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On the Performance of Visible Light Communications Systems with Non-Orthogonal Multiple Access

Hanaa Marshoud, Paschalis C. Sofotasios, Sami Muhaidat, George K. Karagiannidis

arXiv:1612.01114v1cs.IT

TL;DR

The paper studies how channel-state uncertainty affects NOMA-based visible-light communication systems. It derives BER expressions and bounds for perfect, noisy, and outdated CSI, finding that noisy CSI causes slight degradation whereas outdated CSI can be detrimental when mobility changes user-channel ordering.

  • Problem

    The paper examines the limited understanding of NOMA-VLC performance under noisy and outdated channel-state information.

  • Method

    The paper derives a closed-form BER expression, an approximation for noisy CSI, and an upper bound for outdated CSI, validated against Monte Carlo simulations.

  • Results

    Outdated CSI causes dramatic degradation for users with poor channel gains when mobility changes the ordering of users’ channel gains.

  • Takeaways & Limitations

    NOMA-VLC performance is relatively robust to noisy CSI but highly sensitive to outdated CSI that produces unfair power allocation after channel-order changes.

  • Takeaways & Limitations

    The analysis assumes that user-location changes occur between CSI updates.

Abstract

from arXiv · show

Visible light communications (VLC) have been recently proposed as a promising and efficient solution to indoor ubiquitous broadband connectivity. In this paper, non-orthogonal multiple access, which has been recently proposed as an effective scheme for fifth generation (5G) wireless networks, is considered in the context of VLC systems, under different channel uncertainty models. To this end, we first derive a novel closed-form expression for the bit-error-rate (BER) under perfect channel state information (CSI). Capitalizing on this, we quantify the effect of noisy and outdated CSI by deriving a simple approximated expression for the former and a tight upper bound for the latter. The offered results are corroborated by respective results from extensive Monte Carlo simulations and are used to provide useful insights on the effect of imperfect CSI knowledge on the system performance. It was shown that, while noisy CSI leads to slight degradation in the BER performance, outdated CSI can cause detrimental performance degradation if the order of the users' channel gains change as a result of mobility

I. INTRODUCTION · A. Related Literature

The introduction presents VLC as an LED-based indoor broadband technology and reviews multiple-access approaches, emphasizing NOMA’s spectrum efficiency alongside practical challenges from channel uncertainty and dimming. The paper therefore quantifies NOMA-VLC performance under noisy and outdated CSI with analog and digital dimming.

  • I. INTRODUCTION: VLC has emerged as a potential complement to RF communications because growing wireless data demand requires increased connectivity.The introduction frames VLC as a response to the wireless data explosion and a promising indoor broadband technology.
  • I. INTRODUCTION: VLC uses high-rate LED intensity modulation and photodetection to support wireless data transmission while remaining imperceptible to human vision.The transmitter modulates LED light intensity, and a photodetector converts received variations into electrical current for data recovery.
  • A. Related Literature: Existing VLC multiple-access schemes face distinct tradeoffs: CSMA incurs sensing and signalling burdens, OFDMA requires adaptations that reduce spectral efficiency, and OCDMA needs long codes.CSMA can suffer hidden-terminal degradation, while VLC constraints require positive real signals and illumination-compatible OFDMA designs.
  • A. Related Literature: NOMA superposes users in the power domain, assigning more power to weaker channels so users share the full time-frequency resource through SIC.Prior VLC and RF studies associate NOMA with increased spectral efficiency, capacity, fairness, or data rate, although gains can degrade at low SNR.
  • A. Related Literature: Most optical NOMA studies assume perfect CSI, although practical channel estimation errors can cause decoding errors and degrade overall system performance.The literature identifies noisy and outdated CSI as channel uncertainty models requiring explicit performance analysis.
  • A. Related Literature: NOMA-VLC must also satisfy illumination requirements, including dimming for brightness control, energy savings, aesthetics, and comfort.CCR changes LED forward current, whereas PWM keeps current constant and varies duty cycle; prior work reports higher luminous efficacy for CCR and throughput degradation for PWM.
  • A. Related Literature: Dimming is challenging in NOMA-VLC because dividing LED power among users makes performance highly sensitive to transmit-power reductions.This sensitivity motivates analyzing dimming jointly with channel uncertainty rather than treating illumination control separately.
  • A. Related Literature: The present work quantifies channel-estimation effects in indoor NOMA-VLC under noisy and outdated CSI together with analog and digital dimming.This contribution directly addresses the practical limitations identified in prior optical NOMA research.

