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A Comparison of Hybrid Beamforming and Digital Beamforming with Low-Resolution ADCs for Multiple Users and Imperfect CSI

Kilian Roth, Hessam Pirzadeh, A. Lee Swindlehurst, Josef A. Nossek

arXiv:1709.09047v1cs.ITmath.IT

TL;DR

Large mmWave antenna arrays and bandwidth make receiver power consumption a key design challenge. The paper compares hybrid and low-resolution digital beamforming under multiuser conditions and models correlated quantization error. Digital beamforming is more energy efficient in the evaluated multiuser scenarios, while mixed ADC resolutions offer trade-offs near those of uniform resolutions.

  • Problem

    Large bandwidth and antenna counts make the receiver front end power hungry, motivating comparison of hybrid beamforming and digital beamforming with low-resolution ADCs.

    Source: §I · p. 1, A. Related Work · p. 1

  • Method

    The paper compares hybrid and digital beamforming while accounting for channel-estimation error, transmitter impairments, multiple users, and quantization-error correlation.

    Source: B. Contribution · p. 2, B. Modeling the Quantization · p. 7

  • Results

    Digital beamforming is more energy efficient than hybrid beamforming in the evaluated multiuser scenarios, and mixed-resolution ADCs offer rate–energy trade-offs near uniform-resolution configurations.

    Source: §V · p. 9

  • Takeaways & Limitations

    As SNR increases, the energy-efficiency-optimal ADC resolution increases, while energy efficiency decreases above 5 bits in the reported cases.

    Source: B. Energy Efficiency Results · p. 9

Abstract

from arXiv · show

For 5G it will be important to leverage the available millimeter wave spectrum. To achieve an approximately omni- directional coverage with a similar effective antenna aperture compared to state of the art cellular systems, an antenna array is required at both the mobile and basestation. Due to the large bandwidth and inefficient amplifiers available in CMOS for mmWave, the analog front-end of the receiver with a large number of antennas becomes especially power hungry. Two main solutions exist to reduce the power consumption: hybrid beam forming and digital beam forming with low resolution Analog to Digital Converters (ADCs). In this work we compare the spectral and energy efficiency of both systems under practical system constraints. We consider the effects of channel estimation, transmitter impairments and multiple simultaneous users. Our power consumption model considers components reported in literature at 60 GHz. In contrast to many other works we also consider the correlation of the quantization error, and generalize the modeling of it to non-uniform quantizers and different quantizers at each antenna. The result shows that as the SNR gets larger the ADC resolution achieving the optimal energy efficiency gets also larger. The energy efficiency peaks for 5 bit resolution at high SNR, since due to other limiting factors the achievable rate almost saturates at this resolution. We also show that in the multi-user scenario digital beamforming is in any case more energy efficient than hybrid beamforming. In addition we show that if different ADC resolutions are used we can achieve any desired trade-offs between power consumption and rate close to those achieved with only one ADC resolution.

I. INTRODUCTION

Large antenna arrays are needed at mmWave to maintain effective aperture and coverage, but their power demands make receiver design a central challenge. The paper compares low-resolution digital and hybrid beamforming and extends the analysis to practical multiuser constraints.

I. INTRODUCTION

  • At frequencies above 6 GHz, large antenna arrays are needed to preserve effective aperture and achieve high gain while maintaining approximately omnidirectional coverage.The paper relates this requirement to matching the effective aperture of systems operating below 6 GHz.Source: §I · p. 1

A. Related Work

  • Large bandwidth and antenna arrays make receiver power consumption a hardware-design constraint, with high-resolution ADCs identified as a potential bottleneck.High-sampling-frequency ADCs with 6–10 effective bits consume considerable power, according to the cited survey.Source: A. Related Work · p. 1

B. Contribution

  • The work analyzes achievable rate and energy–spectral-efficiency trade-offs for digital and hybrid beamforming with low-resolution ADCs under multiuser, multipath conditions and practical impairments.The considered effects include transmitter impairments, channel-estimation errors, and mixed ADC resolutions.Source: B. Contribution · p. 2
  • The paper generalizes the Additive Quantization Noise Model to account for correlated quantization error, non-uniform quantization, and different ADCs across antennas.It also examines channel-estimation error using reference-signal patterns agreed for 3GPP NR.Source: B. Contribution · p. 2

