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Article

Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems

by
Wenyang Li
,
Zixiong Wang
* and
Jinlong Yu
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
*
Author to whom correspondence should be addressed.
Photonics 2024, 11(12), 1147; https://doi.org/10.3390/photonics11121147
Submission received: 24 October 2024 / Revised: 18 November 2024 / Accepted: 3 December 2024 / Published: 5 December 2024
(This article belongs to the Special Issue New Advances in Optical Wireless Communication)
Browse Figures
Figure 1
<p>Block diagram of RPFRFT-based OFDM system.</p> "> Figure 2
<p>Hammerstein model.</p> "> Figure 3
<p>VLC and VLP link with LOS and NLOS paths.</p> "> Figure 4
<p>Subcarrier allocation in the fractional Fourier domain for 4-LED VLC and VLP systems.</p> "> Figure 5
<p>BER performance comparison at various FRFT orders <span class="html-italic">p</span>.</p> "> Figure 6
<p>BER performance against <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>b</mi> </msub> <mo>/</mo> <msub> <mi>N</mi> <mn>0</mn> </msub> </mrow> </semantics></math> for FRFT-OFDM and FT-OFDM schemes using 4-QAM and 16-QAM.</p> "> Figure 7
<p>Localization errors against <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>b</mi> </msub> <mo>/</mo> <msub> <mi>N</mi> <mn>0</mn> </msub> </mrow> </semantics></math> for FRFT-OFDM and FT-OFDM schemes with 3- or 4-LED positioning.</p> "> Figure 8
<p>Localization errors at various positions throughout the room under the optimal order <span class="html-italic">p</span> of the FRFT.</p> "> Figure 9
<p>Improvement in positioning accuracy at various positions throughout the room under the optimal order <span class="html-italic">p</span> of the FRFT compared to the FT.</p> ">
Versions Notes

Abstract

:
In indoor visible light communication (VLC) and visible light positioning (VLP) systems, the performance of conventional orthogonal frequency-division multiplexing (OFDM) schemes is often compromised due to the nonlinear characteristics and limited modulation bandwidth of light-emitting diodes, the multipath effect in enclosed indoor environments, and the relative positions of transmitters and receivers. This paper proposes an OFDM scheme based on the fractional Fourier transform (FRFT) to address these issues, demonstrating promising results when applied to indoor VLC and VLP systems. The FRFT, a generalization of the conventional Fourier transform (FT) in the fractional domain, captures information in both the time and frequency domains, offering greater flexibility than the FT. In this paper, we first introduce the computation method of the reality-preserving FRFT for an intensity modulation/direct detection VLC system and integrate it with OFDM to optimize system performance. By adopting FRFT-OFDM under the optimal fractional order, we enhance both the bit error ratio (BER) performance and positioning accuracy. Simulation results reveal that the FRFT-OFDM scheme with an optimized fractional order significantly improves the BER and positioning accuracy compared to the FT-OFDM scheme across most receiver positions within the indoor observation plane. For communication, the FRFT-OFDM scheme achieves over 6 dB E b / N 0 gain compared to the FT-OFDM scheme at a BER of 3 × 10 − 4 when the receiver is positioned at most locations in the room. For positioning, the FRFT-OFDM scheme enhances positioning accuracy by more than 1 cm relative to the FT-OFDM scheme at most locations in the room. Notably, both systems maintain the same computational complexity and spectral efficiency.

