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Channel Estimation in Broadband Millimeter Wave MIMO Systems with Few-Bit ADCs

Jianhua Mo, Philip Schniter, Robert W. Heath

arXiv:1610.02735v4cs.IT

TL;DR

Broadband mmWave MIMO channel estimation must operate with few-bit ADCs despite nonlinear quantization and the cost of high-precision conversion. The paper combines angle-delay sparsity with EM-GAMP and EM-VAMP and a low-PAPR FFT-based training design. Simulations show that 1-bit ADCs have small low-SNR losses and 3-4-bit ADCs have small losses through medium SNRs relative to infinite-bit ADCs.

  • Problem

    High-precision ADCs sufficient for broadband mmWave systems can be costly and power-hungry, while few-bit quantization makes MIMO channel estimation challenging and prior AMP methods did not exploit mmWave angle-domain sparsity.

  • Method

    The paper formulates estimation as noisy quantized compressed sensing, exploits joint angle-delay sparsity, and applies EM-GAMP and EM-VAMP with a low-PAPR FFT-based training design.

  • Results

    1-bit ADCs incur only small performance losses at low SNR, while 3-4-bit ADCs incur only small losses up to medium SNRs relative to infinite-bit ADCs.

  • Takeaways & Limitations

    ADC precision and training length can be selected according to SNR: few-bit ADCs are suitable at low SNR, and lower precision favors longer training for achievable rate.

Abstract

from arXiv · show

We develop a broadband channel estimation algorithm for millimeter wave (mmWave) multiple input multiple output (MIMO) systems with few-bit analog-to-digital converters (ADCs). Our methodology exploits the joint sparsity of the mmWave MIMO channel in the angle and delay domains. We formulate the estimation problem as a noisy quantized compressed-sensing problem and solve it using efficient approximate message passing (AMP) algorithms. In particular, we model the angle-delay coefficients using a Bernoulli-Gaussian-mixture distribution with unknown parameters and use the expectation-maximization (EM) forms of the generalized AMP (GAMP) and vector AMP (VAMP) algorithms to simultaneously learn the distributional parameters and compute approximately minimum mean-squared error (MSE) estimates of the channel coefficients. We design a training sequence that allows fast, FFT-based implementation of these algorithms while minimizing peak-to-average power ratio at the transmitter, making our methods scale efficiently to large numbers of antenna elements and delays. We present the results of a detailed simulation study that compares our algorithms to several benchmarks. Our study investigates the effect of SNR, training length, training type, ADC resolution, and runtime on channel estimation MSE, mutual information, and achievable rate. It shows that our methods allow one-bit ADCs to perform comparably to infinite-bit ADCs at low SNR, and 4-bit ADCs to perform comparably to infinite-bit ADCs at medium SNR.

I. INTRODUCTION

Broadband mmWave MIMO channel estimation with few-bit ADCs is challenging because high-bandwidth, high-precision ADCs can be costly, while quantization makes MIMO estimation nonlinear. The paper addresses this with sparsity-aware AMP methods, a low-PAPR FFT-based training design, and simulations spanning performance and complexity tradeoffs.

  • Motivation: High-precision ADCs with sufficient mmWave bandwidth can be unavailable or costly and power-hungry for portable devices.At rates above 100 Msamples/s, ADC power consumption increases quadratically with sampling frequency.
  • Motivation: Few-bit ADCs reduce power consumption and simplify hardware, but quantization makes MIMO channel estimation challenging.The receiver quantizes the inphase and quadrature components, while antenna signals are linearly combined before quantization.
  • Approach: The paper formulates broadband mmWave channel estimation as noisy quantized compressed sensing that exploits sparsity in both angle and delay domains.This formulation is intended to enable accurate estimation with efficient algorithms and relatively short training sequences.
  • Approach: EM extensions of GAMP and VAMP avoid specifying a detailed channel prior while providing nearly minimum-MSE estimates at relatively low complexity.The algorithms are designed for large-scale estimation problems.
  • Training design: A shifted training-sequence design targets low channel-estimation error, low complexity, and low peak-to-average power ratio.Its low-complexity implementation uses FFTs.
  • Evaluation: The study evaluates ADC precision, training type and length, estimation algorithms, MSE, mutual information, achievable rate, and complexity.The experiments report a superior performance-complexity tradeoff for EM-VAMP and support large systems such as 64 × 64 antennas with 16 symbol intervals of delay spread.
  • Findings: 1-bit ADCs incur only small losses at low SNR, while 3-4-bit ADCs incur only small losses up to medium SNRs relative to infinite-bit ADCs.The MSEs of EM-GAMP and EM-VAMP decay exponentially with training length, while achievable rate is usually maximized with relatively short training.

