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From Interference to Diversity: Mode Coloring and Two-Dimensional Modal-Time Rake Combining for Multipath OAM Communications

Runyu Lyu

arXiv:2610.21602v1eess.SP

TL;DR

Multipath scattering spreads OAM energy across modes, which conventional demultiplexing discards. The paper colors modes with orthogonal time codes and jointly combines modal and delay branches; under independent-branch fading, the asymptotic diversity order is LK, while modal covariance rank bounds it otherwise.

  • Problem

    Multipath-induced modal leakage is discarded by conventional OAM demultiplexing, and generic array-domain detection does not exploit the structured mode-order coupling and delay spread.

    Source: §I · pp. 1–2

  • Method

    The receiver assigns each OAM mode an orthogonal time-domain code and jointly combines despread outputs across a mode-delay finger grid.

    Source: §I · p. 2, §VI · p. 12

  • Results

    The asymptotic diversity order is LK under independent-branch conditions and is bounded by modal covariance rank otherwise.

    Source: §I · p. 2, §VI · p. 12

  • Takeaways & Limitations

    The proposed combination can recover multipath-induced modal leakage as a diversity resource when the modes are sufficiently separable.

    Source: M. Comparison with prior art and… · p. 12

    TDOCC requires timing synchronization because fractional timing offsets can disrupt zero-lag code orthogonality.

    Source: J. Timing-offset robustness · p. 11

Abstract

from arXiv · show

Orbital angular momentum (OAM) beams provide a spatial-multiplexing dimension for wireless communications, but multipath scattering broadens their mode spectrum, so that energy leaks across modes as inter-mode interference (IMI), which conventional OAM demultiplexing discards. This paper treats the leaked modal energy not as interference but as a structured, harvestable resource: it attains up to $LK$-fold asymptotic diversity under independent-branch fading and yields harvested energy under the rank-limited per-tap fading of the operating channel. Each active OAM mode is colored with a distinct time-domain orthogonal spreading code, and a two-dimensional (2D) receiver grid spanning the OAM modes and the resolvable multipath delays jointly combines the despread outputs. The design builds on the Bessel order-difference coupling law of tilted OAM beams, summarized by a Gaussian mode-diffusion model with a mode-coherence metric that quantifies the surviving orthogonality, and on an effective diversity metric that separates the combining gain available at finite signal-to-noise ratio (SNR) from the asymptotic diversity order. At 28~GHz, with a four-mode eight-element uniform circular array (UCA) and four resolvable multipath taps, the proposed 2D modal-time Rake lowers the bit-error rate (BER) by one to three orders of magnitude and saves roughly 10--15~dB of required SNR over mode-wise detection and mode-domain zero-forcing (ZF) and minimum mean-square error (MMSE) detection, and it also outperforms an uncolored modal-time MMSE equalizer; the advantage survives the spectral-efficiency cost of spreading as well as channel-estimation and timing imperfections.

I. INTRODUCTION

Multipath scattering leaks OAM energy across modes and delays, while conventional demultiplexing discards that leakage. The paper introduces mode coloring and joint modal-time combining to recover it under separable channel conditions.

I. INTRODUCTION

  • Scattering creates inter-mode interference and delay-spread mixing, while conventional DFT demultiplexing discards leaked energy and incurs an interference floor.The resulting BER floor does not vanish with transmit power.Source: §I · p. 1
  • The design adapts the CDMA 2D-Rake principle to OAM modes and delays, while treating the correspondence as signal-processing analogy rather than channel-model isomorphism.It exploits structured mode-order coupling and delay spread through direct, non-iterative linear processing.Source: §I · p. 2
  • TDOCC assigns each active OAM mode a distinct orthogonal time-domain code, enabling the receiver to recover energy otherwise treated as interference.The proposed L × K modal-time receiver combines the despread outputs using MRC or joint MMSE processing.Source: §I · p. 2, §II · p. 3
  • The contribution combines a Bessel order-difference coupling law, a Gaussian mode-diffusion model, mode coherence, and modal-time diversity analysis.The model and coupling law are validated against free-space UCA field simulation; LK diversity requires independent-branch conditions, otherwise modal covariance rank bounds it.Source: §I · p. 2

II. SYSTEM MODEL

  • The system model uses UCA arrays and a multipath channel with simultaneous modal leakage and inter-symbol interference across resolvable rays.At the receiver, DFT-demultiplexed streams are despread at each mode and delay, then combined by MRC or joint modal-time MMSE.Source: §II · pp. 2–3

III. MODE-TIME CHANNEL MODEL AND PROPOSED 2D-RAKE RECEIVER

The model derives how a tilted OAM ray couples energy between modes, with coupling governed by the difference in mode order. Alignment preserves a diagonal channel, while greater tilt broadens modal coupling.