B. Contribution

The paper studies NOMA for downlink indoor VLC networks, deriving BER results under perfect and imperfect CSI and analyzing dimming support. It presents these topics as previously uninvestigated in the technical literature.

  • Motivation: NOMA is considered for indoor VLC because it efficiently multiplexes the small number of users served by LED small cells and benefits from typically high SNR.The high-SNR suitability is attributed to strong line-of-sight components and short propagation distances.
  • Novelty: The paper states that these topics had not previously been investigated in the open technical literature, including analogous analyses for conventional radio communications.This novelty claim covers the BER, CSI-uncertainty, and dimming analyses described above.
  • Contribution: The paper derives an exact BER expression for NOMA-VLC systems with an arbitrary number of users under perfect CSI.This establishes the error-rate analysis for the perfect-CSI case.
  • Contribution: It quantifies CSI-error effects using a closed-form BER approximation for noisy CSI and a tight BER upper bound for outdated CSI caused by user mobility.The two uncertainty models are noisy CSI and outdated CSI.
  • Contribution: The study analyzes how dimming support affects NOMA-VLC BER using OOK analog dimming and variable OOK dimming.Both dimming schemes are considered in the BER analysis.

C. Structure · II. CHANNEL AND SYSTEM MODEL · A. The VLC Channel

The paper models a single-LED indoor VLC downlink serving multiple users with OOK modulation and direct detection, then specifies a LOS channel model with optical geometry and receiver noise. Its sections cover system modeling, BER analysis under perfect CSI and CSI errors, numerical evaluation, and conclusions.

  • C. Structure: The paper analyzes BER performance under perfect CSI and two CSI-error cases, followed by numerical results and concluding remarks.
  • II. CHANNEL AND SYSTEM MODEL: The system is a single-LED indoor VLC downlink in which the LED provides illumination and communication to N simultaneous users through a PLC backbone.
  • II. CHANNEL AND SYSTEM MODEL: Each user employs one photodetector for direct detection, and the system uses unipolar OOK modulation because of its popularity in VLC systems.
  • A. The VLC Channel: The channel model assumes line-of-sight communication because indoor VLC multipath delays from reflections and diffuse refractions are typically negligible.
  • A. The VLC Channel: The user-to-LED channel is parameterized by photodetector area, distance, emergence and incidence angles, receiver field of view, optical-filter gain, and concentrator gain.
  • A. The VLC Channel: The optical channel also uses Lambertian radiant intensity, with emission order determined by the transmitter semi-angle at half power.
  • A. The VLC Channel: Receiver-site noise is modeled as zero-mean circularly symmetric Gaussian noise combining shot-noise and thermal-noise variances.
  • A. The VLC Channel: Shot noise arises from photoelectric conversion and depends on electronic charge, responsivity, bandwidth, background current, and the noise bandwidth factor, while thermal noise originates in the transimpedance circuitry.

B. NOMA Transmission

The NOMA-VLC downlink sorts users by ascending channel gains and superimposes their signals in the optical power domain. Successive interference cancellation enables decoding, while power allocation favors users with poorer channels.

  • B. NOMA Transmission: Users are ordered by ascending channel gains, h1 ≤ h2 ≤ · · · ≤ hN, before NOMA transmission.
  • B. NOMA Transmission: The LED superimposes real, non-negative user signals in the optical power domain for downlink transmission.
  • B. NOMA Transmission: Each user applies SIC by decoding and subtracting signals with lower decoding order, while treating higher-order residual interference as noise.
  • B. NOMA Transmission: Power allocation assigns higher transmission power to users with poorer channel gains, reflecting their greater power requirements for reliable decoding.

III. NOMA-VLC WITH PERFECT CSI · IV. NOMA-VLC WITH IMPERFECT CSI

The paper derives a closed-form BER expression for NOMA-VLC with perfect CSI and then models imperfect CSI using noisy and outdated channel uncertainty. It characterizes how CSI errors affect detection and SIC-relevant channel ordering.

