PaperlayerView exact revision on arXiv

Source-linked AI summary

Massive MIMO for Communications with Drone Swarms

Prabhu Chandhar, Danyo Danev, Erik G. Larsson

arXiv:1707.01039v2cs.ITmath.IT

TL;DR

Existing wireless links are described as inadequate for long-range, high-throughput, high-mobility UAV communication. The paper evaluates massive-MIMO drone-swarm links and reports high sum throughput, with half-wavelength-multiple antenna spacing optimal for spherical-shell distributions.

  • Problem

    Existing WiFi, ZigBee, and XBee-Pro links are described as too limited for long-range, high-throughput, high-mobility UAV applications, where high-resolution imagery and video also require high-speed communication.

    Source: §I · p. 1

  • Method

    The paper evaluates massive-MIMO drone-swarm communication using low-complexity maximal-ratio processing and ML channel estimation.

    Source: E. Uplink pilot signaling and channel… · p. 9, §VIII · p. 21

  • Results

    2.4 Gbps (with 2194 antennas) and 1.44 Gbps (with 313 antennas) sum throughput in 20 MHz to 20 drones; for spherical-shell distributions, optimal spacing is an integer multiple of half wavelength.

    Source: §VIII · p. 21

  • Takeaways & Limitations

    Pseudo-randomly oriented GS antennas provide diversity against polarization mismatches, facilitating simple cross-dipoles; in LoS, array gain may extend range by factor M or reduce UAV transmit power by factor M.

    Source: §VIII · p. 21

    Detailed zero-forcing performance remains future work, although ZF may add sum-rate gains over MRC at medium and high SNR.

    Source: B. Achievable rate performance with… · p. 15

Abstract

from arXiv · show

We illustrate the potential of massive MIMO for communication with unmanned aerial vehicles (UAVs). We consider a scenario where multiple single-antenna UAVs simultaneously communicate with a ground station (GS) equipped with a large number of antennas. Specifically, we discuss the achievable uplink (UAV to GS) capacity performance in the case of line-of-sight (LoS) conditions. We develop a realistic geometric model which incorporates an arbitrary orientation of the GS and UAV antenna elements to characterize the polarization mismatch loss which occurs due to the movement and orientation of the UAVs. A closed-form expression for a lower bound on the ergodic rate for a maximum-ratio combining receiver with estimated channel state information is derived. The optimal antenna spacing that maximizes the ergodic rate achieved by an UAV is also determined for uniform linear and rectangular arrays. It is shown that when the UAVs are spherically uniformly distributed around the GS, the ergodic rate per UAV is maximized for an antenna spacing equal to an integer multiple of one-half wavelength.

I. INTRODUCTION

This section examines massive MIMO links for UAV swarms using geometric, polarization-aware channel modeling to address mobility and antenna-orientation changes. It characterizes polarization loss for cross-dipole links, including its variation with propagation direction even when antenna orientations are fixed.

I. INTRODUCTION

  • UAV applications can exceed existing WiFi, ZigBee, and XBee-Pro limits in range, throughput, mobility, and simultaneous swarm service, motivating massive MIMO.Massive MIMO uses coherent beamforming to focus power, reduce interference, and improve energy efficiency and latency.Source: §I · p. 1

A. Contributions

  • The analysis derives a closed-form lower bound on MRC uplink ergodic rate with estimated CSI, optimizes GS array geometry, and also studies ZF with perfect CSI.It assumes inverse-SNR power control and averages over UAV positions to analyze ergodic capacity.Source: A. Contributions · p. 2

B. Polarization model with single cross-dipole antenna at the transmitter and the receiver

  • The polarization model captures mismatch using crossed dipoles at the GS and UAV; appropriately oriented large arrays exploit polarization diversity for good communication rates and reliability.Cross-dipoles support different polarizations, while the array's many differently oriented antennas provide polarization diversity.Source: A. Contributions · p. 2

B. Related Works

  • Unlike prior LoS analyses with fixed half-wavelength spacing and perfect CSI, this work optimizes array geometry and accounts for mobility and pseudo-random UAV antenna orientations.The cited prior work did not consider polarization mismatch losses or terminal mobility.Source: B. Related Works · p. 2, A. Contributions · p. 2