A. UCA transceiver and per-ray mode-coupling law

  • The UCA generates an OAM mode using progressive element phases, and an N-point DFT recovers the received modal content.For |ℓ| < N/2 and moderate kR sin θ, the array-field derivation reduces to a single Bessel term.Source: A. UCA transceiver and per-ray… · p. 3
  • For a tilted OAM ray, projection onto the receive-mode basis yields a coupling coefficient governed by a Bessel function of the mode-order difference.The derivation uses the Jacobi–Anger expansion and azimuthal orthogonality.Source: A. UCA transceiver and per-ray… · p. 3
  • A perfectly aligned ray is diagonal, coupling decays with mode-order difference, and increasing tilt broadens the modal coupling distribution.The law is validated against full free-space UCA field simulation.Source: A. UCA transceiver and per-ray… · p. 3

B. Angular-spread average: Gaussian mode diffusion

The channel model approximates angular-spread-induced modal leakage with a Gaussian diffusion kernel and tracks residual modal orthogonality with mode coherence. TDOCC then assigns orthogonal codes to modes for despreading across modal and delay branches.

B. Angular-spread average: Gaussian mode diffusion

  • A Gaussian mode-diffusion amplitude kernel approximates the angular-spread average of the per-ray coupling law, with σℓ denoting modal diffusion bandwidth.This is a modeling assumption rather than an exact law; σℓ scales with kR.Source: B. Angular-spread average: Gaussian… · p. 3

C. Mode-time channel and mode coherence

  • Mode coherence ρ is the ratio of diagonal to off-diagonal coupling power; it indicates interference but is not sufficient to determine diversity.The LoS limit has ρ → ∞, while ρ ≈ 1 indicates an essentially mixed mode domain; ρ decreases with angular spread.Source: C. Mode-time channel and mode coherence · p. 4

D. Mode-colored transmission and received signal

  • TDOCC assigns each active mode an orthonormal spreading code, and a zero guard interval separates successive symbols under the integer-delay channel model.At the receiver, each output-mode stream is despread using each mode’s code at each delay to form modal-time observations.Source: D. Mode-colored transmission and… · p. 4

E. Channel estimation and pilot structure

Orthogonal pilot codes and guard intervals support least-squares estimation of the mode-time channel. The receiver despreads across modes and delays to form a finger bank that isolates modal contributions and collects temporal diversity.

E. Channel estimation and pilot structure

  • The pilot phase spreads a known symbol with each mode’s code, enabling least-squares estimation of each mode-pair, delay-tap channel coefficient.Zero-lag code orthogonality separates modes, while the guard interval separates taps.Source: E. Channel estimation and pilot… · p. 4

F. Two-dimensional finger bank

  • Each receiver finger correlates an output-mode stream with a reference mode’s code at a selected delay tap.Orthogonality at zero lag isolates the reference mode despite inter-mode interference.Source: F. Two-dimensional finger bank · p. 4
  • The L × K finger outputs collect estimates of a symbol across output modes and delays, so the bank captures both modal and temporal diversity.The delay offset aligns each finger with a resolvable tap.Source: F. Two-dimensional finger bank · p. 5

G. MRC combining

MRC coherently combines despread outputs across all output modes, recovering leaked modal energy that conventional demultiplexing discards. Joint MMSE equalization uses the mode-time channel to suppress residual inter-mode and inter-symbol interference.

G. MRC combining

  • MRC sums over all output modes, coherently harvesting leaked energy that a conventional DFT demultiplexer discards as interference.Its weights are the conjugates of the corresponding channel couplings.Source: G. MRC combining · p. 5
  • Under LK independent Rayleigh-faded fingers with equal power, MRC attains diversity order LK.Residual inter-mode and inter-symbol interference at non-zero code lags can saturate MRC, motivating joint MMSE combining.Source: G. MRC combining · p. 5

H. MMSE combining

  • The joint MMSE receiver models each symbol’s response as a column of G, which captures the spreading code and mode-time channel.G is formed from the channel response to unit symbols.Source: H. MMSE combining · p. 5
  • The linear MMSE estimate jointly suppresses inter-mode and inter-symbol interference while coherently recombining the L × K modal-time diversity.When delay spread vanishes (K = 1), the receiver reduces to a mode-domain MMSE receiver.Source: H. MMSE combining · p. 5

I. Complexity

The section distinguishes combining dimensionality from asymptotic diversity, which depends on statistically independent, non-zero-power channel branches. The receiver’s structured modal-time processing also limits its stated computational costs to operations on the symbol-level model.

