PaperlayerView exact revision on arXiv

Source-linked AI summary

Spatial-Code-Domain Grouped Index Modulation: Fluid-Antenna-Assisted System Design and BER Performance Analysis

Peng Zhang, Jian Dang, Yao Ge, Miaowen Wen, Ziyang Liu, Liang Wu, Zaichen Zhang, Yudong Yao

arXiv:2609.07087v1cs.ITmath.IT

TL;DR

Joint mapping and BER analysis across FA port, code, and symbol indices remain limited in prior FA-assisted IM work. This paper introduces grouped code-domain modulation and its spatial-code extension, with corresponding detectors and BER analyses. The schemes show BER and throughput gains over considered IM benchmarks, and CD-IMPD achieves near-ML BER performance with lower complexity.

  • Problem

    Prior FA-assisted IM and error analyses give limited attention to joint Tx-port, code-index, constellation-symbol, and correlated Rx-FA selection decisions.

    Source: C. Motivation and Contributions · p. 2

  • Method

    The paper develops CGIM with grouped orthogonal-code mapping and extends it to SCGIM by associating interleaved Tx-FA port groups with code groups, alongside detectors and BER analyses.

    Source: C. Motivation and Contributions · p. 2

  • Results

    CGIM gains over QCIM are 3.37, 3.77, and 4.07 dB across the three channel types, while the schemes show BER and throughput gains over considered IM benchmarks.

    Source: A. BER Performance of CGIM and… · p. 10, §VIII · p. 12

  • Takeaways & Limitations

    The proposed schemes combine Tx-FA index transmission with Rx-FA selection diversity, and CD-IMPD achieves near-ML performance with substantially lower complexity.

    Source: §VIII · p. 12

Abstract

from arXiv · show

Fluid antenna systems (FASs) provide reconfigurable spatial resources within compact apertures. In this paper, we introduce code-domain grouped index modulation (CGIM) and its spatial-code-domain extension, termed SCGIM, for FA-assisted transceivers. CGIM partitions the available orthogonal spreading codes into multiple subsets and jointly maps information onto their in-phase and quadrature indices and constellation symbols. In an Rx-FAS-assisted single-input multiple-output (SIMO) link, group-wise despreading separates the orthogonal code groups for parallel detection, while receive-port selection provides spatial diversity. SCGIM further associates interleaved Tx-FA port subsets with the code subsets, with the Tx-FAS conveying spatial-index information and the Rx-FAS providing selection diversity in a multiple-input multiple-output (MIMO) link. For SCGIM, we develop maximum-likelihood (ML), staged greedy (GD), and cross-domain index message-passing (CD-IMPD) detectors. CD-IMPD exchanges soft information over a cycle-free factor graph to account for the coupling between the spatial and code indices, requiring only one inward and one outward message pass. For CGIM, the BER is derived from the joint decision regions of the despread-domain observations and averaged over the Rx-FAS selected-gain distribution under Rayleigh, Nakagami-m, and additive white Gaussian noise channels. For SCGIM, an average-BER approximation is derived from a full-pair union bound using the selected-gain density ratio and exponentially tilted quadratic-form Laplace transforms. Simulation results validate the BER analysis and show that the proposed schemes achieve lower BER and higher throughput than the considered IM schemes, while CD-IMPD achieves near-ML BER performance with lower detection complexity.

I. INTRODUCTION

The paper addresses limited joint treatment of FA port, code, and symbol indices, and develops grouped code-domain and spatial-code-domain modulation with associated detection and BER analyses. Its channel setup distinguishes receive-side selection from transmit-side spatial-index signaling.

B. Related Work

  • Existing code-index schemes lack grouped orthogonal-code branches, while FA-assisted IM and BER studies give limited attention to jointly mapping and detecting Tx-port and code indices with correlated Rx-FA selection.The required analysis also includes constellation symbols, Rx-port selection statistics, and bit-label weights.Source: B. Related Work · pp. 1–2, C. Motivation and Contributions · p. 2

C. Motivation and Contributions

  • CGIM partitions orthogonal spreading codes into groups and jointly maps in-phase and quadrature code indices with constellation symbols for parallel groupwise detection.The additional code-index dimensions can increase spectral efficiency at fixed spreading length and QAM order.Source: C. Motivation and Contributions · p. 2
  • SCGIM associates interleaved Tx-FA port groups with orthogonal code groups, and CD-IMPD preserves their coupling through soft-information exchange on a cycle-free factor graph.The paper also develops ML and staged GD detectors; one inward and one outward message pass yield exact variable-node marginals under the factorized likelihood model.Source: C. Motivation and Contributions · p. 2
  • The paper derives CGIM BER expressions for Rayleigh, Nakagami-m, and AWGN channels and an SCGIM average-BER approximation using a full-pair union bound and selected-gain density ratio.The SCGIM analysis uses exponentially tilted quadratic-form Laplace transforms.Source: C. Motivation and Contributions · p. 2

