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Millimeter Wave MIMO with Lens Antenna Array: A New Path Division Multiplexing Paradigm

Yong Zeng, Rui Zhang

arXiv:1507.01699v1cs.IT

TL;DR

The paper tackles the need for large mmWave antenna gains without the hardware and power cost of one RF chain per antenna. It develops lens-array MIMO with OPDM and path grouping, showing reduced complexity and RF-chain cost alongside strong throughput performance, while assuming perfect channel state information.

  • Problem

    MmWave requires large antenna arrays to overcome severe path loss, but conventional beamforming assigns an RF chain to each antenna, creating high hardware and power costs.

  • Method

    The paper derives the lens array’s sinc-shaped, angle-dependent response and uses multipath sparsity for OPDM with antenna selection, adding path grouping and small-scale MIMO processing when paths are insufficiently separated.

  • Results

    The proposed lens MIMO system achieves significant throughput gains over UPA-based MIMO in evaluated setups while reducing signal-processing complexity and RF-chain cost without notable performance degradation.

  • Takeaways & Limitations

    Lens energy focusing enables low-complexity path-based spatial multiplexing with fewer active RF chains, while path grouping addresses interference from closely coupled paths.

  • Takeaways & Limitations

    The study assumes perfect channel state information at both transmitter and receiver, whereas practical systems require channel training, estimation, and feedback.

Abstract

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Millimeter wave (mmWave) communication is a promising technology for 5G cellular systems. To compensate for the severe path loss in mmWave systems, large antenna arrays are generally used to achieve significant beamforming gains. However, due to the high hardware and power consumption cost associated with radio frequency (RF) chains, it is desirable to achieve the large-antenna gains, but with only limited number of RF chains for mmWave communications. To this end, we study in this paper a new lens antenna array enabled mmWave MIMO communication system. We first show that the array response of the proposed lens antenna array at the receiver/transmitter follows a "sinc" function, where the antenna with the peak response is determined by the angle of arrival (AoA)/departure (AoD) of the received/transmitted signal. By exploiting this unique property of lens antenna arrays along with the multi-path sparsity of mmWave channels, we propose a novel low-cost and capacity-achieving MIMO transmission scheme, termed \emph{orthogonal path division multiplexing (OPDM)}. For channels with insufficiently separated AoAs and/or AoDs, we also propose a simple \emph{path grouping} technique with group-based small-scale MIMO processing to mitigate the inter-path interference. Numerical results are provided to compare the performance of the proposed lens antenna arrays for mmWave MIMO system against that of conventional arrays, under different practical setups. It is shown that the proposed system achieves significant throughput gain as well as complexity and hardware cost reduction, both making it an appealing new paradigm for mmWave MIMO communications.

I. INTRODUCTION

The paper addresses mmWave’s severe path loss and costly RF-chain requirements with lens antenna arrays that focus path energy according to angle. It introduces OPDM and related processing to reduce hardware and complexity while preserving communication performance.

  • MmWave suffers substantially greater path loss than lower-frequency cellular systems, motivating large arrays for directional beamforming but making conventional per-antenna RF chains costly.
  • Lens arrays focus signal power on a small antenna subset according to AoA/AoD, enabling antenna selection with fewer RF chains than conventional arrays.
  • In favorable angular conditions, OPDM maps each multipath to a parallel AWGN sub-channel and achieves capacity using only L transmitting and receiving RF chains.
  • The lens array response follows a sinc function whose peak antenna is determined by the received signal’s AoA or transmitted signal’s AoD.
  • Unlike sectorized antennas and conventional SDMA, OPDM provides high spatial resolution without complex array signal processing.
  • For insufficiently separated AoAs or AoDs, path grouping with group-based small-scale MIMO processing mitigates inter-path interference.

B. Channel Model for MmWave Lens MIMO

The channel model represents mmWave propagation as a sparse sum of delayed multipaths whose receive and transmit responses are determined by lens-array focusing. It supports both wide-band and narrow-band signal formulations.

  • The lens MIMO channel is modeled as a sum of L significant multipaths, each with a complex gain, delay, AoA, AoD, and receive/transmit array responses.
  • Under the far-scatterer assumption, each multipath is approximated as a uniform plane wave across the antenna arrays.
  • The nearest receive and transmit focusing antennas are indexed by rounded spatial-frequency products, while fractional terms describe each path’s focusing misalignment.
  • A path transmitted from antenna q = q_l is directed mainly along path l and focused mainly on receive antenna m = m_l.
  • For narrow-band communication, the maximum excess multipath delay is assumed much smaller than the symbol duration; wide-band modeling uses path-specific delays.