A. Geometric Model

  • Including the second distance-expansion term matters when low-altitude UAV distance is comparable to array aperture: in the 25 m example, it reduces approximation error across most angles.Without the term, error is significant for most elevation and azimuth angles; with it, error is small for most angles.Source: A. Geometric Model · p. 4

B. Polarization model with single cross-dipole antenna at the transmitter and the receiver

  • The polarization loss factor is characterized by projecting the incident electric field onto the receiving antenna response; its matrix entries represent mismatch between field components and UAV dipole orientations.The passage identifies the polarization loss factor between the transmitted wave and receive antennas as the relevant quantity.Source: B. Polarization model with single… · p. 6
  • The resultant wave’s polarization depends on the electric-field component magnitudes, their phase difference, and propagation direction; equal-magnitude dipole feeds yield circular polarization along the axis perpendicular to both dipoles.In other directions, the polarization can be elliptical or linear.Source: B. Polarization model with single… · p. 6
  • With fixed transmit and receive antenna orientations, moving the receiver can yield high polarization loss for particular elevation and azimuth angles, irrespective of antenna polarization type.The analysis first considers dipoles aligned with the y- and z-axes, then extends the calculation to antenna rotations at both ends.Source: B. Polarization model with single… · p. 6

C. Single antenna channel transfer function for arbitrary orientation of GS and UAV antennas

The channel-transfer model accounts for arbitrary transmitter and receiver antenna rotations by expressing elevation angles and electric-field orientations in their rotated coordinate systems. It then uses the fields’ projections onto the dipoles to characterize polarization mismatch.

C. Single antenna channel transfer function for arbitrary orientation of GS and UAV antennas

D. Antenna array channel model

The array channel model builds each UAV-to-ground-station channel vector from the single-antenna transfer function, which includes antenna gain and polarization mismatch. The analysis omits UAV-body blockage loss for simplicity.

D. Antenna array channel model

  • The channel vector is formed from the transfer-function elements for the UAV’s position and the ground-station array.The channel-vector definition is given for carrier frequency f0 = fc.Source: D. Antenna array channel model · p. 8
  • Each channel-vector element incorporates free-space path loss and a complex transfer factor combining antenna gains, field patterns, and polarization-mismatch loss.The transmit and receive dipole gains are denoted Gt and Gr; for a half-wavelength dipole, each is approximately 1.643.Source: D. Antenna array channel model · p. 8
  • The model excludes UAV-body blockage loss for simplicity, although the authors say it could be incorporated as a function of azimuth, elevation, roll, pitch, and yaw.Reported blockage measurements cited in the passage are 5–35 dB for fixed-wing aircraft and 5–15 dB for quadcopters.Source: D. Antenna array channel model · p. 8

E. Uplink pilot signaling and channel estimation

The paper describes pilot-based channel estimation within a Doppler-limited coherence interval, then specifies uplink power control and assumptions used in its achievable-rate analysis. The rate analysis considers MRC with estimated CSI and ZF with perfect CSI.

E. Uplink pilot signaling and channel estimation

  • Channel estimates are refreshed after each conservatively designed coherence time because UAV motion produces Doppler shift.The coherence time is approximated by Tcoh ≈ 1/(2fmax), with fmax determined by maximum UAV speed and wavelength.Source: E. Uplink pilot signaling and channel… · pp. 8–9
  • The coherence interval is the product of coherence time and coherence bandwidth, and TDD downlink, pilot, and data symbols must fit within it.The passage notes finite coherence bandwidth in settings with a few multipath components, including over-water and mountainous settings.Source: E. Uplink pilot signaling and channel… · p. 9
  • The ground station estimates the channel from K orthogonal UAV pilot sequences; the paper uses maximum-likelihood estimation because MMSE estimation is nontrivial under its LoS model.Pilot power is selected based on worst-case distance and effective antenna gain.Source: E. Uplink pilot signaling and channel… · p. 9

F. Uplink Data Transmission

  • Channel-inversion power control targets the same average SNR for every UAV, while the power and coherence-interval constraints can place a UAV in outage.The UAV needs the large-scale channel gain for power control, which can be obtained through downlink pilot transmission.Source: F. Uplink Data Transmission · p. 9

III. ACHIEVABLE RATE ANALYSIS

  • The rate analysis derives an estimated-CSI MRC bound and analyzes ZF assuming perfect CSI, using a far-field, identically oriented array approximation for rate analysis.The authors separately examine the reliability impact of arbitrary ground-station element orientations.Source: F. Uplink Data Transmission · pp. 9–10, §III · p. 10

A. MRC receiver

The MRC analysis derives a lower bound on each UAV’s ergodic rate and a closed-form version for independently and spherically uniformly distributed UAV locations. The bound reflects array gain, interference, estimation error, and noise.