II. FLUID ANTENNA SYSTEM AND CHANNEL MODEL

  • The Rx-FAS selects one port per receive-port group, whose candidates are correlated within groups and modeled as independent across sufficiently separated groups.CGIM uses a fixed transmit antenna, whereas SCGIM uses a Tx-FAS with candidate ports and multiple RF chains; the model also specifies transmit-side spatial correlation.Source: §II · p. 3

III. CGIM TRANSCEIVER DESIGN

CGIM divides orthogonal Walsh spreading codes into groups and maps code indices and modulation symbols to each group. The group signals are superimposed for transmission.

A. Code-Domain Grouping and Index Mapping

  • A length-L orthogonal Walsh-code bank supplies F = GP codes divided among G disjoint codebooks of P candidates each.The stated code count satisfies F = GP ≤ L.Source: A. Code-Domain Grouping and Index… · p. 3
  • Each group maps separate in-phase and quadrature code indices and an M-QAM symbol to its transmission tuple.The I- and Q-components select codes independently, and their indices may coincide.Source: A. Code-Domain Grouping and Index… · pp. 3–4
  • The G group signals are superimposed into the transmitted chip vector, with normalization keeping average block energy invariant as G changes.GCIM and QCIM correspond to G = 1 and G = 2, respectively, while CGIM generalizes the number of code groups.Source: A. Code-Domain Grouping and Index… · p. 4

B. CGIM Receiver Design

The CGIM receiver despreads each code group and uses orthogonality to separate its detection metric. For square QAM, a greedy procedure is ML-equivalent.

B. CGIM Receiver Design

  • The receiver noise matrix has independent CN(0, N0) entries, with N0 set by block energy, payload bits, and Eb/N0.The stated variance is N0 = Es/(BCGIMγb).Source: B. CGIM Receiver Design · p. 4
  • Groupwise despreading produces an observation vector for each code label, with I- and Q-components confined to their selected code columns.When the two code indices coincide, both components occupy the same despread vector.Source: B. CGIM Receiver Design · p. 4
  • Code orthogonality decouples the group-wise ML metrics, allowing independent minimization over each group's P^2M candidates.The detector jointly recovers two code indices and the QAM label for each group.Source: B. CGIM Receiver Design · p. 4
  • For square M-QAM, the GD detector is ML-equivalent because code orthogonality and Cartesian PAM structure separate the metric across I/Q branches.It channel-matched combines despread vectors, selects indices from matched-filter magnitudes, then slices to nearest PAM levels.Source: B. CGIM Receiver Design · pp. 4–5

IV. SCGIM TRANSCEIVER DESIGN

SCGIM assigns each code-domain group a subset of candidate Tx-FA ports. An interleaved mapping spaces the candidates within each group across the aperture.

A. Interleaved Tx-Port Grouping and Index Mapping

  • Assuming Kt is divisible by G, each RF chain is associated with Kt,g = Kt/G candidate ports and one code-domain group.The groups use disjoint port sets.Source: A. Interleaved Tx-Port Grouping and… · p. 5
  • The k-th candidate in group g maps to physical port πg(k) = g + (k − 1)G, interleaving each group's port subset across the Tx-FAS aperture.Compared with contiguous partitioning, this mapping increases separation between adjacent candidates within a group.Source: A. Interleaved Tx-Port Grouping and… · p. 5
  • SCGIM appends a log2 Kt,g-bit port label to the code-domain tuple, selecting kg and transmitting the code-domain waveform through physical port πg(kg).The active-channel matrix collects the selected channels for the G groups.Source: A. Interleaved Tx-Port Grouping and… · p. 5

B. SCGIM Receiver Design

The receiver uses group-wise despreading to isolate orthogonal code groups while preserving the coupled port, code-index, and QAM-symbol information. The proposed SCGIM detectors use these observations to recover the indices and symbols.

