PaperlayerView exact revision on arXiv

Source-linked AI summary

Spatial- and Frequency-Wideband Effects in Millimeter-Wave Massive MIMO Systems

Bolei Wang, Feifei Gao, Shi Jin, Hai Lin, Geoffrey Ye Li

arXiv:1708.07605v5cs.IT

TL;DR

The paper addresses the failure of conventional massive MIMO designs to account for propagation delay across large apertures alongside frequency selectivity. It develops sparse angle-delay channel models and uplink/downlink estimation strategies, with simulations showing effective handling of these dual-wideband effects and reduced training overhead without pilot contamination.

  • Problem

    Conventional massive MIMO studies consider frequency selectivity but ignore non-negligible propagation delays across large array apertures, motivating dual-wideband analysis.

  • Method

    The paper models mmWave channels using sparse parameters in the angle and delay domains and develops uplink and downlink channel-estimation strategies.

  • Results

    Numerical examples show that the proposed transmission design effectively handles dual-wideband effects, whereas existing designs cannot.

  • Takeaways & Limitations

    The proposed channel estimation requires significantly less training overhead, avoids pilot contamination, and suits both TDD and FDD massive MIMO systems.

Abstract

from arXiv · show

When there are a large number of antennas in massive MIMO systems, the transmitted wideband signal will be sensitive to the physical propagation delay of electromagnetic waves across the large array aperture, which is called the spatial-wideband effect. In this scenario, transceiver design is different from most of the existing works, which presume that the bandwidth of the transmitted signals is not that wide, ignore the spatial-wideband effect, and only address the frequency selectivity. In this paper, we investigate spatial- and frequency-wideband effects, called dual-wideband effects, in massive MIMO systems from array signal processing point of view. Taking mmWave-band communications as an example, we describe the transmission process to address the dual-wideband effects. By exploiting the channel sparsity in the angle domain and the delay domain, we develop the efficient uplink and downlink channel estimation strategies that require much less amount of training overhead and cause no pilot contamination. Thanks to the array signal processing techniques, the proposed channel estimation is suitable for both TDD and FDD massive MIMO systems. Numerical examples demonstrate that the proposed transmission design for massive MIMO systems can effectively deal with the dual-wideband effects.

I. INTRODUCTION

Massive MIMO and mmWave systems motivate large arrays, but their propagation delays across the aperture invalidate conventional spatial-narrowband assumptions. The paper therefore develops an array-signal-processing treatment of spatial- and frequency-wideband effects.

  • Motivation: Massive MIMO uses hundreds or thousands of base-station antennas to serve many users while improving spectrum efficiency, energy efficiency, spatial resolution, and coverage.mmWave communication further supports large arrays because its small antenna size enables dense packing and massive MIMO helps combat path loss and fading.
  • Research gap: Existing massive MIMO models account for frequency selectivity from multipath delays but ignore the non-negligible propagation delay across a large array aperture.This neglected spatial-selective effect is an inherent large-array property previously studied in array signal processing.
  • Approach: The paper models dual-wideband channels using limited path parameters, including complex gain, DOA or DOD, and time delay.The model exploits channel sparsity in both angle and delay domains.
  • Approach: The proposed uplink and downlink channel-estimation techniques require significantly less training overhead and avoid pilot contamination.The strategies are designed for massive MIMO systems under dual-wideband effects.
  • Implications: Angle and delay domains, together with their spatial- and frequency-domain Fourier counterparts, exhibit a duality that supports training and data-transmission design.The paper also states that the strategy suits both TDD and FDD systems through angle and time-delay reciprocities.
  • Technical basis: The spatial-narrowband approximation requires the propagation delay across the array to be much smaller than the symbol duration.Under this condition, the received channel can be represented with a conventional spatial steering vector.

B. Large Array and Spatial-Wideband Effect

Large arrays and wide bandwidth jointly produce spatial- and frequency-wideband effects that require a coupled channel model. The paper formulates this model for OFDM mmWave systems and uses its angle-delay structure to guide transmission and estimation.