C. Notation

  • The paper uses bold lowercase and uppercase letters for vectors and matrices, respectively, and defines correlation matrices using E[ab^H].It also specifies notation for conjugation, transpose, Hermitian, inverse, and Fourier transforms.Source: C. Notation · p. 2

II. SIGNAL MODEL

The signal model represents multiple users transmitting through multipath channels to a basestation, with receiver combining and transmitter impairments included. It also specifies a sub-array hybrid receiver and a channel model used in the analysis.

II. SIGNAL MODEL

  • The received signal combines each user's power-scaled, multipath-transmitted signal and transmitter impairment, plus receiver noise.The model treats transmitter impairments as circularly symmetric complex Gaussian noise with covariance tied to EVM.Source: §II · p. 3
  • In hybrid beamforming, each RF chain connects exclusively to the same number of antennas; for digital beamforming, the analog-combining matrix is replaced by an identity matrix.For more than one antenna per chain, the analog block applies phase shifts and combines the signals.Source: §II · p. 3
  • Mixed-resolution digital beamforming assigns higher-resolution ADCs to a subset of antennas and lower-resolution ADCs to the rest.The model assumes equal average receive power across antennas and permits the higher-resolution ADCs to be assigned to any subset.Source: §II · p. 3

A. Channel Model

  • The channel model assumes one arriving ray per delay, with complex Gaussian path gains and an exponential power-delay profile parameterized by channel length, tap count, and decay rate.Only P of the L channel-variance entries are nonzero.Source: A. Channel Model · p. 3
  • The per-user SNR is defined as an average at each antenna, with the expectation accounting for channel and transmitted-signal realizations.Source: A. Channel Model · p. 3

B. Analytic MSE of frequency domain channel estimation with time-frequency interpolation

The section derives channel-estimation MSE using time-frequency-space interpolation and evaluates it for a 3GPP NR reference-signal pattern. The analysis uses modeled channel statistics and accounts for model mismatch in the interpolation parameters.

B. Analytic MSE of frequency domain channel estimation with time-frequency interpolation

  • The channel-estimation analysis assumes perfect timing and carrier-frequency synchronization and a channel impulse response shorter than the cyclic prefix.Source: B. Analytic MSE of frequency domain… · p. 3
  • A 3-D time-frequency-space filter smooths the channel estimate, assuming knowledge of statistical channel parameters including user signal power and spatial correlation.The analysis evaluates theoretical MSE; the cited 2-D time-frequency MMSE method is identified as having the best performance among the methods discussed.Source: B. Analytic MSE of frequency domain… · p. 4
  • Separate interpolation matrices for time, frequency, and space reduce complexity, with minimal performance loss compared with the theoretical optimum.The covariance may be non-separable or unknown, requiring matrices based on a covariance model whose parameters must also be estimated.Source: B. Analytic MSE of frequency domain… · p. 4

KLM E

  • Interpolation and spatial smoothing matrices are selected using estimated SNR, delay spread and its PDP model, and Doppler spread, so mismatch from actual channel statistics is included.The channel model assumes a single arriving ray, uniformly distributed direction of arrival, and a ULA with λ/2 element spacing; for hybrid beamforming, spatial correlation after analog combining is treated as unknown and Rs is set to identity.Source: KLM E · p. 5
  • For up to four UEs, the NR reference-signal pattern places DMRS in separate OFDM symbols, with different DMRS groups on adjacent subcarriers.The MSE uses the same channel statistics as the rate calculation; the plotted SNR degradation assumes SISO and estimation error independent of channel realizations.Source: KLM E · p. 5, Figure 2 · p. 5

C. Power Model

The receiver power model uses component data reported for 60 GHz WiGig equipment and combines those component powers to model the overall receiver front end.