B. SCGIM Receiver Design

  • Group-wise despreading removes inter-group components under exact code orthogonality while retaining the coupled port index, code indices, and QAM symbol.The despread observations provide the basis for group-wise detection.Source: B. SCGIM Receiver Design · p. 5
  • The group-wise ML detector jointly estimates all four indices and evaluates P 2MKt,g candidates in each group.Its inputs are the group-wise despread observations and the corresponding noiseless response.Source: B. SCGIM Receiver Design · p. 5
  • Greedy detection first scores every candidate transmit port using all despread code observations, then obtains the I/Q code indices and PAM amplitudes given the selected port.The latter decisions use the effective channel associated with that port.Source: B. SCGIM Receiver Design · p. 6

C. Cross-Domain Index Message-Passing Detection

CD-IMPD preserves coupling between spatial and code indices by passing soft information across a cycle-free factor graph. The section also compares scheme spectral efficiencies and detector complexity.

C. Cross-Domain Index Message-Passing Detection

  • CD-IMPD exchanges soft information between spatial-index and code-domain nodes, requiring one inward and one outward pass over a cycle-free factor graph.This retains cross-domain coupling that sequential Tx-port and code-index detection may discard.Source: C. Cross-Domain Index Message-Passing… · p. 6

1) Factor-graph construction:

  • The factor graph links I/Q branch states through a shared port variable, with the two despread observations conditionally independent given that port.Each branch factor is its local likelihood for a candidate port and branch state.Source: 1) Factor-graph construction · p. 6

2) Cross-domain message updates:

  • The outward pass returns each branch the extrinsic message from the opposite branch rather than slicing the port belief.The port node combines the two incoming messages before information is sent back to the branches.Source: 2) Cross-domain message updates · p. 6

3) Marginal decisions:

A. Spectral Efficiency Analysis

  • Table I compares information-bearing domains and spectral efficiencies across the proposed schemes and representative code- and spatial-domain IM counterparts.The Rx-FAS provides receive-port selection and does not convey additional index bits.Source: A. Spectral Efficiency Analysis · p. 6, Table I · p. 7

B. Detection Complexity Analysis

  • CD-IMPD replaces each P 2M search with two P M branchwise state spaces while preserving spatial-code coupling; SCGIM ML adds a Tx-FA port search factor Kt,g.The complexity accounting uses real multiplications and excludes exponential and logarithmic evaluations for message normalization.Source: B. Detection Complexity Analysis · pp. 6–7

VI. BER PERFORMANCE ANALYSIS

This section derives BER analyses for CGIM and SCGIM, beginning with CGIM’s code-domain decision regions. For CGIM, conditional error probabilities are combined with the Rx-FAS selected-gain distribution to obtain average BER.

A. CGIM Performance Analysis

  • The conditional error probability is analyzed for a fixed combining gain, then averaged over the Rx-FAS selected-gain distribution to obtain CGIM BER.The section also introduces a PEP-based BER analysis for SCGIM.Source: A. CGIM Performance Analysis · p. 7, §VI · p. 7

1) Conditional error probability:

  • By branch and group symmetry, the conditional BER of the complete CGIM block follows from one real-valued branch of an arbitrary group.The branch bit error probability weights decisions by Hamming distance, and N0 is defined through Es, B_CGIM, and γb.Source: A. CGIM Performance Analysis · p. 7, 1) Conditional error probability · p. 8
  • Conditioned on combining gain Ω = ω, the transmitted-code despread observation has mean αi, unselected-code observations have mean zero, and their shared variance is N0/(2ω).These distributions underpin the ML score and decision region for each candidate amplitude.Source: 1) Conditional error probability · p. 7
  • The CGIM conditional BER analysis partitions joint code-index and PAM-amplitude ML decisions, accounting for competition among all P despread code outputs without a pairwise-error approximation.For P > 2, each decision region is a convex polyhedron whose Gaussian probability is evaluated through a maximum-score CDF rather than direct P-dimensional integration.Source: 1) Conditional error probability · p. 7

2) Rx-FAS selection statistics:

The Rx-FAS selection statistics characterize selected gains under correlated Rayleigh and integer-m Nakagami-m fading. Convolving the receive-group gain densities supplies the total selected-gain density used to average CGIM BER.

