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Performance Analysis and Low-Complexity Design for XL-MIMO with Near-Field Spatial Non-Stationarities
Kangda Zhi, Cunhua Pan, Hong Ren, Kok Keong Chai, Cheng-Xiang Wang, Robert Schober, Xiaohu You
TL;DR
Near-field XL-MIMO requires analysis that accounts for spherical wavefronts and spatially varying power across discrete arrays. The paper develops an EM-based analysis and VR-informed detection and user partitioning methods; simulations report performance very similar to whole-array designs with much lower complexity.
Problem
Near-field spherical wavefronts and array-wide power variation complicate analysis of discrete-aperture XL-MIMO.
Source: §I · p. 1Method
The paper analyzes discrete-aperture XL-MIMO with an EM channel model and proposes VR-based detection and graph-theory user partitioning with partial zero-forcing.
Source: §I · p. 2Results
The proposed algorithms achieve very similar performance to conventional WA-based design with much reduced complexity.
Source: §I · p. 2Takeaways & Limitations
Simulations validate the effectiveness of the two low-complexity algorithms for XL-MIMO with near-field spatial non-stationarities.
Source: §VI · p. 14
Abstract
from arXiv · showhide
Extremely large-scale multiple-input multiple-output (XL-MIMO) is capable of supporting extremely high system capacities with large numbers of users. In this work, we build a framework for the analysis and low-complexity design of XL-MIMO in the near field with spatial non-stationarities. Specifically, we first analyze the theoretical performance of discrete-aperture XL-MIMO using an electromagnetic (EM) channel model based on the near-field spherical wavefront. We analytically reveal the impact of the discrete aperture and polarization mismatch on the received power. We also complement the classical Fraunhofer distance based on the considered EM channel model. Our analytical results indicate that a limited part of the XL-array receives the majority of the signal power in the near field, which leads to a notion of visibility region (VR) of a user. Thus, we propose a VR detection algorithm and leverage the acquired VR information to devise a low-complexity symbol detection scheme. Furthermore, we propose a graph theory-based user partition algorithm, relying on the VR overlap ratio between different users. Partial zero-forcing (PZF) is utilized to eliminate only the interference from users allocated to the same group, which further reduces computational complexity in matrix inversion. Numerical results confirm the correctness of the analytical results and the effectiveness of the proposed algorithms. It reveals that our algorithms approach the performance of conventional whole array (WA)-based designs but with much lower complexity.
I. INTRODUCTION
XL-MIMO arrays operate in the near field, where spherical wavefronts and spatially varying power make channel analysis and detection more complex. This work models discrete-aperture XL-MIMO electromagnetically and develops VR-based detection and user partitioning methods.
I. INTRODUCTION
- Near-field phase, amplitude, and incident-direction variations matter, while prior work leaves VR detection, general UPA VR overlap, and nonzero out-of-VR power insufficiently addressed.Continuous-aperture models simplify analysis but entail fabrication complexity and complicated interantenna coupling; discrete-aperture studies cited here did not use an EM model to analyze polarization mismatch.Source: §I · pp. 1–2
- The paper derives single-user SNR insights for discrete-aperture XL-MIMO, including aperture and polarization effects, and complements the Fraunhofer distance for arbitrary incident directions and array-wide power variation.It also proposes a sub-array-based VR detector, a VR-based low-complexity linear detector, and conventional MRC, ZF, and MMSE whole-array detectors for multi-user systems.Source: §I · p. 2
- The proposed algorithms achieve very similar performance to conventional WA-based design with much reduced complexity.The paper also reports simulation validation of its analytical results.Source: §I · p. 2
II. SYSTEM MODEL
- The system model considers uplink transmission from K single-antenna users to an XL-MIMO array, with the array center at the coordinate origin.The array lies in the xoy plane and the z-axis is perpendicular to it.Source: §II · p. 3
A. XL-MIMO Array Structure
- The discrete array has M = MxMy elements with effective area A = AxAy and spacing greater than each element's corresponding side length; η = A/(∆x∆y) describes its occupation ratio.The channel model uses a dyadic Green's-function EM formulation, accounts for element-specific power and phase, and assumes low-cost uni-polarized elements with current in the positive y direction.Source: A. XL-MIMO Array Structure · p. 3, B. Channel Modeling · pp. 3–4
B. Channel Modeling
- The EM channel model represents element-wise channel power and phase, including projected aperture and polarization effects; vertically incident signals simplify to the case without polarization mismatch.The model approximates each element's response using its center point when the antenna element is much smaller than the user-to-element distance.Source: B. Channel Modeling · p. 4
III. SNR ANALYSIS FOR SINGLE-USER SCENARIO
The single-user analysis derives an MRC SNR for the near-field EM channel and examines how discrete aperture and polarization mismatch affect it. The analysis finds that polarization mismatch matters more in the near field and depends on the user’s position relative to the array.