  • III. NOMA-VLC WITH PERFECT CSI: Under perfect CSI and ideal OOK timing, the paper derives a closed-form BER expression for NOMA-VLC.The result assumes accurate channel coefficients and ideal time synchronization.
  • III. NOMA-VLC WITH PERFECT CSI: The BER theorem accounts for successive cancellation, where user U_k cancels the first k−1 detected signals before decoding its own.The formulation explicitly conditions the kth-user error probability on preceding SIC detection stages.
  • III. NOMA-VLC WITH PERFECT CSI: The perfect-CSI error expression incorporates residual interference caused by detection errors and possible interference combinations from transmitted OOK vectors.These terms capture error propagation across earlier detection stages.
  • IV. NOMA-VLC WITH IMPERFECT CSI: Imperfect CSI is practically important because channel knowledge determines both receiver recovery and transmitter power allocation according to users’ channel-gain ordering.That ordering enables effective SIC, but perfect CSI is not realistic even indoors.
  • IV. NOMA-VLC WITH IMPERFECT CSI: Indoor VLC channel uncertainty arises from downlink and uplink noise, user mobility, and quantization errors introduced during channel-estimate conversion.The paper focuses on stochastic uncertainty rather than quantization errors beyond its scope.
  • IV. NOMA-VLC WITH IMPERFECT CSI: To quantify imperfect-CSI effects on NOMA-VLC performance, the paper considers two realistic stochastic models: noisy CSI and outdated CSI.These models represent distinct uncertainty mechanisms in channel knowledge.

A. Noisy CSI · B. Outdated CSI

The paper models noisy CSI with Gaussian channel-estimation errors and derives an approximate conditional decoding-error expression. For outdated CSI caused by mobility or shadowing, it models bounded errors and derives a tight upper bound on conditional error probability.

  • A. Noisy CSI: Noisy CSI is modeled through an estimated channel ˆh_k and a zero-mean Gaussian channel-estimation error ϵ_n.The error variance is denoted σ^2_ϵn.
  • A. Noisy CSI: The Gaussian estimation-error model is adopted as a reasonable model for indoor VLC systems.
  • A. Noisy CSI: For noisy CSI, the paper derives an approximation for the conditional error probability of decoding the k-th signal at user U_k.This result is stated in Proposition 1 and conditions on previous detections.
  • B. Outdated CSI: Outdated CSI is attributed to channel variations caused by user mobility and shadowing after the latest channel-estimate update.
  • B. Outdated CSI: Outdated CSI is modeled with a deterministically bounded random error ϵ_o satisfying ϵ_o ≤ E.E represents the error bound when a mobile user moves at maximum velocity between pilot-signal reception and data transmission.
  • B. Outdated CSI: For outdated CSI, the paper derives a tight upper bound on the conditional error probability at user U_k.This result is stated in Proposition 2.

1) Determination of the value of E:

The section derives the CSI error bound E for a horizontally moving user under fixed LED–PD height and vertical alignment. The resulting algebraic expression is tractable to compute and particularly accurate.

  • 1) Determination of the value of E:: The channel-gain model assumes a fixed LED–PD height z and vertical alignment of the LEDs and photodetectors.Under these assumptions, the channel gain h_i is expressed using the simplified distance geometry.
  • 1) Determination of the value of E:: The error bound E is calculated from the user’s horizontal movement between two positions at maximum velocity v.The derivation uses the movement from (x1, y1) to (x2, y2) and the associated distances involving the fixed height z.
  • 1) Determination of the value of E:: The resulting algebraic representation of E is tractable for straightforward computation and particularly accurate.Its accuracy is demonstrated in Fig. 3.

V. NOMA-VLC WITH DIMMING CONTROL

This section analyzes NOMA-VLC under analog and digital dimming control. Analog dimming is simpler but may shift chromaticity, whereas digital dimming avoids shifts at the cost of spectral efficiency and data rate.

  • V. NOMA-VLC WITH DIMMING CONTROL: Analog dimming adjusts LED driving current proportionally to dimming factor γd using unipolar OOK, but may cause chromaticity shifts.Its BER expressions reuse the perfect-, noisy-, and outdated-CSI formulations after changing transmit power.
  • V. NOMA-VLC WITH DIMMING CONTROL: Increasing redundant bits in VOOK improves BER, while digital dimming reduces achievable data rate linearly with codeword length.The BER improvement is attributed to redundant bits, whereas digital dimming’s spectral-efficiency penalty grows with the number of codeword bits.

VI. RESULTS AND DISCUSSION

The results validate the BER analysis for NOMA-VLC and show that performance depends strongly on power allocation and channel-CSI quality. Outdated CSI causes greater degradation than noisy CSI, while dimming produces a BER–throughput trade-off.