2) Rx-FAS selection statistics:

  • For correlated Nakagami-m fading with positive integer m, the selected-gain analysis uses a Gamma-distributed scaled reference-port power, and m = 1 recovers the Rayleigh result.The Rayleigh and Nakagami-m statistics are derived by conditioning on the reference port.Source: 2) Rx-FAS selection statistics · p. 8

3) Average BER:

  • The total Rx-FAS gain density is the convolution of the independent receive-port-group gain densities, and combining it with the conditional BER kernel gives CGIM BER.This averaging separates code-domain decision geometry from channel dependence and requires no channel-realization sampling.Source: 3) Average BER · p. 8

P CGIM

The CGIM analysis separates code-domain decision geometry from channel dependence, while the SCGIM analysis uses pairwise error probabilities to handle coupling between spatial and code indices. In normalized AWGN, FA-Rx and FPA-Rx have identical performance under matched RF-chain count and power normalization.

P CGIM

  • The CGIM result separates code-domain decision geometry, contained in K_CGIM, from channel dependence, represented by f_Ω, without channel-realization sampling.For fixed-position reception, K_r = 1.Source: P CGIM · p. 8

P CGIM,AWGN

  • In normalized AWGN, FA-Rx and FPA-Rx have identical performance under the same RF-chain count and power normalization.The normalized channel has |h_r,q|^2 = 1, giving Ω_r = 1 and Ω = N_r.Source: P CGIM,AWGN · p. 8, 4) AWGN special case · p. 8

B. SCGIM Performance Analysis

1) Conditional PEP:

  • The SCGIM pairwise distance and Rx-FAS selection statistic remain two quadratic forms of the same Gaussian vector, rather than independent gain and direction variables.The one-branch transform incorporates the selection-induced energy-density ratio, and exponential tilting supplies a base quadratic-form transform with a moment-matched Gamma approximation.Source: 1) Conditional PEP · p. 9
  • Conditional-PEP expectations are evaluated by deterministic quadrature without generating channel realizations; for K_r = 1, the expression reduces to the exact FPA result.Source: 1) Conditional PEP · p. 9

3) Average PEP and BER:

The average-PEP and BER procedure combines independent receive-port transforms to obtain exact UPEP, then uses it in a classical full-pair union bound. A numerical approximation is also introduced for evaluation.

3) Average PEP and BER:

  • Since the N_r receive-port sets are mutually independent, their combined transform is the one-branch transform raised to N_r, and the exact UPEP is an expectation of Q.Source: 3) Average PEP and BER · p. 9
  • A classical approximation to Q(x) is applied for numerical evaluation.Source: 3) Average PEP and BER · p. 9
  • The exact UPEP is substituted into the classical full-pair union bound to obtain the average BER expression.The bound uses the bit label of candidate v.Source: 3) Average PEP and BER · p. 9

P SCGIM

Simulations validate the CGIM analysis and compare CGIM and SCGIM across channel conditions, index-modulation benchmarks, port groupings, and detector tradeoffs. SCGIM and interleaved grouping improve BER, while CD-IMPD approaches ML performance at lower complexity.

P SCGIM

  • For SCGIM, replacing PU(v → bv) in (54) with (53) gives the tractable approximation plotted in the work.The simulations are described as validating the analytical results.Source: P SCGIM · p. 9, §VII · p. 9

A. BER Performance of CGIM and Code-Domain IM Scheme Benchmarks

B. SCGIM Versus Code-, Spatial-, and Joint-Domain IM Scheme Benchmarks

C. Impact of Tx-FA Port Grouping on SCGIM

  • At BER 10^-4, interleaved Tx-port grouping gains approximately 1.26, 2.48, and 2.98 dB for Nr = 2, 4, and 6, respectively.Interleaving increases within-group port spacing, reducing spatial correlation and enlarging the decision distance between Tx-index states.Source: C. Impact of Tx-FA Port Grouping on… · p. 11, Figure 8 · p. 12

D. Detection Performance and Complexity Comparison

  • At BER 10^-4, CD-IMPD and ML differ by less than 0.01 dB, while SCGIM-GD incurs a 2.29-dB loss relative to ML.CD-IMPD preserves coupling between Tx-port and spreading-code indices through soft message exchange.Source: D. Detection Performance and… · p. 11
  • For SCGIM, CD-IMPD uses 37.9% of ML's theoretical RMs and 5.14% of its CPU time, while requiring 1.98 times the RMs and 2.56 times the CPU time of GD.The reported overall complexity ordering is GD lowest, ML highest, and CD-IMPD between them.Source: D. Detection Performance and… · p. 12

VIII. CONCLUSION

  • The conclusion reports that numerical results confirm the analysis and show BER and throughput gains over representative IM benchmarks, with interleaving mitigating spatial correlation.It also identifies CD-IMPD as achieving near-ML performance with substantially lower complexity.Source: §VIII · p. 12
Loading 2609.07087v1…