C. Power Model

  • The component power model is based on literature-reported WiGig equipment operating in the 60 GHz ISM band.The authors use these components as representative of mature, low-cost, low-power CMOS designs.Source: C. Power Model · p. 6
  • The overall receiver-front-end power is computed from the power consumption of its components.The component inventory is presented in Table I.Source: C. Power Model · p. 6
  • The power formula covers digital, analog, and hybrid beamforming cases, with 1-bit quantization modeled using limiting-amplifier power and negligible quantizer power.The formula switches between analog combining and digital beamforming configurations based on the number of RF chains and antennas.Source: C. Power Model · p. 6

A. Allocation of RF chains for hybrid beamforming with multiple users

The hybrid-beamforming method searches candidate beams for each user and subarray, then allocates RF chains through a resource-fair procedure. Its simplifications address a non-convex configuration problem and reflect practical systems such as WiGig.

A. Allocation of RF chains for hybrid beamforming with multiple users

  • The method simplifies a non-convex phase-shifter configuration problem by connecting adjacent antennas to one RF chain, reflecting practical systems such as WiGig.Source: A. Allocation of RF chains for hybrid… · p. 6
  • For each user and subarray, the algorithm tests candidate directions and records the direction yielding the largest receive power.The beam codebook has 4MC uniformly spaced directions; the cited prior work reports 10% error for this codebook size under the stated ULA and single-direction assumptions [28].Source: A. Allocation of RF chains for hybrid… · p. 6
  • RF chains are assigned to the highest-receive-power subarray-user combination, while cycling through users to distribute subarrays fairly and avoid starving users with weaker channels.The authors state that selecting high-power RF chains without this allocation procedure could starve users with the worst channels.Source: A. Allocation of RF chains for hybrid… · p. 6

B. Modeling the Quantization

Quantization is modeled by decomposing the quantized signal into a signal component and an uncorrelated error, then incorporating the error covariance into the effective system model. The analysis includes correlated errors and supports different quantizers across antennas.

B. Modeling the Quantization

  • The Bussgang decomposition represents the quantized signal as a signal component plus an uncorrelated quantization error.Quantization is applied separately to each vector element and to its real and imaginary parts, allowing different ADC resolutions at each element.Source: B. Modeling the Quantization · p. 7
  • The quantization model computes the signal scaling and error covariance from the receive covariance and quantization functions, then incorporates them into the effective channel and noise covariance.Source: B. Modeling the Quantization · p. 7
  • Unlike evaluations using only a diagonal quantization-error covariance approximation, this analysis includes off-diagonal elements and generalizes the formulas to different quantization functions.The authors report that including off-diagonal elements can dramatically affect overall performance.Source: B. Modeling the Quantization · p. 7

C. Modeling the Channel Estimation Error

The paper models channel-estimation error as additional noise independent of the receive channel. Its covariance is assembled from per-user, spatially white error terms whose variances depend on channel-estimation error and channel power.

C. Modeling the Channel Estimation Error

  • Channel-estimation error is modeled as additional noise independent of the receive channel, rather than reducing useful signal power as in.The authors argue this better reflects practical low-SNR communication, where their alternative could drive received signal power and rate to zero.Source: C. Modeling the Channel Estimation Error · p. 7
  • The overall error covariance Rww is the sum of the per-user error covariances.Each per-user covariance is modeled as spatially white and diagonal; its element variance depends on estimation error and channel power at frequency bin f.Source: C. Modeling the Channel Estimation Error · p. 7
  • The per-user error variances are determined using average SNR per antenna and user.The supplied passage ends before describing the rest of this calculation.Source: C. Modeling the Channel Estimation Error · p. 7

D. Combined Rate Expression

The rate calculation uses Gaussian signaling and per-frequency-bin mutual information over an effective channel. It incorporates transmit noise, quantization, and channel-estimation effects, with approximations to make the expression tractable.

