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Fluid Antenna Multiple Access for Noise Modulation

Hadi Zayyani, Felipe A. P. de Figueiredo

arXiv:2608.30104v1cs.IT

TL;DR

The paper asks how FAMA can manage same-dimension co-channel interference in two-level NoiseMod, where shared random interferer bits alter the variance statistic used for detection. It derives exact finite-sample interference and port-selection results, analyzes independent and correlated spatial operation, and evaluates load and sensing. The results show increased admissible load with spatial degrees of freedom, while finite-sample probing retains substantial overhead under favorable frozen-state assumptions.

  • Problem

    Two-level NoiseMod co-channel users all modulate variance, so shared interferer bits create bit-state-dependent interference and cross-port dependence that complicate exact FAMA port selection.

  • Method

    The paper combines exact finite-sample ML energy detection with random-bit interference MGFs, conditional order-statistic analysis, a full-pairwise Jakes-correlated integer-μ κ-μ benchmark, conditional Monte Carlo, and a frozen-state sensing diagnostic.

  • Results

    Under the stated benchmark, FAMA increases admissible co-channel load as spatial degrees of freedom grow, while probing more ports improves selection but incurs substantial acquisition overhead.

  • Takeaways & Limitations

    The exact BEP-optimal oracle ranks ports by H_k/(σ_w^2 + J_k), and practical fast NoiseMod-FAMA remains a selection-oriented sensing problem.

  • Takeaways & Limitations

    The sensing diagnostic freezes interferer states, and the load results depend on stated powers, BEP target, synchronous activity, ideal switching, and oracle state knowledge.

Abstract

from arXiv · show

Noise modulation (NoiseMod) encodes information in the variance of a noise-like waveform. Its noncoherent structure enables low-complexity links, but co-channel users are challenging because interference affects the same variance statistic used for detection. We study fluid antenna multiple access (FAMA) for two-level NoiseMod with independent random bits from each interferer, producing two possible nonzero interference variances. We adopt exact finite-sample maximum-likelihood energy detection from prior NoiseMod/TNM work. We derive the exact single-port moment-generating function and moments of the random-bit aggregate interference, including heterogeneous user powers, and prove that the port minimizing conditional BEP is $\arg\max_k H_k/(σ_w^2+J_k)$, where $H_k$ is desired-link power and $J_k$ is instantaneous bit-state-aware interference variance. For independent fading ports, the common interferer-bit vector couples per-port decision metrics; conditioning on the number of high-state interferers yields an exact binomial-mixture order-statistic representation and avoids the generally invalid rule $F_{Z_{\max}}=F_Z^{N_p}$. For spatially correlated operation, an integer-$μ$ $κ$-$μ$ benchmark uses full pairwise Jakes correlation in scattered fields, with channel-averaged BEP evaluated by conditional Monte Carlo. A standard-error-aware load study shows increased admissible co-channel users as spatial degrees of freedom grow. Finally, a finite-sample variance-domain sensing diagnostic quantifies the oracle gap under frozen channels and interferer states, providing a lower-bound warning on fast-FAMA sensing difficulty rather than a deployable acquisition protocol.

I. Introduction

The paper addresses same-dimension co-channel interference in two-level NoiseMod using FAMA, where shared interferer bits create port-coupled variance fields. It derives exact port selection and interference statistics, evaluates load scaling, and diagnoses the gap between oracle selection and practical sensing.

  • Problem: Two-level NoiseMod interferers are always present, but their random bits change the interference variance seen across spatially varying ports.The common interferer-bit vector couples per-port decision metrics even under independent fading.
  • Exact selection: The exact finite-sample BEP-minimizing port is k⋆ = arg max_k H_k/(σ_w^2 + J_k), rather than an interference-only SIR heuristic.The result follows from strict monotonicity of conditional BEP in the variance ratio.
  • Independent-port analysis: Conditioning on common interferer-bit states yields an exact order-statistic reduction and a binomial mixture for identical interferers.This avoids the generally invalid unconditional independence rule for complete per-port metrics.
  • Load evaluation: The study evaluates a load metric under full-pairwise Jakes-correlated integer-μ κ-μ fading against SIR-FAMA, max-H FAS, min-interference, and fixed-port baselines.A near–far stress test is also included.
  • Practicality: A frozen-state finite-sample probing diagnostic quantifies the oracle-to-sensing gap but is not a symbol-by-symbol fast-FAMA reconstruction protocol.The deliberately favorable setup exposes sensing overhead before fast interferer-state variation is introduced.