1. Introduction

The rapid advancement of the information age has significantly increased the demand for high-speed, real-time communication. However, conventional wireless communication technologies are facing a serious bottleneck due to the exhaustion of the available wireless spectrum. As a result, the development of new high-speed communication technologies has become a critical focus for the future of wireless communication. To meet the growing bandwidth demands, particularly for multimedia services, it is becoming essential to expand the wireless spectrum from lower to higher frequencies. This has led many researchers to explore the largely untapped visible light spectrum, which holds great potential for high-speed transmission. Unlike conventional wireless bands, the visible light spectrum is not subject to spectrum regulation, offering additional advantages such as enhanced communication security and freedom from electromagnetic interference. These characteristics suggest that visible light communication (VLC) is a promising solution for a broad range of applications across various environments, including underwater [1], atmospheric [2], and indoor settings [3].
Providing high-speed wireless communication indoors is a primary focus of VLC applications, primarily leveraging indoor light-emitting diode (LED) lighting systems to simultaneously deliver lighting and communication services. Thanks to the high response rate of LED semiconductor materials, intensity modulation/direct detection (IM/DD) of white light signals can be employed to enable wireless communication. Currently, the modulation bandwidth of most commercial LEDs is limited to around 5 MHz [4]. However, with the increasing transmission rate requirements in the era of big data, researchers are exploring ways to enhance this limited bandwidth. Techniques such as pre- and post-equalization [5,6], higher-order quadrature amplitude modulation (QAM) [7], discrete multitone (DMT) modulation [8], wavelength-division multiplexing (WDM) [9], and efficient multiple-input multiple-output (MIMO) [10] have all been investigated to extend the modulation bandwidth and increase transmission rates. Among these methods, orthogonal frequency-division multiplexing (OFDM)-based schemes have gained significant attention [11]. Due to the constraints of IM/DD in optical communication, improved OFDM schemes have been introduced into VLC, including widely used variants such as direct-current-biased optical OFDM (DCO-OFDM), asymmetrically clipped optical OFDM (ACO-OFDM), pulse-amplitude-modulated discrete multitone modulation (PAM-DMT), and unipolar OFDM (U-OFDM) [11,12]. In [13], the authors propose a novel direct time-domain waveform equalization method for indoor VLC using a bidirectional gated recurrent unit (BiGRU) neural network within a DCO-OFDM framework. Experimental results show that the BiGRU-based method offers low complexity and strong nonlinear channel learning capabilities. In [14], a carrierless amplitude-phase (CAP) modulation method is introduced for VLC systems to enhance the performance of MIMO structures, effectively reducing the peak-to-average power ratio (PAPR) of the system. The authors of [15] present an adaptive least squares (LS) channel estimation algorithm for VLC systems using a DCO-OFDM scheme, achieving improved performance in terms of the received constellation diagram, mean square error (MSE), and bit error ratio (BER).
In parallel, traditional Global Positioning System (GPS) technologies suffer from low positioning accuracy in complex indoor environments due to multipath effects. This has spurred interest in LED-based indoor visible light positioning (VLP) systems, especially in spaces equipped with numerous LED lighting devices [16,17]. In [18], the authors introduce a machine learning approach for accurately predicting the two-dimensional indoor position of a mobile receiver using measured received signal strength (RSS) values from four photodiodes (PDs) arranged in a star architecture. In [19], the authors propose a lightweight artificial neural network architecture designed to predict human blockages in hybrid indoor visible light/radio frequency (RF) communication systems. This approach optimizes beamforming matrices and time allocation parameters, achieving a near-optimal uplink sum rate. Consequently, VLP will play a significant role in future wireless networks [20]. And, the integration of OFDM-based schemes into LEDs enables them to provide illumination while supporting both wireless communication and indoor positioning, further demonstrating the potential and importance of VLC and VLP technology.
However, OFDM has several significant drawbacks, such as its sensitivity to the carrier frequency offset (CFO), Doppler shift, phase noise, nonlinear effects, and chromatic dispersion (CD). These factors can introduce inter-carrier interference (ICI), reducing the reliability of system transmission. This leads to an error floor, where increasing the transmitter’s power fails to significantly reduce the BER at the receiver, ultimately degrading system performance. To enhance the interference immunity of OFDM schemes, researchers have extended the concept of the Fourier transform (FT) and proposed the FRFT-OFDM scheme, which combines the fractional Fourier transform (FRFT) with OFDM. Unlike exponential signals, chirp signals offer more flexible and variable modulation frequencies. As a result, the FRFT-OFDM scheme can adapt to different channel conditions by adjusting the tuning frequency specifically by changing the order p of the FRFT. This adaptability enables improved BER performance and positioning accuracy compared to FT-OFDM schemes.
It is worth noting that the computational complexity of the FRFT is comparable to that of the conventional fast Fourier transform (FFT) [21]. As a result, FRFT-OFDM has seen rapid development since FRFT was first introduced into OFDM schemes [22]. In [23], the author analyzed the ICI issue in MIMO-OFDM systems in high-mobility, fast-fading, double-dispersion, and non-stationary environments. By replacing the FFT with the FRFT, the correlation coefficient was increased, the effects of ICI were mitigated, and BER performance was improved. In [24], a novel orthogonal time–frequency space (OTFS) modulation technique based on FRFT-OFDM was proposed, demonstrating superior resilience to the severe Doppler effect in double-dispersion channels compared to conventional OTFS systems. Furthermore, a multi-block sparse Bayesian learning channel estimation method for hydroacoustic communication based on FRFT-OFDM was introduced in [25]. In this case, the FRFT effectively compensated for the Doppler shift, resulting in enhanced channel estimation performance and an improved signal-to-noise ratio (SNR).
In conclusion, FRFT-OFDM schemes have been widely applied in various fields, including RF communication [26], fiber optics [21,27], and hydroacoustic communication systems [5]. LEDs are essential components in indoor VLC and VLP systems, but their limited modulation bandwidth and nonlinear power–current characteristics can lead to substantial performance degradation, creating bottlenecks in both the BER and positioning accuracy when using FT-OFDM schemes. The FRFT-OFDM scheme offers enhanced performance in nonlinear or frequency-selective systems, effectively mitigating the LED’s nonlinear effects and limited bandwidth constraints on the BER and positioning accuracy. However, the potential of FRFT-OFDM for VLC and VLP applications remains largely unexplored.
Motivated by the above, we propose an indoor VLC and VLP system based on the FRFT-OFDM scheme. The main contributions of this paper are summarized as follows:
  • To ensure the real-valued signal in VLC and VLP systems, we derive a reality-preserving FRFT (RPFRFT) method and implement it within these systems. The architecture of the proposed VLC and VLP systems based on the FRFT-OFDM scheme is described in detail, along with the derivation of relevant mathematical formulas.
  • We simulate and validate the proposed VLC and VLP system architectures based on the FRFT-OFDM scheme. The results demonstrate the significant performance improvement of the FRFT-OFDM scheme over the FT-OFDM scheme. For communication, the FRFT-OFDM scheme achieves over 6-dB E b / N 0 gain compared to the FT-OFDM scheme at a BER of 3 × 10 − 4 when the receiver is positioned at (1, 1, 0.8). For positioning, the FRFT-OFDM scheme enhances positioning accuracy by more than 1 cm relative to the FT-OFDM scheme at most locations within the room.
The remainder of this paper is organized as follows. Section 2 outlines the basic algorithm of the FRFT, details the system architecture of the FRFT-OFDM scheme, and presents the relevant mathematical formulations. Section 3 analyzes and discusses the simulation results. Finally, the conclusions are summarized in Section 4.