I. Complexity

  • Per symbol, modal-time MMSE costs O(L^2KNc) for finger processing and O(L^2) for combining, after per-block G construction and an O(L^3) inverse.It inverts an L × L Gram matrix rather than the full observation-dimension matrix required by a conventional array-domain space-time MMSE equalizer.Source: §I · p. 5

IV. DIVERSITY AND SPECTRAL-EFFICIENCY ANALYSIS

  • The finger count Nf = LK is combining dimensionality, not an unconditional diversity order; diversity depends on independent, non-zero-power branches or covariance rank.The section separates asymptotic diversity, finite-SNR effective diversity, and finger-grid dimensionality.Source: §IV · p. 5

A. Channel fading models

  • In the physical per-tap model, mode pairs on each tap share one Rayleigh coefficient, so the fading dimension is K rather than LK.The idealized independent-branch model instead assigns each coefficient its own independent Rayleigh factor.Source: A. Channel fading models · pp. 5–6

B. Diversity order under independent modal-delay branches

  • With LK mutually independent, non-zero-power finger coefficients, both MRC and joint MMSE attain asymptotic diversity order LK.This result relies on independence across output modes and delay taps.Source: B. Diversity order under independent… · p. 6

C. Effective diversity metric under unequal branch powers

The effective diversity metric describes finite-SNR usability of unequal-power independent branches, not the high-SNR diversity exponent. Under correlated modal diffusion, covariance rank instead bounds the diversity available from the modal-time grid.

C. Effective diversity metric under unequal branch powers

  • At ρ ≈ 4.4 with L = 4 and K = 4, the effective diversity metric is ≈ 5.9 of the maximum 16.It measures usable combining dimensionality when cross-mode branches are weak; the asymptotic order remains LK for independent, non-zero-power branches.Source: C. Effective diversity metric under… · p. 6
  • As cross-mode power vanishes, the effective metric tends to K; with equally powered branches, it tends to KL.The latter case corresponds to γ = 1 and ρ = 1/(L − 1).Source: C. Effective diversity metric under… · p. 6

D. Diversity under correlated modal diffusion

  • For correlated modal diffusion, asymptotic diversity is bounded by covariance rank, and a rank-one tap contributes one independent fading dimension.In the factorized example, modal branches add energy but no independent diversity despite the L-fold finger count.Source: D. Diversity under correlated modal… · pp. 6–7
  • For full-rank but correlated modal diffusion, finite-SNR behavior is governed by K times the target-mode effective rank averaged over taps.The unequal-power metric remains applicable.Source: D. Diversity under correlated modal… · p. 7

E. When is IMI diversity, and when is it interference?

Leaked modal energy yields diversity only when it is sufficiently strong, separable by the codes, supported by adequate covariance rank, and combined with adequate channel estimates. The section quantifies an outage benefit and the spectral-efficiency cost of spreading.

E. When is IMI diversity, and when is it interference?

  • Recovering modal leakage as diversity requires sufficient cross-modal energy, code separability, covariance rank, and adequate channel-state information.If any ingredient fails, the leaked energy turns back into interference.Source: E. When is IMI diversity, and when is… · p. 7

F. Outage probability

  • At a 10% outage target, the 2D-Rake requires mean branch SNR ≈ 0.09γth versus 0.57γth for the time-only Rake, a saving of about 8 dB.This comparison uses the idealized independent-branch model.Source: F. Outage probability · p. 7

G. Spectral-efficiency trade-off

  • Spreading reduces information rate by Nslot/(Nc + Ng) within the fixed frame, while the same-power comparison assigns an ≈8.5 dB penalty for Nc = 4 and Ng = 3.The passage identifies that shift as an artifact of the fixed-frame comparison, not a claim about other normalizations.Source: G. Spectral-efficiency trade-off · p. 7

H. Adaptive coloring and conditioning

Adaptive coloring uses estimated mode coherence to target rich-multipath conditions, while the MMSE combiner’s reported conditioning is limited to the tested angular-spread range. The beam-coupling model and its Gaussian approximation are validated against free-space simulation over the stated operating regimes.