II. Signal and Spatial Channel Model

The signal model contains one desired NoiseMod transmitter and multiple co-channel interferers transmitting independent Gaussian samples per bit. Two-level interferer variances are held constant over each observation block, while channels vary across fluid-antenna ports.

  • Signal model: The model considers one desired transmitter and N_I co-channel NoiseMod interferers, each sending N_s independent complex Gaussian samples per bit.The received waveform is therefore modeled at the sample level within each bit.
  • Interferer states: Each interferer uses equiprobable bits and two nonzero variance levels with common ratio α > 1, parameterized by its average transmitted variance.The parameterization preserves E[P_u,B_u] = P̄_u.
  • Received signal: At port k, the received sample combines the desired waveform, interfering waveforms, and thermal noise through port-dependent channel gains.Desired and interfering channel powers are represented by H_k and G_i,k.
  • Block assumption: Desired and interfering users have independent channel processes, while each interferer bit remains constant across the N_s samples of one desired bit.Asynchronous state changes inside the observation window are outside the model.
  • Conditional distribution: Conditioned on channels and interferer bits, the aggregate received sample is exactly Gaussian, and the same transmitted bit vector enters the variance at every port.This shared state produces cross-port dependence even when fading is independent.

B. Jakes-Correlated Integer-µ κ-µ Benchmark

The paper uses an integer-μ κ-μ benchmark with full pairwise Jakes correlation to model spatially correlated ports, then applies exact finite-sample energy detection and conditional Monte Carlo evaluation.

  • Spatial correlation: The correlated benchmark models an N_p-port linear aperture spanning Wλ with full pairwise scattered-field Jakes correlation.The construction is intended to represent closely spaced port dependence more accurately than a reference-port approximation.
  • Correlation matrix: For N_p = 1, the correlation matrix is R = [1], while multiport links use the specified pairwise correlation matrix.The benchmark is defined consistently for single-port and multiport cases.
  • κ-μ construction: Independent complex Gaussian cluster variables combined with a common dominant-component phase generate κ-μ power marginals with mean power Ω.The same construction is applied independently to each interfering-link power field.
  • Scope: The benchmark restricts μ to positive integers and is presented as a transparent joint model rather than a universal correlated κ-μ process.Its full pairwise Jakes structure targets the correlation-modeling concern motivating the benchmark.
  • Detection: Exact Gamma-likelihood energy detection supplies the threshold and conditional BEP under the complex-Gaussian sample model.The resulting BEP is exact for the finite-sample model and scale-invariant under common variance scaling.

IV. Bit-State-Aware Interference Statistics

The paper derives exact single-port interference statistics for random two-level user states, including heterogeneous powers, and explains why the aggregate generally does not remain a single κ-μ variate.

  • Single-user statistics: The integer-cluster κ-μ construction provides a single-user normalized-power MGF for deriving interference statistics.This MGF is the starting point for the aggregate random-bit interference transform.
  • Aggregate interference: Independent equiprobable interferer bits produce an exact marginal MGF for the aggregate interference.The transform accounts for each user's random state scaling before summation.
  • Heterogeneous powers: Heterogeneous received average powers are supported, with the valid positive-s domain determined by the intersection of component-MGF domains.Laplace-transform checks with s < 0 automatically satisfy the stated domain condition.
  • Distributional scope: Random state scaling generally prevents replacing the aggregate interference by one common-scale κ-μ variate.Closure under noncentral chi-square combinations holds only when common scales are conditioned on; the unconditional random mixture differs.

V. Exact BEP-Optimal Port Selection

The exact finite-sample NoiseMod detector makes conditional BEP decrease with the variance ratio, yielding a noise-aware desired-to-interference port-selection rule. Finite sample size also imposes an intrinsic BEP floor, and the selection result is explicitly oracle-based.