III. SPARSITY OF THE MMWAVE CHANNEL MODEL

The mmWave MIMO channel is modeled with few multipath clusters and large UPAs, producing joint sparsity in angle and delay domains, though leakage makes this sparsity approximate.

  • The channel impulse response is modeled using multipath clusters characterized by path gains, delays, and arrival and departure angles.
  • The transmitter and receiver use uniform planar arrays, including configurations with more than 100 antennas.A cited 5G prototype used a 256-element 16 × 16 base-station UPA.
  • B. Angle-Delay Representation: Angle-domain coefficients provide an alternative to antenna-aperture coefficients, with entries representing gains between discrete transmit and receive angles.
  • Few clusters and many antenna elements make the number of clusters much smaller than the number of scalar channel coefficients.
  • Fig. 2 shows angle-domain energy concentrated in two locations, each corresponding to one multipath cluster.The example uses delay spread L = 16, Ncl = 2 clusters, 8 × 8 UPAs, and 10 paths per cluster.
  • The angle-delay channel is jointly sparse, but leakage means its coefficients are not expected to be exactly zero.The proposed GAMP approach is described as robust to leakage effects.

IV. PROPOSED CHANNEL ESTIMATION ALGORITHM

The proposed estimator recovers approximately sparse angle-delay channel coefficients from noisy few-bit measurements using a known training sequence, treating the task as noisy quantized compressed sensing.

  • The algorithm estimates an approximately sparse angle-delay channel from few-bit measurements and a known training sequence.
  • Training is transmitted as a block of length Np with a cyclic prefix of length L.
  • The received training block consists of an unquantized noiseless signal block plus additive Gaussian noise before quantization.
  • The estimation problem recovers x ∈ C^(NtNrL) from y ∈ C^(NrNp×1) under a known linear transform A.
  • Because angle-delay sparsity is approximate due to leakage, the problem is formulated as noisy quantized compressed sensing.Existing methods are strongly dependent on the assumed channel sparsity rate.

B. EM-AMP Algorithms

The paper applies EM-augmented GAMP and VAMP to few-bit quantized compressed sensing, using learned mixture priors to avoid specifying the channel distribution while retaining computational efficiency.

  • The proposed channel estimators combine expectation maximization with GAMP and VAMP to avoid specifying a prior distribution.
  • The proposed GAMP and VAMP approaches are selected because they offer nearly minimum-MSE performance with computational efficiency.
  • GAMP approximates MMSE estimation through scalar estimation problems and matrix multiplications, with state-evolution guarantees for large IID sub-Gaussian matrices.
  • EM-GAMP learns Gaussian-mixture prior parameters from GAMP posterior approximations instead of requiring the true channel distribution.
  • The EM-GAMP iteration alternates message-passing updates with an EM update of the distributional parameters.

2) EM-VAMP:

EM-VAMP combines scalar and linear MMSE estimations with EM parameter updates, using sparse Bernoulli-Gaussian or Gaussian-mixture priors for angle-delay coefficients.

  • 2) EM-VAMP:: VAMP targets the same estimation problem as GAMP but supports right-rotationally invariant sensing matrices through scalar state evolution.
  • When the prior and likelihood are unavailable, VAMP can learn approximations through an EM methodology.
  • EM-VAMP alternates nonlinear scalar estimations, linear vector estimations, and EM updates.
  • Its scalar steps compute conditional means and variances for x and z under pseudo-measurements, priors, and likelihoods.
  • The coefficient prior is modeled as either a Bernoulli Gaussian-mixture or Bernoulli-Gaussian distribution with unknown parameters.
  • The Gaussian-mixture prior can fit the true channel distribution better than the Bernoulli-Gaussian prior, while having somewhat higher complexity.
  • The model assumes an IID sparse prior and does not exploit possible correlation among coefficients.Exploiting such correlation may improve estimation accuracy.

D. Computational Issues and Training Sequence Design

The paper addresses the memory and computational challenges of large mmWave channel-estimation operators by designing training signals that enable fast implicit implementations while preserving AMP-relevant properties and low transmitter PAPR.

  • Computational challenges: For example, Nt = Nr = 64, Np = 1024, and L = 16 produce Ny = Nx = 65536, while explicitly storing A would require 32 GB of memory.The large operator dimensions make explicit matrix storage inconvenient and matrix-vector multiplication computationally demanding.
  • Training-design requirements: The training signal must make A, A∗, UA, U∗A, and VA fast operators, support GAMP convergence, and maintain low transmitter PAPR.These requirements jointly determine practical implementability, matrix conditioning for AMP, and power-amplifier efficiency.
  • Training structure: Each transmit antenna sends an nL-place circular shift of the first antenna’s sequence, with Np chosen as an integer multiple of NtL.This structure generates the required shifted training rows and enables their reordering into a Toeplitz matrix.
  • Fast implementation: The resulting matrix-vector products with A and A∗ use FFTs and fast convolution, reducing the operator memory footprint to that of t ∈ C^Np.The implementation decomposes the multiplications into FFTs across antenna, delay, and training dimensions.
  • Fast implementation: The construction supports FFT-based operations for the matrix factors and avoids storing the full sensing matrix explicitly.The described implementation uses multiple FFTs and inverse FFTs, including an Np-point FFT for fast convolution.