H. Adaptive coloring and conditioning

  • For LoS fractions above ≈0.8, the conventional detector comes within about 15 times of the 2D-Rake, while switching coloring off saves the full spreading penalty η.The transmitter colors when estimated mode coherence is below a threshold and switches coloring off otherwise; the optimal threshold is left to a one-dimensional implementation search.Source: H. Adaptive coloring and conditioning · p. 7
  • The MMSE Gram-matrix condition number κ stays between 2.5 and 2.9 over tested angular spreads, where regularization σ2 nI suffices.The authors do not claim this conditioning result beyond the tested range.Source: H. Adaptive coloring and conditioning · p. 7

B. OAM beam generation and coupling-law validation

  • NMSE of the per-ray coupling law (7) is below 1.2% for β ≤30°, but rises to 1.85 at β = 45°.At β = 30°, full-field propagation differs from the Bessel prediction by less than 0.13 per entry over d = −4, . . . , 4; the 30° boundary is for a single ray, distinct from the operating RMS spread ∆θ = 35°.Source: B. OAM beam generation and… · p. 8, Figure 3 · p. 8

A. Simulation setup

  • The 16-element UCA at z = 30λ validates free-space beam fields, while communication simulations use an 8-element UCA and assume perfect receiver CSI unless stated otherwise.The communication simulation uses the mode-time channel with a Gaussian mode-diffusion kernel and a Rician LoS/non-LoS split.Source: A. Simulation setup · p. 7, B. OAM beam generation and… · p. 8

B. OAM beam generation and coupling-law validation

  • For ∆θ ≥15°, the Gaussian mode-diffusion kernel’s NMSE against the Monte-Carlo angular average is 3.5%–5.5%, with fitted bandwidth within 1.4% at the operating point.At that point, the fitted and reference bandwidths are 1.645 and 1.622, respectively; the model is adopted as a tractable approximation over the operating range.Source: B. OAM beam generation and… · p. 8

C. Mode coherence

The section tracks surviving modal orthogonality, compares receiver BER and required SNR, and distinguishes empirical reliability slopes from asymptotic diversity under the adopted per-tap fading model.

C. Mode coherence

  • Mode coherence ρ falls from effectively infinite at ∆θ = 0 to ≈2.7 at 60°, and is ≈4.4 at the operating point ∆θ = 35°.The monotonic decline quantifies degradation, but not destruction, of modal orthogonality.Source: C. Mode coherence · p. 8

D. BER performance

  • The 2D-Rake stays ahead of the uncolored modal-time MMSE equalizer at every SNR, by up to two orders of magnitude at 4–8 dB.At ∆θ = 35°, L = K = Nc = 4 with perfect CSI, it also beats delay-collapsed mode-domain ZF/MMSE by 6–12 dB.Source: D. BER performance · pp. 8–9, Figure 5 · p. 9

E. SNR gain at target BER

  • At BER 5×10^-2, the 2D-Rake needs ≈0 dB, versus ≈10 dB for mode-domain MMSE/ZF and 15 dB for mode-wise detection.At BER 5×10^-3, mode-wise detection cannot reach the target at finite SNR, while the 2D-Rake reaches it at ≈2 dB.Source: Figure 6 · p. 9, E. SNR gain at target BER · p. 9

F. Diversity order under block fading

  • The 2D-Rake’s empirical BER slope is ≈3.4, but the adopted per-tap channel has asymptotic order K, not LK.Each tap’s mode pairs share one fading coefficient; the LK-order result applies to independent-branch fading, while finite-sample resolution also limits slope estimation.Source: F. Diversity order under block fading · pp. 9–10

G. Parameter analysis

The section tests parameter sensitivity, when modal leakage supplies independent diversity, and robustness to channel-estimation and timing errors. It finds that harvested energy and reliability depend on channel rank and operating assumptions.