  • Power allocation: ρ = 0.3 provides the best average BER because it allocates sufficient power to users with low channel gains, whereas larger ρ harms early SIC stages.As ρ increases, power shifts toward users with good channel conditions, causing errors that propagate through SIC decoding.
  • Perfect CSI: Under perfect CSI, the analytic BER results closely match Monte Carlo simulations, and all users achieve satisfactory BER above 120 dB transmit SNR.The lowest decoding-order user performs best, while BER worsens as decoding order increases; 120 dB transmit SNR corresponds to about 40 dB receive SNR.
  • Channel uncertainty: Noisy CSI produces accurate approximations but disproportionately degrades U1, while its error generally preserves channel-gain ordering and can create an irreducible high-SNR error floor.U3 is least affected by channel uncertainty, whereas U1’s lowest channel gain and early decoding position make noisy CSI more detrimental.
  • Dimming: Analog intensity dimming degrades BER, whereas digital VOOK dimming substantially improves BER near dimming factor 0.5 but reduces achievable throughput.Analog dimming lowers received SNR through reduced transmit power, while VOOK increases the effective received signal component.
  • Channel uncertainty: Outdated CSI causes greater BER degradation than noisy CSI, especially when mobility changes the users’ channel-gain ordering and produces unfair power allocation.Users with poor channel gains suffer dramatically, whereas users receiving erroneously high power can benefit.

VII. CONCLUSIONS

The paper analyzes BER performance in downlink VLC networks using NOMA under perfect and imperfect CSI. It finds that outdated CSI causes substantially greater degradation than noisy CSI, especially when user-channel ordering changes between updates.

  • VII. CONCLUSIONS: The study analyzes BER in a downlink VLC network where NOMA provides multiple access under perfect and imperfect CSI.The analysis covers both perfect and imperfect channel-state information.
  • VII. CONCLUSIONS: Noisy CSI degrades system performance, but less severely than outdated CSI.The comparison is stated qualitatively in the conclusions.
  • VII. CONCLUSIONS: Outdated CSI causes detrimental BER performance loss when user-channel ordering changes between channel updates.This outdated CSI results from user-terminal mobility between CSI updates.
  • VII. CONCLUSIONS: Derived analytic results were validated through extensive comparisons with corresponding Monte Carlo simulations.The conclusions state that these comparisons justified the validity of the analytic results.
  • VII. CONCLUSIONS: The results provide insights expected to support future VLC system design and deployment.The conclusion identifies future design and deployments as the intended practical relevance.

APPENDIX A PROOF OF THEOREM 1 · APPENDIX B PROOF OF PROPOSITION 1

Appendix A derives the BER expressions for ML detection under successive interference cancellation, including residual interference from cancellation errors. Appendix B shows that the exact integral evaluation is infeasible and derives a simple, accurate closed-form approximation instead.

  • APPENDIX A PROOF OF THEOREM 1: Theorem 1 derives ML detection by selecting the candidate vector minimizing Euclidean distance from the received signal.The proof formulates the decoder at user U_k through the ML decision rule and potential received signals.
  • APPENDIX A PROOF OF THEOREM 1: Under successful cancellation of preceding signals, conditional error probabilities for detecting s_m are expressed in closed form using the Gaussian Q-function.The derivation treats both transmitted-symbol cases and rewrites the resulting expressions algebraically.
  • APPENDIX A PROOF OF THEOREM 1: Theorem 1 extends the error analysis to residual interference caused by detection errors during successive cancellation.The total decoding error probability at U_k is obtained by summing conditional error probabilities from previous detections.
  • APPENDIX B PROOF OF PROPOSITION 1: Proposition 1 begins from the ML decision rule at user U_k and represents the conditional error probability using detection errors e_j = ŝ_j − s_j.The resulting expression follows after algebraic manipulation of the error representation.
  • APPENDIX B PROOF OF PROPOSITION 1: An exact closed-form evaluation for Proposition 1 is infeasible because it requires analytically evaluating two integrals unavailable in the technical literature and tables.The appendix therefore replaces exact evaluation with an approximation.
  • APPENDIX B PROOF OF PROPOSITION 1: A relatively simple closed-form approximation is derived by approximating the one-dimensional Gaussian Q-function with fitted parameters a, b, and c.The fitting parameters are selected according to fitting criteria, with values available in.
  • APPENDIX B PROOF OF PROPOSITION 1: The approximation is particularly accurate across all considered scenario values and produces the closed-form expression in (16).The stated accuracy follows from small absolute and relative errors over the full range of x.

APPENDIX C PROOF OF PROPOSITION 2

The appendix proves Proposition 2 by deriving an integral representation under ML detection, converting it to Q-function closed form, and deducing equation (22) after algebraic manipulation.

  • The proof uses ML detection to establish the initial result.
  • The resulting integral representation is converted into an equivalent closed-form expression involving the Q-function.
  • With e_j = ŝ_j − s_j and further algebraic manipulation, equation (22) is deduced, completing the proof.
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