C. Benchmark System: MmWave MIMO with Uniform Planar Array

The benchmark uses conventional UPAs without EM lenses, matched to the lens array's physical aperture, and assumes perfect channel knowledge. Under ideal angular separation, the lens system resolves paths into parallel AWGN sub-channels that support OPDM.

  • Benchmark configuration: The UPA benchmark uses conventional antenna arrays with half-wavelength element spacing and the same physical dimensions as the lens array.The receiver and transmitter use UPAs with d_U = 0.5λ, while their dimensions are selected for equal array apertures.
  • Benchmark configuration: Matching the lens array aperture generally requires more UPA antennas because lens energy focusing reduces the required element count.The benchmark states M_U = 4A_R > M and Q_U = 4A_T > Q.
  • Benchmark configuration: The UPA array responses use phase shifts across antenna elements, and its narrow-band and wide-band input-output relationships follow the conventional MIMO formulation.The response vectors are defined for the transmitter and receiver, with relative phase shifts for successive elements.
  • Assumptions: Both the lens and UPA systems are evaluated assuming perfect channel knowledge at the transmitter and receiver.This assumption applies to both proposed and benchmark systems.
  • Ideal-path operation: With distinct, sufficiently separated AoAs and AoDs, lens-array paths are spatially resolved and the channel becomes L parallel SISO AWGN sub-channels.Each sub-channel corresponds to one propagation path, and the decomposition holds for both narrow-band and wide-band communications.
  • Ideal-path operation: Water-filling over the path gains achieves the lens MIMO capacity in both narrow-band and wide-band communications.The relevant gains are {|α_l|^2 A_R A_T} for l = 1, …, L.

B. Capacity Comparison

The paper compares lens MIMO using OPDM with conventional UPA-based MIMO under narrow-band and wide-band settings. The lens system matches narrow-band capacity and exceeds the wide-band UPA benchmark in the reported simulations.

  • Simulation setup: The comparison uses lens apertures A_T = A_R = 20 and effective azimuth dimensions ˜D_T = ˜D_R = 10, giving M = Q = 21 lens antennas.The equal-aperture UPA benchmark uses M_U = Q_U = 80 antennas.
  • Simulation setup: The channel model contains L = 3 paths at 73 GHz, with 500 MHz bandwidth and path-gain variations modeled through attenuation, fractional power, phase, delay, and shadowing terms.The path-loss model uses c_1 = 86.6, c_2 = 2.45, and ϵ = 8 dB; the delay and shadowing parameters are r_τ = 3 and ζ = 4.
  • Narrow-band comparison: In narrow-band simulations over 10^4 random channel realizations, OPDM achieves almost the same capacity as UPA eigenmode transmission under ideal AoAs and AoDs.The UPA capacity benchmark uses SVD with water-filling power allocation.
  • Wide-band comparison: In wide-band communication, the lens MIMO achieves higher capacity than the UPA-based MIMO, mainly because it saves cyclic-prefix transmission overhead.The UPA benchmark uses MIMO-OFDM with N = 512 sub-bands and a 100 ns cyclic prefix.

C. Complexity and Cost Comparison

Lens MIMO reduces processing, channel-estimation, and RF-chain requirements by exploiting multipath sparsity, while retaining near-optimal full-MIMO performance. Its required RF chains depend primarily on the number of paths rather than deployed antenna count.

  • MIMO processing: Lens MIMO achieves capacity for narrow-band and wide-band channels with OPDM at signal-processing complexity O(L).UPA-based MIMO requires eigenmode or MIMO-OFDM processing with substantially higher complexity.
  • Channel estimation: Lens MIMO requires estimating L parallel SISO channels with complexity O(L), whereas UPA-based MIMO estimates one or N full MIMO channels.The wide-band UPA case uses MIMO-OFDM and therefore requires channel estimation across N sub-carriers.
  • Hardware cost: Only L focusing-point antennas need active RF chains at a time, allowing the remaining lens-array antennas to be deactivated.This reduces hardware cost because RF chains contain mixers, amplifiers, and data converters.
  • Sparsity structure: The channel is nearly block-sparse, with each path contributing an approximately 2∆ × 2∆ nonzero block, although supports can overlap for different paths.For practical applications, ∆ = 1 is described as a reasonable approximation.
  • RF-chain scaling: Only |MS| ≪ M receiving and |QS| ≪ Q transmitting RF chains are generally needed for near-optimal full-MIMO performance.The active subsets are formed from antennas supporting the non-negligible multipath components.