  • Exact selection rule: Conditional BEP is strictly decreasing in ρ = V1/V0, so maximizing the exact port metric minimizes BEP.The theorem applies to equiprobable two-level NoiseMod with positive low- and high-state powers and the finite-Ns ML detector.
  • Exact selection rule: The exact BEP-minimizing port maximizes Hk/(σ_w^2 + Jk), where the denominator includes noise and instantaneous interference variance.The monotonicity proof links the detector’s variance ratio to the port statistic Zk.
  • Relation to SIR: When thermal noise is negligible, the exact rule approaches SIR selection arg maxk Hk/Jk; when interference is weak, omitting σ_w^2 is not generally optimal.Thus the classical SIR rule is a limiting case rather than the general finite-noise optimum.
  • Finite-sample floor: No amount of channel gain or port diversity removes the finite-Ns, finite-α detector floor.For Ns = 120 and α = 10, the floor is 1.97 × 10^-34 and is invisible on plotted scales.
  • Scope: The selection theorem assumes instantaneous knowledge of Hk and Jk at every candidate port, so it serves as an oracle benchmark.A separate finite-sample sensing baseline quantifies the gap between this ideal and practical acquisition.

VI. Independent-Port Theory With Common Interferer

Independent fading ports remain dependent through the common interferer-bit vector. Conditioning on that vector restores conditional independence and gives the exact order-statistic representation, whereas mixing before maximization is generally invalid.

  • Common-bit dependence: Even with independently fading ports, the shared interferer-bit vector makes port metrics independent only conditional on the transmitted bits.The common bit states alter interference at every port while channel powers vary independently.
  • Exact order statistics: Conditioning on a fixed bit vector enables conditional iid order statistics for the maximum port metric.This conditioning is the key step before averaging over the common interferer state.
  • Exact averaging: The exact iid-port average BEP is obtained by integrating the conditional detector BEP against the conditional maximum-metric distribution.The function ψ(z) represents the exact detector BEP after substituting the metric-dependent variances.
  • Ordering of operations: Mixing over common bit states before taking the port maximum generally differs from the correct conditional order-statistic calculation.Therefore the tempting identity F_{Z_max} = F_Z^{N_p} is generally invalid under shared interferer bits.

VII. Correlated Evaluation

The correlated benchmark uses full pairwise Jakes correlation and evaluates channel-averaged BEP by conditional Monte Carlo. It compares noise-aware oracle and alternative port-selection rules under the correlated fading model.

  • Correlated model: A tractable closed-form joint distribution for the correlated port ratios Zk is not assumed.The evaluation instead samples channel and interferer states from the stated correlated model.
  • Evaluation: Conditional Monte Carlo evaluates the exact conditional BEP after selecting a port under each rule.The estimator integrates out received-sample randomness analytically rather than relying on direct error counting.
  • Evaluation: Rao–Blackwellized conditional Monte Carlo resolves small BEPs more efficiently than error counting, with uncertainty reported as the standard error.A separate received-sample simulation validates the detector implementation.
  • Selection baselines: The compared rules are the exact noise-aware oracle, SIR-FAMA, max-H FAS, min-J, and fixed-port selection.Random-port performance is not plotted separately because it has the same ensemble-average marginal performance as a fixed port under the symmetric stationary model.

B. Variance-Domain Partial Probing Under Frozen States

The sensing diagnostic estimates the theorem’s variance-domain metric from a subset of probed ports under deliberately frozen channels and interferer states. It exposes substantial acquisition overhead and is not a deployable fast-FAMA protocol.

  • Frozen-state setup: The experiment probes M ≤ Np uniformly spaced ports while assuming channels and interferer states remain fixed through sensing and the following data bit.This favorable coherence assumption makes the experiment a lower bound on sensing difficulty.
  • Estimator: The procedure is a direct variance-domain estimator of the theorem’s metric, not a channel-reconstruction algorithm.It estimates the desired and interference-related variance quantities only over the probed subset.
  • Calibration: Threshold uncertainty can be addressed by optional calibration using known low- and high-state samples with an ordered two-variance constrained ML fit.Boundary cases are retained and their frequency is recorded rather than silently clipped.
  • Overhead boundary: Changing interferer bits changes Jk, so sensing-state reuse across multiple bits does not automatically amortize acquisition overhead in fast FAMA.The one-bit accounting is deliberately stringent, although reuse would reduce overhead if the state remained valid.

A. Why the Correct Modeling Choices Matter

Exact detector and interference-aware selection materially affect BEP and admissible load, while spatial degrees of freedom increase supported co-channel users under uncertainty-aware admission.