A. MRC receiver

  • The authors average over UAV positions because the channel vectors and their interference change as UAVs move.The interference power depends on antenna count, spacing, and UAV velocity and moving direction.Source: A. MRC receiver · p. 10
  • The desired-signal term in the rate bound grows proportionally with the number of ground-station antennas, while its denominator includes interference, channel-estimation error, and noise.The passage gives 100 antennas as an example: a 20 dB array gain and 10-times range extension in LoS at the same radiated power as a single-antenna system.Source: A. MRC receiver · p. 10
  • For independently and spherically uniformly distributed UAV locations in a shell, the paper gives a closed-form lower bound on each UAV’s MRC ergodic rate using ML channel estimates.The general rate expression can also be applied to arbitrary UAV-location distributions, though numerical evaluation may be needed.Source: A. MRC receiver · pp. 10–11
  • The expected polarization-dependent gain factor must be calculated numerically because hkl depends on rotation angles, field patterns, and polarization mismatch.The passage directs discussion of this calculation to Section V.Source: A. MRC receiver · p. 11
  • For UAVs on the sphere’s surface, the lower bound simplifies, with the spacing-dependent quantity Ω1 entering the expression.The authors identify antenna spacing as a factor affecting ergodic rate.Source: A. MRC receiver · p. 12

B. Zero forcing (ZF) receiver

The section derives an uplink-rate lower bound for a zero-forcing (ZF) receiver, including a tractable expectation for the two-UAV case. The ZF detector separates the received signal into user streams using the channel matrix.

B. Zero forcing (ZF) receiver

  • ZF detection separates the received signal into K streams by multiplying it with G†.The derivation then forms the covariance of the detector output and uses it to bound achievable uplink rate.Source: B. Zero forcing (ZF) receiver · p. 13
  • Jensen’s inequality yields a lower bound on the achievable uplink rate with ZF.The bound is derived from the output covariance and its associated expectation.Source: B. Zero forcing (ZF) receiver · p. 13
  • For the expectation in the ergodic-rate bound, the analysis specializes to K = 2 because the general-K expectation is difficult to obtain.The two-UAV case allows the determinant of G^H G to be evaluated before substituting the expectation into the bound.Source: B. Zero forcing (ZF) receiver · p. 13

IV. ERGODIC RATE PERFORMANCE WITH OPTIMAL ANTENNA SPACING

This section analyzes ergodic-rate performance with optimal antenna spacing for uniform linear and uniform rectangular arrays.

IV. ERGODIC RATE PERFORMANCE WITH OPTIMAL ANTENNA SPACING

  • The analysis considers ergodic-rate performance with optimal antenna spacing for both ULA and URA structures.The passage states the section’s scope but does not report a result.Source: §IV · p. 13

A. Optimal antenna spacing

The section selects antenna spacings for ULA and URA arrays, then evaluates achievable throughput with MRC and discusses power scaling and ZF gains. Under the stated parameters, the required GS antenna count for 20 Mbps rises with the number of UAVs.