III. SNR ANALYSIS FOR SINGLE-USER SCENARIO
- Theorem 1 gives the single-user SNR for the near-field channel: it is zero when u_k,z = 0 and otherwise follows the stated expression.The analysis assumes a single user and uses an MRC detector.Source: §III · p. 4
- The SNR increases with the array occupation ratio η, which represents the effective array aperture.The ratio is defined as η = A/(ΔxΔy) ≤ 1.Source: §III · p. 4
- Polarization mismatch plays a more important role in the near field, with its impact dependent on Ly and the user’s y-coordinate u_k,y.For large u_k,z, the cited analysis gives the far-field condition for assessing mismatch.Source: §III · p. 5
A. Asymptotic Limit
The near-field EM model limits the SNR as array size grows, unlike the far-field expression’s linear growth with antenna count. The asymptotic limit depends on the array’s view geometry, polarization mismatch, and effective aperture.
A. Asymptotic Limit
- The far-field SNR expression grows linearly with M, but the passage states that this scaling no longer holds as M tends to infinity under the near-field model.Source: A. Asymptotic Limit · p. 5
- With polarization mismatch, the infinite-array SNR limit falls from 1/2 to 1/3; for discrete aperture, it is additionally constrained by occupation ratio η.The cited passage attributes the reduction to additional amplitude attenuation as Ly increases.Source: A. Asymptotic Limit · p. 5
- The near-field SNR cannot increase without bound because the user’s view angles remain limited as the array aperture expands.When both Lx and Ly tend to infinity, the angles approach π/2 and π/4, respectively.Source: A. Asymptotic Limit · pp. 5–6
B. XL-ULA
The section simplifies the XL-UPA to an XL-ULA to examine polarization mismatch. Theorem 2 gives the resulting SNR expression for the case My = 1.
B. XL-ULA
- The analysis specializes the XL-UPA to an XL-ULA by setting Mx = 1 or My = 1.Source: B. XL-ULA · p. 6
- For My = 1, Theorem 2 states a simplified expression for the XL-ULA SNR.Source: B. XL-ULA · p. 6
- The stated result is conditional on the XL-ULA case with My = 1.Source: B. XL-ULA · p. 6
F ULA
For the XL-ULA, the asymptotic analysis relates polarization mismatch to the user’s y-coordinate and shows that the infinite-length SNR limit is independent of the x-coordinate. A geometric view-angle description also explains why the SNR remains limited as the array grows.
F ULA
- As Mx tends to infinity, the asymptotic SNR is independent of the user’s x-coordinate u_k,x.The passage explains that an infinitely long array makes the x-coordinate inconsequential.Source: F ULA · p. 6
- The SNR gap due to polarization mismatch first increases and then decreases with u_k,y, tending to zero at u_k,y = 0 and as u_k,y grows large.The passage attributes the first limit to vanishing mismatch and the large-coordinate limit to the user being in the far field.Source: F ULA · p. 6
- For a user centered perpendicular to the array, the half view angle γ tends to π/2 as Mx grows, and the SNR is therefore limited.The passage defines γ as half the user-to-ULA view angle.Source: F ULA · p. 6
C. Near-Field/Far-Field Boundary
The classical Fraunhofer boundary assumes perpendicular incidence and accounts only for phase error, while the paper extends the boundary to arbitrary incident directions and also considers power variation. The phase-based boundary recovers the classical distance for perpendicular incidence, but varies with array size and signal direction.