3) Choice of the training sequence t:

The proposed training sequence uses a Zadoff–Chu sequence to shape the sensing matrix’s singular values and provide fast VAMP operators, while its constant modulus keeps transmitter PAPR low.

  • Zadoff–Chu design: Choosing t as a Zadoff–Chu sequence gives eT eT∗ = PtNp and equal singular values for the relevant training structure.The resulting singular-value pattern yields the minimum possible peak-to-average squared-singular-value ratio for A.
  • AMP properties: A consequently has the minimum possible peak-to-average squared-singular-value ratio, which is beneficial for GAMP.The same singular-value structure permits fast choices for the singular-vector matrices used by VAMP.
  • AMP properties: The singular-vector matrices can be chosen as fast operators, making the training design suitable for efficient VAMP implementation.The construction supports VA = I and a fast implementation of UA.
  • Transmitter efficiency: The constant-modulus property of Zadoff–Chu sequences ensures that T has low peak-to-average power ratio.This property is also why Zadoff–Chu sequences are used as reference signals in LTE.

4) Other choices of T:

The study compares alternative training sequences and benchmark estimators, showing that alternatives can achieve comparable recovery but lack the proposed design’s low-complexity matrix operations.

  • Alternative training designs: Golay, IID QPSK, and IID Gaussian training designs have recovery performance comparable to the proposed design.Their main disadvantage is computational rather than recovery performance.
  • Alternative training designs: The alternative training designs lead to substantially higher GAMP and VAMP complexity because fast matrix-vector products are unavailable.The comparison highlights the practical value of the proposed training structure beyond estimation accuracy.
  • Benchmark estimators: The benchmark set includes least squares, approximate LMMSE based on Bussgang linearization, SPGL1 sparse reconstruction, and quantized iterative hard thresholding.These methods respectively provide linear, linearized, sparse-reconstruction, and quantized iterative baselines.
  • Benchmark assumptions: The effective-noise approximation used by the linearized LMMSE benchmark is reported to be especially accurate at low SNR, but the effective noise remains non-Gaussian.Thus, the approximation has a stated operating regime and a distributional caveat.

E [Axx∗A∗] + E [ww∗]

This section addresses channel-norm estimation, sparse-reconstruction benchmarks, and practical limitations of the evaluated methods, including approximate quantizer-noise modeling and solver scalability.

  • Modeling limitation: The Bussgang-based effective-noise approximation is reported accurate especially at low SNR, but the resulting noise is non-Gaussian.This limits the interpretation of the corresponding linearized benchmark as an exact Gaussian-noise model.
  • Benchmark methods: The SPGL1 benchmark applies sparse reconstruction to the linearized model and uses the same fast A and A∗ implementations as the proposed methods.Its computational cost is dominated by matrix-vector multiplications.
  • Benchmark methods: QIHT retains only the largest 1% of elements per iteration in the simulations, with its step size selected using the spectral norm of A.The algorithm uses hard thresholding to enforce a sparse estimate.
  • Channel-norm estimation: With one-bit ADCs, amplitude information is lost, making precise recovery of the channel norm difficult, especially at high SNR.The paper notes that prior work assumed known channel variance, while an AGC circuit can estimate norm-related received power before quantization.
  • Channel-norm estimation: The channel estimate is normalized using measured total received power, and this normalization reduced channel-estimation error in all cases.The procedure relies on the assumption that total received power across antennas can be measured.
  • Benchmark limitation: An ℓ1-based method encountered numerical problems, and alternative implementations could not be used because the matrix A was too large to fit in memory.This constrains the benchmark comparison for large problem instances.
  • Illustrative example: A representative Fig. 3 setting uses a 4 × 16 MIMO channel, 16-sample delay spread, 512-symbol training, 10 dB SNR, and 4-bit ADCs.The figure compares true channel magnitudes, EM-GM-VAMP estimates, and estimation-error magnitudes.

V. SIMULATION RESULTS

The simulations evaluate training designs, computational complexity, convergence, and recovery performance for broadband mmWave channel estimation. Shifted-ZC training enables low-PAPR, fast matrix operations, while EM-VAMP and EM-GM-VAMP offer favorable estimation-complexity trade-offs.