A. Optimal antenna spacing

  • For a ULA, antenna spacings at the nonzero integer sinc zero crossings make Ω zero and maximize the ergodic rate.The optimal spacing follows from the zero crossings of sinc(u).Source: A. Optimal antenna spacing · p. 13
  • For a URA, Ω has local minima rather than vanishing at cross-diagonal elements; Ω is approximately 0.053 when n = My and m = Mx.The allowable n and m values are limited by the array’s maximum aperture size.Source: A. Optimal antenna spacing · pp. 13–14

B. Achievable rate performance with optimal antenna spacing

  • For 20 Mbps, the required GS antenna counts are approximately 27, 68, and 136 for K = 20, 50, and 100 UAVs, respectively.The example uses ρu = 0 dB and ρp = 10 dB; Fig. 6(a) plots per-UAV ergodic throughput against the number of GS antennas.Source: Figure 6 · p. 15, B. Achievable rate performance with… · p. 15
  • For a given K, the rate grows without bound as M increases, while sum throughput eventually decreases as K grows because the coherence-interval pre-log term shrinks.The example reports per-UAV antenna requirements for a 20 Mbps target; the sum-throughput trend depends on the number of simultaneously communicating UAVs.Source: B. Achievable rate performance with… · p. 15
  • ZF may provide additional sum-rate gains over MRC at medium and high SNR, but its detailed study is left for future work.The section’s rate discussion otherwise focuses on MRC.Source: B. Achievable rate performance with… · p. 15
  • With finite K and perfect CSI, scaling UAV transmit power as ρu = εu/M makes each UAV’s lower-bound rate tend to Λ log2(1 + εu) for large M.The limiting rate matches the single-UAV case.Source: B. Achievable rate performance with… · p. 15

V. IMPACT OF POLARIZATION MISMATCH AND ANTENNA PATTERN ON THE LINK RELIABILITY

The analysis shows that antenna orientation and polarization can strongly affect effective gain and required UAV transmit power. Pseudo-random GS element orientations and circularly polarized cross-dipoles mitigate some of the low-gain outcomes described for aligned elements.

V. IMPACT OF POLARIZATION MISMATCH AND ANTENNA PATTERN ON THE LINK RELIABILITY

  • With identically oriented GS elements, effective gain falls below −50 dB at some UAV orientations, risking signal loss at those positions.The analysis varies elevation and azimuth angles under the stated circularly polarized antenna configuration.Source: §V · p. 17
  • For an example with χkl = −40 dB and χwc = −50 dB, the required UAV transmit power is approximately 94 W, and outage probability will be high.The passage attributes the high required power to low gain when all GS elements are identically oriented.Source: §V · p. 17
  • Pseudo-random GS element orientations can cancel gain nulls through polarization diversity; the reported average gain is approximately −8 dB.The comparison uses an array with Mx = 100 and a specified UAV direction.Source: §V · p. 17
  • Increasing the antenna count from 1 to 50 increases gain by a factor of 50, or approximately 17 dB.This gain comparison is reported for the CDF observations in Figures 7(c) and 7(d).Source: §V · p. 17
  • With identical GS orientation, gain below 10 dB occurs with probability 0.045 for circularly polarized cross-dipoles and 0.26 for linearly polarized dipoles.With pseudo-random GS orientations, the corresponding probabilities are zero and 0.16, respectively.Source: §V · p. 17
  • For circularly polarized cross-dipoles, χwc is around −17 dB with identical GS orientation and −3.5 dB with arbitrary orientation.In the stated example with χkl = −12 dB and χwc = −20 dB, required transmit power is 0.15 W; the passage says arbitrary orientations can achieve 100% coverage with very low uplink power.Source: §V · p. 17

VI. SURVEILLANCE USE CASE

The surveillance use case estimates the antenna resources needed for 20 drones to transmit imaging data, then discusses the operational and system-level implications of massive MIMO. The discussion and conclusions address mobility assumptions, array gain, receiver processing, polarization, and antenna spacing.

VI. SURVEILLANCE USE CASE

  • For the chosen trajectories, throughput fluctuates as drone motion changes interference; 5λ spacing produces more frequent capacity fluctuations than smaller spacing.The reported comparison concerns the corner drones and the cumulative interference from the other 19 drones.Source: §VI · pp. 17–18
  • The use case sizes antenna counts from image and video data-rate requirements, assuming Ω = 0 and specified uplink, pilot, bandwidth, and carrier parameters.For image transmission, the example uses 20 drones, 2664 × 1496 resolution, 24 bits per pixel, and 70% front overlap.Source: §VI · p. 20
  • At 5 cm GSD and 30 m/s, the mission takes 6 minutes 16 seconds with 20 drones, versus more than two hours with one drone.The example uses 2664 × 1496 camera resolution and 60% side overlap.Source: §VI · p. 20