C. Near-Field/Far-Field Boundary
- The general near-field region combines phase- and power-based boundaries: their union marks where either effect is evident, and their intersection where both are pronounced.Beyond the union, both power variation and phase error are slight enough for a planar-wavefront approximation with negligible errors.Source: C. Near-Field/Far-Field Boundary · p. 7
1) Near-field region characterized by phase error with arbitrary incident directions:
- The phase-error-based 3D boundary gives a near-field distance for all incident directions, and that distance increases with array size.For the square-array case, the stated near-field distance is smaller than the classical Fraunhofer distance and changes with incident direction.Source: 1) Near-field region characterized by… · p. 7
- The phase-based boundary recovers the classical Fraunhofer distance for perpendicular incidence, while the extended boundary accounts for arbitrary signal incident directions.The classical criterion is limited to perpendicular incidence and phase error; the paper also identifies power variation as a relevant near-/far-field distinction.Source: C. Near-Field/Far-Field Boundary · pp. 6–7, 1) Near-field region characterized by… · p. 7
2) Near-field region characterized by power variations:
The power-based boundary identifies near-field locations by measuring power variation across the array. The paper computes that variation and compares it with a threshold for acceptable far-field power difference.
2) Near-field region characterized by power variations:
- The power-variation measure v(u_k) is approximately 1 in the far field and decreases toward 0 as the user approaches the array.The near-field region is identified by locations where v(u_k) falls below a threshold for the maximal acceptable far-field power difference.Source: 2) Near-field region characterized by… · p. 7
- Received power decreases with y-coordinate separation because both distance and polarization mismatch increase, but its variation with x-coordinate separation is non-monotonic.Along x, increasing separation raises distance while reducing relative polarization mismatch.Source: 2) Near-field region characterized by… · p. 8
- A user’s power-based near-/far-field boundary is found by fixing its x and y coordinates and searching over z until v(u_k) equals the threshold.The variation measure is calculated using the antenna elements that maximize and minimize received channel power.Source: 2) Near-field region characterized by… · p. 8
3) The general phase- and power-based near-/far-field boundary:
The paper combines phase-error and power-variation boundaries to characterize near-field behavior, then motivates low-complexity multi-user detection by the interference and matrix-inversion costs of whole-array designs. In near-field spherical-wave channels, conventional MRC’s favorable interference behavior does not hold.
3) The general phase- and power-based near-/far-field boundary:
- The general boundary’s union marks locations where phase errors or power variations are evident, while its intersection marks where both are pronounced.Outside the union, both effects are slight and the spherical wavefront can be approximated by a planar wavefront with negligible errors.Source: 3) The general phase- and power-based… · p. 8
IV. MULTI-USER SCENARIO
- The multi-user section applies near-field properties to propose low-complexity symbol detectors.It follows the single-user analysis of XL-MIMO performance and properties in the near field.Source: §IV · p. 8
A. Whole Array-Based Design
- Near-field spherical-wave channels lack the far-field condition under which MRC interference tends to zero for users at different angles.Near-field channel amplitudes differ across antenna elements and phases are not linearly scaled; the cited analysis says MRC does not work well because of severe interference.Source: A. Whole Array-Based Design · p. 9
- ZF or MMSE can eliminate interference, but their required matrix inversion significantly increases computational complexity.This motivates low-complexity ZF/MMSE schemes that exploit near-field spatial non-stationarity.Source: A. Whole Array-Based Design · p. 9
B. VR-Based Low-Complexity Design
The VR-based design uses the fact that a limited part of the array receives a large portion of a user’s signal power. It selects sub-arrays contributing a chosen fraction of the user’s SNR and uses the resulting VR channel for detection.
B. VR-Based Low-Complexity Design
- A user’s VR is the limited part of the array that receives a large portion of its signal power, and it can be approximated by a combination of sub-arrays.Because these sub-arrays may be considerably smaller than the whole array, the VR can support lower-complexity detector design.Source: B. VR-Based Low-Complexity Design · p. 9
- The VR detector selects sub-arrays contributing a chosen fraction ϖ of the user’s SNR, collecting their indices in B_k.The described procedure sorts sub-array powers and selects them until the accumulated power reaches the target fraction.Source: B. VR-Based Low-Complexity Design · pp. 9–10
- For each user, the VR-based detector substitutes the channel matrix restricted to that user’s VR for the whole-array channel matrix.The passage identifies VR-based ZF as an example of this substitution.Source: B. VR-Based Low-Complexity Design · p. 10
C. User Partition-Based PZF
The user-partitioning method groups users according to VR overlap and applies PZF only within groups. Its complexity advantage is strongest when VRs occupy a small fraction of the array and users can be divided into many groups.