A. Transceiver Design Based on PDM

The PDM design transmits independent streams over multipath-supported antenna subsets using per-path beamforming and receiver-side stream detection. It applies to both narrow-band and wide-band mmWave communications, with interference analyzed through the achievable sum-rate.

  • A. Transceiver Design Based on PDM: PDM transmits L independent data streams, each through one of the L multipaths using transmit beamforming or precoding.The scheme is based on the reduced-size channel matrix obtained after deactivating antennas with negligible channel power.
  • A. Transceiver Design Based on PDM: Each path uses a unit-norm per-path MRT transmit beamforming vector directed toward its AoD.The transmit vector is formed from the selected transmitting antenna subset.
  • A. Transceiver Design Based on PDM: At the receiver, a unit-norm beamforming vector is applied over the selected receiving antennas to detect each data stream.The received signal is processed on a per-stream basis.
  • A. Transceiver Design Based on PDM: For wide-band analysis, path delays are approximated as integer multiples of the symbol interval, separating desired signal, same-stream ISI, and inter-stream interference.The achievable sum-rate is R = Σ_l log2(1 + γ_l).

1) MRC Receive Beamforming:

With MRC reception, PDM’s interference is governed by transmitter- and receiver-side inter-path contamination. Sufficient angular separation or lens resolution removes these interference terms and makes PDM capacity-achieving.

  • 1) MRC Receive Beamforming:: MRC selects each receive beamformer to maximize the desired signal power from its corresponding path.The resulting SNR expression includes desired-path power and interference contributions.
  • 1) MRC Receive Beamforming:: ISI is double attenuated through transmitter- and receiver-side inter-path contamination coefficients, while inter-stream interference is attenuated through one side’s coefficient.These coefficients quantify leakage between path-focused responses.
  • 1) MRC Receive Beamforming:: Transmitter-side contamination vanishes when |φ̃T,l − φ̃T,l′| > 2∆/D̃T, equivalently when the transmitting support subsets do not overlap.The analogous receiver-side condition follows for sufficiently separated AoAs.
  • 1) MRC Receive Beamforming:: When both AoAs and AoDs are sufficiently separated, ISI and inter-stream interference vanish and MRC achieves channel capacity for narrow-band and wide-band communications.The per-stream SNR becomes γ_l = p_l|α_l|^2A_RA_T/σ^2.
  • 1) MRC Receive Beamforming:: The transmitter-side contamination coefficient decreases with greater AoD separation and higher transmitter lens-array resolution.Fig. 9 plots this coefficient against AoD difference for different D̃T values.

2) MMSE Receive Beamforming:

MMSE reception mitigates interference in general PDM operation by balancing desired-signal enhancement against interference suppression. Under favorable angular separation, it reduces to the same receive beamformer as MRC.

  • 2) MMSE Receive Beamforming:: Nonzero inter-path contamination caused by limited angular separation or insufficient lens resolution produces both ISI and inter-stream interference in PDM.MMSE beamforming is introduced as a mitigation method for this general case.
  • 2) MMSE Receive Beamforming:: MMSE beamforming generally outperforms MRC because it balances maximizing desired signal power with minimizing interference.The comparison follows from the inequality used to bound the corresponding SNR.
  • 2) MMSE Receive Beamforming:: When receiver-side contamination is negligible under sufficiently separated AoAs and AoDs, MMSE and MRC receive beamforming vectors are identical.Thus MMSE does not alter the favorable-propagation solution.

B. Path Grouping

Path grouping jointly processes paths with overlapping antenna support, converting the effective channel into smaller parallel MIMO channels and mitigating inter-path interference. With sufficiently separated AoAs or AoDs, PDM can eliminate ISI, while grouped eigenmode transmission achieves capacity under the stated conditions.

  • B. Path Grouping: Path grouping jointly processes paths whose supporting antenna subsets overlap and partitions them into groups.For example, paths 2 and 3 form one group when their relevant antenna supports overlap.
  • B. Path Grouping: With sufficiently separated AoAs, each receiving antenna receives signals through only one path, enabling per-path delay compensation.Nonoverlapping receiving subsets create path-specific supports, although transmitting subsets may still overlap.
  • B. Path Grouping: With either sufficiently separated AoAs or AoDs, PDM can completely eliminate inter-symbol interference.The condition corresponds to negligible inter-path coupling coefficients.
  • B. Path Grouping: The grouped channel decomposes into G parallel MIMO AWGN channels with smaller dimensions.Each group uses the union of its supporting transmitting and receiving antenna subsets.
  • B. Path Grouping: Eigenmode transmission over the grouped channels achieves capacity with lower complexity than processing the original channel directly.The reduced dimensions are the direct source of the complexity reduction.