VII. DISCUSSIONS

  • The random-within-a-sphere mobility model enables closed-form capacity results; other position distributions yield similar rate expressions with a different Ω.The authors expect different mobility models to give similar final performance when codewords span many realizations of spatial-signature inner products.Source: §VII · p. 20
  • Changing UAV speed from 0 to 30 m/s changes the pre-log factor by only 0.05 in the stated example; known trajectories may allow channel prediction and less frequent pilot transmission.The example uses τ_dl = T_len/8, B_c = 2 MHz, f_c = 5 GHz, and K = 100; the pre-log values are 0.875 and 0.825.Source: §VII · pp. 20–21

VIII. CONCLUSIONS

  • A drone-swarm surveillance case study reports sum throughput of 2.4 Gbps with 2194 antennas and 1.44 Gbps with 313 antennas, in 20 MHz bandwidth to 20 drones.Source: §VIII · p. 21
  • In LoS, array gain provides range extension by a factor of √M and can reduce UAV transmit power by a factor of M, potentially reducing the need for multi-hop links.The surveillance links between the ground station and corner drones are several kilometers long.Source: §VIII · p. 21, §VII · p. 21
  • Low-complexity maximal-ratio processing is reported as sufficient for good performance; differently oriented ground-station antennas provide polarization diversity, and spherical-shell placement favors half-wavelength-multiple spacing.The spacing result is exact for a linear array and approximate for a rectangular array; further zero-forcing gains are noted as future work.Source: §VIII · p. 21

APPENDIX A APPENDIX A: 3D ROTATION MODEL

Appendix A describes rotations of the coordinate frame around the three reference axes and notes that rotation order affects the resulting matrix. Appendix B derives equation (35) by calculating the variances of three terms.

APPENDIX A APPENDIX A: 3D ROTATION MODEL

  • Roll, pitch, and yaw denote rotations around the x-, y-, and z-axes, respectively.The stated angle ranges are [−π/2, π/2] for roll and pitch, and [0, 2π] for yaw.Source: Appendix A · p. 21
  • The rotation matrix depends on axis order; the example for z-, y-, then x-axis rotations is R(α_z, α_y, α_x) = R_x(α_x)R_y(α_y)R_z(α_z).The appendix also gives rotation matrices for the individual axes.Source: Appendix A · p. 21

APPENDIX B APPENDIX B: PROOF OF EQUATION (35)

  • The proof of equation (35) calculates the variances of the three terms in (31) and substitutes those results to obtain (35).The proof labels the calculations as var(a2), var(a1), and var(a3).Source: Appendix B · pp. 21–22

APPENDIX C APPENDIX C: PROOF OF EQUATIONS (49) AND (50)

Appendix C proves equations (49) and (50) through expectation and integral manipulations. The supplied proof steps express an expectation in real and imaginary parts and then evaluate the resulting integral.

APPENDIX C APPENDIX C: PROOF OF EQUATIONS (49) AND (50)

  • The proof rewrites the expectation in equation (49) as the cosine expectation plus i times the sine expectation.This separates the real and imaginary parts before evaluating the integral.Source: Appendix C · p. 22
  • The derivation evaluates the last integral in (77) and substitutes it into (77).These steps lead to the expectation expression given in (49).Source: Appendix C · pp. 22–23

2) Proof of Equation (50):

The proof of equation (50) evaluates an angular integral by changing variables and using trigonometric symmetries. It then relates the integral to roots of a function and a Fourier-transform step.

2) Proof of Equation (50):

  • The proof first rewrites the angular expression using α1 cos y + α2 sin y = p and selects an angle ξ from α1 and α2.It then applies trigonometric symmetries to the integral.Source: 2) Proof of Equation (50) · p. 23
  • A delta-function integral is evaluated by summing over roots in the specified set, with zero result when there are no roots.The proof applies this root-based integral rule to g(x).Source: 2) Proof of Equation (50) · p. 23
  • The derivation changes variables in a double integral, evaluates an inner integral, and uses an inverse Fourier transform to obtain h(α).The steps include deriving H(f) and then recovering the original integral.Source: 2) Proof of Equation (50) · p. 24
Loading 1707.01039v2…