  • A. Choice of Training Matrix: Four training designs have very similar EM-GM-VAMP NMSE, while constant-modulus sequences reduce PAPR and shifted-ZC enables efficient VAMP matrix operations.The compared designs are IID Gaussian, IID QPSK, Golay complementary, and shifted-ZC sequences.
  • A. Choice of Training Matrix: Fast implementations of Abx and A∗bs consume much less runtime than baseline implementations that avoid explicitly storing the high-dimensional matrix A.The fast implementation uses the design described for shifted-ZC training; hardware FFTs could further reduce runtime.
  • B. Algorithm Complexity: VAMP converges in ≈3 iterations, GAMP in ≈8 iterations, and QIHT in ≈35 iterations under 4-bit ADCs at 0 dB SNR.The comparison uses shifted-ZC training with Np = 2048.
  • B. Algorithm Complexity: EM-GM-VAMP provides the best NMSE-complexity trade-off for runtimes > 0.4 s, while EM-BG-VAMP is best between 0.17 s and 0.4 s.EM-BG-GAMP is near the EM-GM-VAMP frontier, whereas LS and ALMMSE are far from optimal.
  • B. Algorithm Complexity: FFT-like matrix-vector multiplication gives EM-AMP, SPGL1, and QIHT lower complexity scaling than LS and ALMMSE when transmit and receive antenna counts are similar.LS and ALMMSE are dominated by inversion of an Np × Np matrix, while the other methods are dominated by fast multiplies with A and A∗.

C. Effect of SNR, ADC resolution, and Training Length

The study examines how SNR, ADC resolution, training length, mutual information, and achievable rate interact. Few-bit ADCs are most competitive at low SNR, while training length should be selected jointly with ADC precision and coherence time.

  • C. Effect of SNR, ADC resolution, and Training Length: At SNR < 0 dB, the 1-bit versus infinite-bit NMSE gap is only 2 dB, but the gap grows as SNR increases.Higher-resolution ADCs therefore provide significant estimation benefits mainly at higher SNRs.
  • C. Effect of SNR, ADC resolution, and Training Length: At SNR = 10 dB, EM-GAMP/VAMP gain little from increasing ADC resolution above 4 bits, while SPGL1 benefits from higher resolution and performs worse with few bits.SPGL1 matches EM-BG-GAMP/VAMP for > 7 bits but is significantly worse with few bits.
  • D. Mutual Information and Achievable Rate Bounds: Estimated-channel mutual information has relatively small loss versus perfect CSI, and few-bit ADCs have small loss versus infinite-bit ADCs at low SNR.The mutual-information lower bound is evaluated under EM-GAMP/VAMP channel estimates.
  • D. Mutual Information and Achievable Rate Bounds: At SNR = 10 dB, mutual information saturates above 5 ADC bits, while EM-GM-GAMP/VAMP achieve the highest values and EM-BG-GAMP/VAMP are close behind at low resolutions.This comparison uses the mutual-information lower bound across channel-estimation algorithms.
  • D. Mutual Information and Achievable Rate Bounds: For Nco = 10240, achievable rate is maximized at Np = 1536 with 1- or 2-bit ADCs, whereas Np = 1024 is preferred with 3- or 4-bit ADCs.The longer training at lower precision reflects the different cost of training symbols in the achievable-rate bound.
  • D. Mutual Information and Achievable Rate Bounds: With Nco = 10240, training overhead is around 10%-20%, and 1024-2048 training symbols are described as appropriate for broadband mmWave communication.The proposed training is shorter than the total training length described for the 802.11ad comparison.

VI. CONCLUSION

The paper concludes that joint angle-delay sparsity and AMP-based estimation support broadband mmWave channel estimation with few-bit ADCs. ADC precision and training length should be selected according to SNR and precision, although angle-delay coefficient correlation remains unmodeled.

  • VI. CONCLUSION: The proposed methodology uses EM-GAMP and EM-VAMP with separate channel-norm estimation to estimate broadband mmWave channels from few-bit ADC observations.It exploits joint angle-delay sparsity and uses shifted-ZC training for accurate, efficient, low-PAPR estimation.
  • VI. CONCLUSION: The study concludes that exploiting joint angle-delay sparsity is important for channel estimation.The conclusion also links this structure to the paper’s few-bit ADC setting.
  • VI. CONCLUSION: At low SNR, few-bit ADCs incur very small MSE or achievable-rate loss relative to infinite-bit ADCs, while lower ADC precision favors longer training.Training length should therefore be chosen based on ADC precision.
  • VI. CONCLUSION: Correlation among angle-delay channel coefficients was neglected, and exploiting it is identified as a possible way to improve estimation accuracy.This is the paper’s stated limitation and future direction.
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