2) Sufficiently Separated AoDs:

When AoDs are sufficiently separated, transmitting antenna supports become disjoint even if some AoAs are close. Transmitter-side delay pre-compensation enables an equivalent MIMO AWGN channel, which path grouping divides into smaller capacity-achieving channels.

  • 2) Sufficiently Separated AoDs:: Sufficiently separated AoDs make the transmitting antenna subsets for different paths disjoint.This setting can arise when the transmitter has accurate AoD resolution but the receiver has only moderate AoA resolution.
  • 2) Sufficiently Separated AoDs:: Each transmitting antenna then carries signals through only one path, enabling path-delay pre-compensation at the transmitter.The transmitted signal on path l is advanced by its path delay τ_l.
  • 2) Sufficiently Separated AoDs:: After pre-compensation, the lens MIMO system is equivalent to a |M_S| × |Q_S| MIMO AWGN channel for narrow-band and wide-band communications.Its capacity is achievable through eigenmode transmission with water-filling power allocation.
  • 2) Sufficiently Separated AoDs:: Grouping paths with overlapping receiving supports decomposes the equivalent channel into G parallel MIMO AWGN channels.The resulting group channels have reduced sizes.
  • 2) Sufficiently Separated AoDs:: Eigenmode transmission over each reduced-size group channel achieves the channel capacity.Processing each smaller channel reduces the dimensionality of the capacity-achieving operation.

C. Numerical Results

Numerical evaluations compare lens-MIMO PDM schemes with UPA-based MIMO-OFDM under wide-band settings and limited RF chains. Lens systems provide higher rates, while interference mitigation becomes more important as angular separation decreases and SNR increases.

  • C. Numerical Results: Q = 41 and M = 21 lens-array antennas use the same apertures as UPA arrays with QU = 400 and MU = 200 antennas.Both systems use M_RF = Q_RF = 2ΔL RF chains, with L = 3 paths and Δ = 1 in the example.
  • C. Numerical Results: Lens-MIMO PDM schemes achieve significant rate improvement over UPA-based MIMO-OFDM with the same number of RF chains or selected antennas.The comparison is reported for wide-band mmWave communication with antenna selection.
  • C. Numerical Results: At low SNR, MMSE and MRC PDM achieve the same performance as path grouping.The passage attributes this to negligible inter-path interference in the low-SNR regime.
  • C. Numerical Results: At higher SNR, the PDM schemes diverge because their interference-mitigation capabilities differ.The performance gaps become more pronounced when AoA separations are smaller.
  • C. Numerical Results: Smaller AoA separations increase the need for path grouping or other sophisticated interference-mitigation techniques.This comparison is made between Φ_R = 10° and Φ_R = 150° settings.

APPENDIX A PROOF OF LEMMA 1

The appendix derives lens-array focusing from the EM lens phase profile and shows that the focal-arc field is a Fourier-transform representation of the incident signal. For plane waves, this yields the lens response used to characterize antenna focusing.

  • APPENDIX A PROOF OF LEMMA 1: The EM lens phase profile is designed so rays from the aperture arrive with a common phase at the focal point.The lens uses spatial phase shifters to compensate for differing propagation distances.
  • APPENDIX A PROOF OF LEMMA 1: For uniform plane-wave incidence with negligible elevation AoA, the focal-arc signal is derived from the incident aperture signal.The derivation assumes the incident signal is uniform along the elevation dimension.
  • APPENDIX A PROOF OF LEMMA 1: The spatially phase-shifted focal-arc signal is represented as the Fourier transform of the scaled incident signal.The transform uses spatial frequency and spatial position variables defined in the derivation.
  • APPENDIX A PROOF OF LEMMA 1: For an incident plane wave with azimuth AoA φ, substituting the aperture field into the focal-arc expression yields the effective lens response.The spatial frequency is ˜φ = sin(φ).
  • APPENDIX A PROOF OF LEMMA 1: Evaluating the focal-arc response at the antenna positions gives the lens-array response, which simplifies under the critical antenna spacing.The appendix concludes the proof after this array-response reduction.
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