  • Detector and interference modeling: At 10 dB desired average variance, the CLT/harmonic detector overestimates exact BEP by 16.1%, while equal-power interference underestimates it by 31.3%.The exact result is 2.43 × 10−4, compared with 2.82 × 10−4 and 1.67 × 10−4, respectively.
  • Detector and interference modeling: With four weak interferers at −15 dB each, noise-aware selection achieves 8.12 × 10−7 versus 2.44 × 10−5 for SIR-FAMA.At higher interference levels, SIR approaches the noise-aware rule because the system becomes interference limited.
  • Load scaling and uncertainty: The load metric N max I is evaluated with a conservative upper-confidence rule rather than by thresholding Monte Carlo point estimates.The rule retains loads whose estimated BEP plus 1.96 standard errors remains below the target.
  • Load scaling and uncertainty: At target 10−2, admissible load rises from N max I,95 = 2 for one port at W = 2 to 16 for Np = 16 at W = 4.This is an eightfold increase under the stated desired and per-interferer variance normalization.
  • Load scaling and uncertainty: Additional spatial degrees of freedom, rather than port count alone, determine the robust load gain because Jakes correlation can oscillate with port separation.The detailed dependence need not be monotone at every discrete aperture–port-count pair.
  • Common-bit coupling: Conditioning on common interferer-bit states matches direct iid multiport simulation within 1.1%, whereas unconditional order statistics become increasingly optimistic.At Np = 16, the naive rule predicts 4.71 × 10−5 instead of the correct 2.75 × 10−4, an optimism factor of 5.83.

D. Near–Far Robustness

Near–far behavior favors noise-aware or SIR selection over desired-signal-only selection, while sensing and correlated-port analyses define the practical and statistical boundaries of the oracle benchmark.

  • Near–far robustness: As one interferer becomes dominant, noise-aware and SIR rules remain near 1.2–1.8 × 10−3, while max-H FAS degrades to 9.92 × 10−3.The fixed-port baseline remains around 5 × 10−2 across the sweep.
  • Sensing trade-offs: The sensing experiment is a diagnostic lower bound because frozen channels and interferer states extend coherence beyond the synchronous one-bit model.It therefore does not constitute a complete symbol-by-symbol fast-FAMA acquisition protocol.
  • Selection regimes: Figure 2 separates noise-limited and interference-limited selection behavior rather than treating SIR selection as universally exact.Thermal noise causes a large SIR penalty for weak interference, while SIR converges toward noise-aware selection as interference dominates.
  • Sensing trade-offs: All-port probing with Ls = 64 reduces BEP from 8.82 × 10−2 at one probed port to 8.51 × 10−3 at all 16 ports.The same configuration consumes 2048 acquisition samples, leaving only ξ ≈ 5.5% data samples under one-bit accounting.
  • Sensing trade-offs: With Lc = 512 calibration samples per desired state, the plug-in receiver reaches 1.32 × 10−2 but remains above the full oracle and same-selected-port genie detector.The data fraction is again about 5.5%, showing the overhead required to approach oracle behavior even under favorable assumptions.
  • Common-bit coupling: Independent fading ports still have dependent decision metrics when the same interferer bits illuminate every port, requiring conditioning before the order statistic.Mixing first with an unconditional F_Z^Np is overly optimistic.

B. Oracle, Sensing, and Channel Scope

Theorem 1 is an oracle benchmark requiring instantaneous full-port knowledge, while the sensing experiment deliberately freezes interference states and therefore understates acquisition difficulty. The correlated fading benchmark and load metric are scope-limited, but the analysis reports FAMA gains under near–far imbalance and increased admissible load.

  • Sensing scope: The sensing experiment freezes channels and interferer states across sensing and the following bit, making it a diagnostic lower bound rather than an implementable fast-FAMA protocol.Its poor sample efficiency therefore understates the full acquisition challenge.
  • Oracle benchmark: The BEP-optimal oracle metric is H_k/(σ_w^2 + J_k), based on strict conditional BEP decrease with the variance ratio.The result assumes instantaneous knowledge of H_k and J_k at every candidate port.
  • Channel and load scope: Under the stated Jakes-correlated benchmark, FAMA increases admissible co-channel load and remains useful under near–far imbalance.The benchmark uses Jakes-correlated scattered cluster fields, so alternative spatial models can change the numerical curves.
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