  • Large Array and Spatial-Wideband Effect: For wideband massive MIMO, the conventional approximation that all antennas receive the same baseband signal fails for antennas farther across the aperture.The resulting spatial-wideband effect can make different antennas observe different transmitted symbols.
  • Large Array and Spatial-Wideband Effect: For M = 128 and d = λc/2, aperture propagation delay reaches 0.58Ts in a typical LTE setting and 0.92Ts in a typical 60-GHz, 1-GHz-bandwidth system.These examples show that aperture delay can be comparable to the symbol duration.
  • Large Array and Spatial-Wideband Effect: When the narrowband assumption fails, algorithms that ignore spatial-wideband effects suffer performance loss as either the antenna count or transmission bandwidth increases.The paper also notes degradation of phased-array hybrid structures as bandwidth increases.
  • Channel Modeling with Dual-Wideband Effects: The OFDM model represents each user’s channel through physical paths with path gains, directions of arrival, and path delays.The formulation includes antenna-dependent path delays and corresponding spatial-frequency channel responses.
  • Channel Modeling with Dual-Wideband Effects: The SFW channel couples the spatial steering vector, frequency-domain steering vector, and a path-dependent phase-shift matrix.This coupling explicitly accounts for both spatial- and frequency-wideband effects and is associated with beam squint.
  • Channel Modeling with Dual-Wideband Effects: Angle and delay domains have a dual relationship, mirrored by spatial and frequency domains in their Fourier transforms.Angular and temporal beamforming can therefore be implemented by weighting antennas and subcarriers in analogous ways.

B. Essential Requirement of CP Length for OFDM

The OFDM cyclic prefix must cover both ordinary multipath delay and the additional propagation delay across the large array aperture. This spatial-wideband contribution grows with bandwidth and antenna count, although typical mmWave examples require only modest extra samples.

  • The SFW channel model requires an OFDM cyclic prefix long enough to overcome the delay at every antenna.
  • For each path, the spatial-wideband CP component equals the propagation duration across the full array, while the path’s earliest arrival adds the ordinary delay component.
  • The overall CP must accommodate the largest required delay across all users and incident paths, rather than depend on an instantaneous direction of arrival.
  • The additional CP term scales with transmission bandwidth fs and antenna count M, and inversely with carrier frequency fc.
  • 2 and 9 extra CP samples arise at fs = 1 GHz and fc = 60 GHz for ULAs with M = 128 and M = 1024, respectively.The paper states that this extra burden does not seem problematic for existing OFDM protocols.
  • Even with sufficient CP length, algorithms that ignore the spatial-wideband channel suffer severe performance loss.

C. Asymptotic Characteristics of SFW Channels

Asymptotically, the spatial-frequency response becomes localized in angular-delay coordinates, producing sparse square regions associated with individual paths. Spatial-wideband phase diffusion changes each path from an impulse into a DOA-dependent square region.

  • Under the stated asymptotic conditions, the 2D-IDFT of Θ(ψ) is block-sparse, with nonzero elements confined to a square region.
  • The angular-delay channel Gp is asymptotically sparse and contains only Lp nonzero square regions, one for each channel path.
  • Without spatial-wideband effects, each path appears as an impulse in the angular-delay domain; with them, Θ(ψp,l) diffuses its energy into a square region.
  • Different paths produce square regions of different sizes, determined by their path directions of arrival.
  • The asymptotic sparsity analysis assumes the signal bandwidth is below the carrier frequency and uses the proposed CP design for the second condition.

D. Sparse Channel Representation and Angular-Delay Orthogonality

The SFW channel is represented with angular-delay basis vectors, whose asymptotic orthogonality supports simultaneous scheduling and channel estimation without pilot contamination or mutual interference.