C. User Partition-Based PZF
- An edge connects users whose VR overlap ratio meets the threshold; the algorithm seeks a maximum independent set, then groups each selected user with its neighbors.The overlap ratio is |Bk∩Bi|/min{|Bk|,|Bi|}, and the threshold controls which pairs receive an edge.Source: C. User Partition-Based PZF · p. 10
- The degree-based reduction finds the independent set in linear O(K) complexity, after which PZF eliminates interference only within each user group.The method treats inter-group interference as weak and neglects it during detector design.Source: C. User Partition-Based PZF · p. 10
D. Complexity Analysis
- Under ideal equal-size VRs and uniform user groups, the PZF inversion dimension falls from K to K/|eI|, and complexity reductions are strongest for small VRs and many groups.The analysis assumes |Bk| = |eB| for each user and divides K users uniformly into |eI| groups.Source: D. Complexity Analysis · p. 10
A. Single-User Case
The single-user analysis finds that near-field SNR saturates as arrays grow, with saturation occurring sooner for a ULA than a UPA. In multi-user settings, VR-based detection and user-partitioned PZF approach whole-array performance while reducing complexity, with performance depending on VR selection and overlap thresholds.
A. Single-User Case
- For XL-UPA, near-field SNR initially rises but ultimately saturates as M approaches infinity, unlike the far-field result, which increases linearly with M.The proposed EM model includes additional loss from the discrete aperture and polarization mismatch.Source: A. Single-User Case · p. 11
- The XL-ULA SNR saturates at roughly 10^3 antenna elements, versus 10^6 for the XL-UPA, as near-field effects become more pronounced across the ULA.The passage attributes the difference to the ULA's larger dimension at a given antenna count, increasing variation in amplitudes, incident angles, and polarization mismatch.Source: A. Single-User Case · p. 11
- The phase-based near-/far-field boundary is smaller than the classical Fraunhofer distance and shrinks as the incident angle increases; the power-based boundary has a different shape.The power-based boundary shrinks as the user approaches the array center and expands with a stricter power-variation threshold.Source: A. Single-User Case · pp. 11–12, Figure 7 · p. 12
B. Multi-User Case
- For large M, VR-based MMSE and ZF perform very close to whole-array designs, and with M = 10^4, 25% of antennas receive 80% of the total power.Increasing the VR detection ratio includes more sub-arrays and trades complexity for performance; low ratios can cause greater rate loss.Source: B. Multi-User Case · pp. 12–13
- With moderate VR-overlap thresholds, user-partitioned PZF achieves almost the same performance as VR-based detectors; large thresholds can leave strongly interfering users in different groups.Only intra-group interference is nulled, reducing the matrix inversion burden.Source: B. Multi-User Case · pp. 13–14, Figure 11 · p. 13, Figure 12 · p. 13
- VR-based ZF complexity grows sublinearly with antenna count, while user-partitioning PZF complexity practically saturates for large M, and the conclusion reports both algorithms as effective low-complexity designs.The conclusion says simulations validated the effectiveness of the two proposed algorithms.Source: B. Multi-User Case · p. 14, §VI · p. 14, Figure 13 · p. 14
APPENDIX A
The appendices derive an approximation to the double sum using small normalized antenna spacings and provide a separate expression for the one-dimensional array case.
APPENDIX A
- When normalized spacings are small, the double sum is approximated by a double integral divided by 1/(ϵxϵy), using a change of variables and solving the integrals successively.The derivation treats each small integration-domain area as having approximately the objective value at its center.Source: Appendix A · pp. 14–15
APPENDIX B
- For My = 1, the appendix obtains a separate expression and completes its proof with cited integral identities and algebra.Source: Appendix B · p. 15