D. Combined Rate Expression

  • The rate expression is made tractable through approximations, including no collaboration among users.The rate analysis is then formulated in the frequency domain.Source: D. Combined Rate Expression · p. 8
  • The calculation assumes Gaussian input signals and models quantization as additive Gaussian noise while retaining off-diagonal quantization-error correlations.The method also selects beamforming vectors from a finite set separately for each antenna group using an SNR criterion.Source: D. Combined Rate Expression · p. 8
  • The rate is calculated separately for each frequency bin by maximizing mutual information over the input covariance, using the equivalent channel H′(f).The band limits f1 and f2 specify the frequency range included; the effective channel and noise covariance depend on the input covariance.Source: D. Combined Rate Expression · p. 8
  • The sum-rate calculation procedure is summarized in Algorithm 2.The supplied passages identify its inputs and frequency-domain operations but do not provide a complete readable algorithm.Source: D. Combined Rate Expression · p. 8

IV. SIMULATION RESULTS

The evaluation compares digital and hybrid beamforming, including mixed-resolution ADCs, for a 64-antenna base station receiving from four users. Digital beamforming is more energy efficient in the multiuser evaluation, while mixed ADC configurations provide intermediate rate–energy-efficiency operating points.

IV. SIMULATION RESULTS

  • The evaluation uses a 64-antenna base station and four users, with HBF RF-chain counts and ADC resolutions varied across configurations.Uniform ADC resolutions range from 1 to 8 bits; mixed DBF uses 5-bit high-resolution ADCs and lower-resolution ADCs of 1 to 4 bits.Source: §IV · p. 8

A. Average Achievable Rate Results

  • At high SNR, achievable rate improves only slightly above 5-bit ADC resolution as transmitter impairments and channel-estimation error cause saturation.The reported average-rate curves cover 30 channel realizations; the figure compares DBF, HBF, and mixed DBF configurations.Source: A. Average Achievable Rate Results · p. 8, Figure 4 · p. 9

B. Energy Efficiency Results

  • Digital beamforming is more energy efficient than hybrid beamforming in the evaluated multiuser scenarios, and retains more degrees of freedom.The paper attributes the difference to the degrees of freedom available to digital beamforming; hybrid subarrays must focus on a single user.Source: B. Energy Efficiency Results · p. 9
  • The energy-efficiency-optimal ADC resolution increases with SNR, but energy efficiency decreases above 5 bits.The paper links the high-resolution decline to the limited rate improvement and increased front-end power consumption.Source: B. Energy Efficiency Results · p. 9

V. CONCLUSION

  • Mixed-resolution DBF provides rate and energy-efficiency operating points around those of uniform-resolution DBF, spanning intermediate trade-offs.The average-rate results show intermediate rates, and the energy-efficiency curves cluster around uniform-ADC curves.Source: A. Average Achievable Rate Results · p. 8, §V · p. 9, B. Energy Efficiency Results · p. 9
  • The HBF evaluation assumes that its beams are already aligned; the conclusion identifies beam-alignment overhead as a future consideration.The authors note that alignment can require large overhead.Source: §V · p. 9

APPENDIX

The appendix generalizes quantized-output correlation calculations to non-uniform quantizers and combines them with rate analysis. It also describes a precomputed correlation grid and interpolation procedure to reduce calculation complexity.

APPENDIX

  • The quantization model generalizes output-correlation calculations to non-uniform quantizers and incorporates quantization effects into rate calculations.The calculation applies to two zero-mean Gaussian variables and supports different quantizers at each antenna.Source: APPENDIX · p. 9
  • The quantizer bin representatives and step positions are adapted to input power, which a practical system handles through an AGC loop.The quantization levels for the two Gaussian variables are denoted Na and Nc.Source: APPENDIX · p. 9
  • A non-uniform input-correlation grid stores numerically calculated output correlations, with cubic-spline interpolation between grid points.Grid spacing is chosen to limit output-correlation changes between adjacent points; the authors report sufficient accuracy with reduced complexity.Source: APPENDIX · p. 10
  • The output correlation matrix Rrr is calculated from the pre-quantization correlation matrix Ryy, including diagonal distortion variances and off-diagonal terms.The procedure is combined into the operator T(·).Source: APPENDIX · pp. 10–11
  • For a different quantizer at each antenna, the Bussgang decomposition matrix F remains diagonal, with each diagonal element defined for its quantizer.The appendix combines this operation with an operator dependent on the quantization step functions.Source: APPENDIX · p. 11
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