  • Sparse channel representation: Each path contributes an angular-delay basis vector formed from the spatial and frequency steering structure, including the phase-shift matrix.The basis vector spans the vectorized SFW channel.
  • Angular-delay orthogonality: When two paths do not share both DOA and time delay, their vectorized SFW channels are asymptotically orthogonal.Partial overlap between angular-delay energy regions does not destroy orthogonality unless their positions are identical.
  • Implications: Angular-delay orthogonality allows users with distinct signatures to be simultaneously scheduled without pilot contamination or mutual interference.This property is used in the subsequent channel-estimation design.
  • Estimation procedure: The proposed estimation procedure uses array signal processing to handle dual-wideband effects in a redesigned SFW channel model.The paper presents a simple channel-estimation algorithm for this purpose.
  • Estimation procedure: The preamble applies conventional least-squares MIMO-OFDM estimation per base-station antenna to obtain initial DOAs and path delays.These initial signatures facilitate later uplink and downlink estimation with fewer pilot resources.
  • Estimation procedure: Finite antenna and subcarrier dimensions cause power leakage, expanding the nominal nonzero region and requiring a more sophisticated signature-extraction method.The algorithm is listed in Table I and uses a neighborhood search to identify angular-delay signatures.

C. Uplink Channel Estimation by Soft Grouping

Uplink estimation exploits slowly varying angular-delay signatures and soft grouping, allowing users to train together while updating only path gains.

  • Signature reuse: Because users’ angular-delay signatures remain unchanged over tens of channel coherence times, subsequent estimation can focus on channel gains.The channel state information itself still requires re-estimation every coherence time.
  • Soft grouping: Soft grouping adjusts users’ training times so their angular-delay signatures become distinct while preserving simultaneous training.Guard intervals are imposed between adjusted signatures for finite antenna and subcarrier dimensions.
  • Soft grouping: Unlike hard grouping, soft grouping permits interlocked transmissions rather than requiring completely non-overlapped user intervals.Hard grouping is a special case when users are postponed into fully non-overlapped intervals.
  • Gain estimation: The training block uses pilot “1” on all carriers, producing an M × N received-signal matrix under a training-power constraint and additive white Gaussian noise.This is the stated simplified training assumption.
  • Gain estimation: Angular-delay orthogonality and the users’ signatures allow each complex path gain to be updated, after which the uplink channel is reconstructed.The sparse representation is determined by path gain, angle, and delay parameters.

D. Downlink Channel Representation

Downlink representation transfers physical angle and delay information through angular-delay reciprocity, so the channel retains an SFW structure with path-specific gains.

  • Angular-delay reciprocity: Physical DOAs and path delays are approximately reciprocal between uplink and downlink, including in FDD systems.The stated condition is that the uplink-downlink frequency separation remains within several GHz.
  • Downlink signatures: The downlink angular parameter is obtained using angular-delay reciprocity, while soft grouping adjusts path delays to create separated downlink signatures.The adjusted signatures obey the same guard-interval rule as in uplink estimation.
  • SFW representation: The downlink channel is modeled as a spatial-frequency channel with a downlink path-gain vector and corresponding angular-delay structure.The model is expressed for the channel from the base station toward each user.
  • SFW representation: Only the corresponding downlink path gains need to be estimated to obtain the downlink channel.The angular-delay signatures are reused rather than fully re-estimated.

E. Downlink Channel Estimation

Downlink training forms path-directed beams and estimates one gain per physical multipath, reducing feedback because users need not know their angular-delay signatures.

  • Training design: Training uses an LM × LM matrix across LM blocks, with each user summing received signals over subcarriers to form an LM × 1 observation vector.The received-signal construction is specified per subcarrier, block, and antenna.
  • Channel parameterization: The downlink channel is parameterized by a gain vector whose columns are assembled from path-dependent steering components.The matrix Pp contains the path-associated channel structure.
  • Beamforming: Choosing the beamforming matrix from Pp forms separate beams toward each path and yields optimal estimation of the downlink gains.The construction is applied across users through an overall beamforming matrix.
  • Gain estimation: The downlink gain estimator uses the received training observations while the second interference-related term becomes asymptotically zero.The resulting gain estimate is then used to rebuild the downlink SFW channel.
  • Overhead reduction: The number of estimated downlink parameters equals each user’s number of physical multipaths, and users need not know their own signatures.The scheme therefore reduces feedback cost by handling signature information at the base station.