G. Parameter analysis

  • The 2D-Rake BER falls from 3.1 × 10^-3 at 5° to 2.5 × 10^-4 at 60°, while the mode-wise and 1D-Rake baselines degrade.Under the power-normalized mode-diffusion model, wider angular spread redistributes energy into separable cross-mode branches; this benefit is limited to the tested range.Source: G. Parameter analysis · p. 10
  • The 2D-Rake holds at ≈10^-3 or below, while the mode-wise detector degrades from 0.08 (L = 2) to 0.24 (L = 6) as more modes inject more IMI.Increasing Nc from 4 to 16 lowers BER to below 10^-4, at the cost of a larger spreading penalty. As the LoS fraction rises from 0.1 to 0.8, the 2D-Rake degrades mildly from below finite-sample resolution to ≈10^-2.Source: G. Parameter analysis · p. 10

H. When does IMI become diversity: energy versus rank

  • For per-tap fading, target-mode effective rank stays at 1 across angular spread, whereas it grows from 1 to ≈2.9 for independent-branch fading.The 2D-Rake BER improves across the tested range even at low rank, because leaked energy can be harvested without creating independent modal diversity.Source: H. When does IMI become diversity:… · pp. 10–11

I. Imperfect CSI

  • With up to 40% relative channel-estimation error, BER rises from 10^-3 to 4.7 × 10^-3; at data Eb/N0 = 8 dB, pilot SNR ≈20 dB matches perfect-CSI performance.At pilot SNR 10 dB, BER is 6.1 × 10^-2; below 0 dB, performance is interference-dominated.Source: §I · p. 11, Figure 12 · p. 11

J. Timing-offset robustness

  • At a fractional timing offset of 0.1Tc, BER rises from 9.0×10^-4 to 2.5 × 10^-3; timing synchronization is a structural requirement of TDOCC.The receiver is unaware of the offset and uses the nominal integer-delay model; degradation becomes substantial at larger offsets.Source: J. Timing-offset robustness · p. 11, Figure 13 · p. 11

K. Fair comparison at equal energy per information bit

At equal energy per information bit, the 2D-Rake retains a BER advantage over conventional receivers, including uncolored modal-time MMSE. Spreading makes the 10^-3 target reachable at a quantified spectral-efficiency cost, while the proposal combines modal leakage recovery with prior receiver techniques and extensions.

K. Fair comparison at equal energy per information bit

  • At equal energy per bit, the 2D-Rake stays ahead of uncolored modal-time MMSE at every normalized Eb/N0 and beats the mode-wise and delay-collapsed mode-domain MMSE detectors.At the 10^-3 crossing, the uncolored equalizer requires ≈12.7 dB normalized Eb/N0; the 2D-Rake is ahead by about 1.6 times there and by up to an order of magnitude where colored structure matters most. The mode-domain MMSE comparison is ≈12 dB for 2 × 10^-2 versus ≈8.5 dB normalized, about 3.5 dB better.Source: K. Fair comparison at equal energy per… · pp. 11–12

L. Spectral-efficiency trade-off

  • At Nc = 8, required normalized Eb/N0 is ≈11.6 dB, while spectral efficiency falls from 1.14 bit/s/Hz at Nc = 4 to 0.42 bit/s/Hz at Nc = 16, versus 8 bit/s/Hz unspread QPSK.The 10^-3 target remains unreachable for the DFT mode-wise detector, whose floor is ≈1.8×10^-2; longer spreading improves processing gain before overhead yields a sweet spot.Source: L. Spectral-efficiency trade-off · p. 12, Figure 15 · p. 12

M. Comparison with prior art and discussion

  • The proposal’s distinct combination uses per-mode time-domain orthogonal spreading to separate modes at zero delay and a joint mode×delay Rake to recover some multipath-induced leakage under sufficiently separable conditions.The paper describes the ingredients as having precedents, while distinguishing their combined use.Source: M. Comparison with prior art and… · p. 12
  • At 0 dB, Walsh–Hadamard and Zadoff–Chu coloring yield essentially identical 2D-Rake BER, 2.15 × 10−2 versus 1.92×10−2; code family matters only for MRC.Source: M. Comparison with prior art and… · p. 12
  • Transmit-side beam shaping and the receiver are described as complementary; a user-specific scrambling tier could extend the design to a user×mode×delay finger grid.Source: M. Comparison with prior art and… · p. 12

VI. CONCLUSIONS

  • The conclusion reports that the 2D-Rake outperforms mode-wise, mode-domain ZF/MMSE, uncolored modal-time MMSE, and time-only Rake processing across the parameter space, with robustness to channel-estimation error and fractional timing offsets.It notes the Gaussian mode-diffusion model is not an exact law and calls for broader geometry, Rician-factor, and code-family sweeps plus validation against measured or ray-traced channels.Source: §VI · p. 12
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