the stated comparison.

B. Energy efficiency

The paper evaluates achievable-rate and power-consumption trade-offs across hybrid and fully-digital receivers, varying ADC resolution and RF-chain count. Coarse quantization generally offers the strongest energy-rate balance under the studied configurations.

  • 4–5-bit ADCs substantially increase achievable rate over 1-bit ADCs with negligible receiver-power growth.The comparison is reported for all four cases in Fig. 7.
  • 4–5-bit ADCs provide a better rate-power trade-off than 7–8-bit ADCs, whose spectral-efficiency gains are slight but power costs are significant.
  • Below 11 Gbps, one RF chain has the best power-rate trade-off; between 11 and 18 Gbps, two RF chains are best; above 18 Gbps, fully digital is best.
  • With four RF chains, the hybrid receiver consumes more power than the fully-digital receiver at the same rate.The paper attributes this to the high power consumption of the larger number of RF components.
  • At SNR = 10 dB, the energy-efficiency trend is similar, but channel inversion achieves the maximum; four bits still give optimal energy efficiency.

VIII. CONCLUSION

The paper develops and evaluates a generalized hybrid architecture with few-bit ADC receivers using channel inversion and SVD-based transmission. Simulations find comparable low- and medium-SNR rates to fully digital systems and favorable energy-rate trade-offs with coarse quantization.

  • The paper derives achievable spectral efficiencies and energy-rate trade-offs for a generalized hybrid architecture with few-bit ADC receivers.
  • The transmission methods jointly address analog and digital precoding, transmit-signal distribution, and quantizer setup.
  • At low and medium SNRs, the proposed architecture and precoding achieve rates comparable to the fully-digital solution.
  • Coarse quantization with 4–5 bits normally achieves maximum energy efficiency across various system configurations.

APPENDIX A PROOF OF PROPOSITION 1

The appendix proves Proposition 1 by bounding the mutual information under the quantized receiver model and transmit-power constraint. The argument uses entropy bounds and properties of the effective noise and quantizer-related functions.

  • The proof seeks an upper bound on mutual information subject to the transmit-power constraint.
  • It bounds conditional entropy using the quantizer output cardinality and the effective noise distribution.The argument invokes at most 2^N possible outputs and independent Gaussian in-phase and quadrature noise components.
  • The derivation uses monotonicity and convexity properties of the quantizer-related function to obtain the stated bound.
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