V. SIMULATION RESULTS

The simulations evaluate the proposed channel-estimation strategies across bandwidth, antenna count, carrier frequency, SNR, and subcarrier count. Results show that conventional methods can degrade under spatial-wideband conditions, whereas the proposed approach handles dual-wideband effects.

  • Bandwidth and antenna comparisons: The experiments compare the proposed strategy with conventional frequency-wideband-only methods under mmWave transmission conditions.The comparisons use on-grid CS and gridless CS baselines, including settings with carrier frequency 60 GHz, M = 128, and N = 128.
  • Bandwidth and antenna comparisons: At fs = 1 GHz, both conventional frequency-wideband-only algorithms fail, while the gridless method approaches the proposed method at fs = 20 MHz.At 20 MHz, the spatial-wideband effect is negligible; propagation delays across the array are 0.021Ts at 20 MHz and 1.06Ts at 1 GHz.
  • Bandwidth and antenna comparisons: As M increases beyond 16 under fs = 1 GHz, the spatial-wideband effect becomes more severe and gridless CS performance degrades.The on-grid method remains poor because of power leakage, while propagation delays increase from 0.125Ts at M = 16 to 1.06Ts at M = 128.
  • Carrier-frequency effects: At fs = 20 MHz and fc = 1.9 GHz, the spatial-wideband effect is observed for M = 128, while delays reach 5.38Ts for M = 1024.The comparison indicates that conventional spatial-narrowband algorithms may not estimate channels well under these array conditions, whereas the effect can be ignored for fewer than 16 antennas.
  • SNR and estimation behavior: Increasing SNR improves estimation accuracy until error floors caused by finite angular-delay non-orthogonality are reached.The proposed uplink procedure estimates angular-delay signatures, schedules users by soft grouping, and updates channel gains using one subsequent training block.
  • SNR and estimation behavior: More antennas improve uplink and downlink estimation at high SNR, while larger N improves downlink estimation when OFDM-block power is not normalized.Both uplink and downlink estimates ultimately meet error floors determined by finite angular-delay non-orthogonality.

APPENDIX A PROOF OF LEMMA 1

The appendix proof analyzes the stated cases separately and concludes the result after deriving the corresponding asymptotic relations.

  • Proof of Lemma 1: The proof first considers ϑ ∈ [0, π/2] and derives the required expression for that angular range.It then identifies non-zero matrices in the resulting distributions and asymptotic structure.
  • Proof of Lemma 1: The argument uses relations involving (14), (46), and (53) to establish the stated asymptotic structure.The proof explicitly notes that U, L, and C are non-zero matrices where introduced.
  • Proof of Lemma 1: The analogous result for ϑ ∈ [−π/2, 0) completes the proof.The appendix states this complementary angular case directly before concluding.

APPENDIX B PROOF OF THEOREM 1

The appendix proof establishes the theorem by treating distinct and equal angular parameters separately and combining the resulting relations.

  • Proof of Theorem 1: For ϑ1 ≠ ϑ2, equivalently ψ1 ≠ ψ2, the proof derives the corresponding relation using the stated parameter ranges and equivalences.The argument explicitly separates this unequal-parameter case from the equal-parameter case.
  • Proof of Theorem 1: When ψ1 = ψ2, equation (19) is simplified from equations (58)–(60).This provides the special-case expression needed for the theorem’s proof.
  • Proof of Theorem 1: Combining the relations for the considered cases completes the proof.The appendix also states that a similar result holds when ψp,l < 0.
Loading 1708.07605v5…