2. Principle

2.1. Fractional Fourier Transform

The FRFT is a generalized form of the FT in the fractional domain. If the FT is considered a 90 ° counterclockwise rotation from the time axis to the frequency axis, the FRFT can be viewed as a rotation of the signal by an angle α = p π / 2 counterclockwise on the time axis [21], where p represents the order of the FRFT.
There are several ways to define the FRFT, each offering a different physical interpretation [28]. The fundamental definition of the FRFT, from the perspective of a linear integral transform, is as follows:
X p ( u ) = { F p [ x ( t ) ] } ( u ) = ∫ − ∞ ∞ K p ( u , t ) x ( t ) d t
where F p denotes the FRFT operator, and K p ( u , t ) , the kernel of the FRFT, is defined as
K p ( u , t ) = 1 − j cot α exp [ j π ( u 2 cot α − 2 u t csc α + t 2 cot α ) ] , α ≠ n π δ ( u − t ) , α = 2 n π δ ( u + t ) , α = ( 2 n + 1 ) π
Note that when p = 1 , the FRFT reduces to the FT [29].
Equation (1) can be rewritten as
X p ( u ) = 1 − j cot α 2 π ∫ − ∞ ∞ exp ( j u 2 + t 2 2 cot α − j u t csc α ) x ( t ) d t , α ≠ n π x ( t ) , α = 2 n π x ( − t ) , α = ( 2 n + 1 ) π
Additionally, the inverse FRFT (IFRFT) of order p is equivalent to the FRFT of order − p and is expressed as
x ( t ) = ∫ − ∞ ∞ X p ( u ) K − p ( u , t ) d u
In recent years, researchers have proposed various discrete algorithms for the FRFT. The current discrete fractional Fourier transform (DFRFT) algorithms can be categorized into three main types: sampling-type DFRFT [30,31], eigenvector decomposition-type DFRFT [32,33], and linear combination-type DFRFT [34].
In this thesis, the sampling-type discretization algorithm proposed by Soo-Chang Pei is utilized [31]. This algorithm has computational complexity comparable to that of the FT, with the computation for N point data being O ( N log 2 N ) . Additionally, it ensures the validity of the transform kernel matrix, which strictly satisfies invertibility. When applying this algorithm for modulation and demodulation at the transmitter and receiver of the FRFT-OFDM scheme, no errors will arise from the algorithm itself.
By applying this algorithm, the DFRFT can be expressed in matrix form as
X p = F p x
where x = [ x ( 0 ) , x ( 1 ) , ⋯ , x ( N − 1 ) ] T and X p = [ X p ( 0 ) , X p ( 1 ) , ⋯ , X p ( N − 1 ) ] T are the vectors before and after passing through the pth-order FRFT, respectively. F p is the N × N DFRFT matrix, with its components defined as
F p ( m , n ) = A p ρ 1 m 2 W m n ρ 2 n 2 , p ≠ 2 n I N × N , p = 4 n J N × N , p = 4 n + 2
where ρ 1 m 2 = exp ( j 0.5 Δ u 2 m 2 cot α ) , ρ 2 n 2 = exp ( j 0.5 Δ t 2 n 2 cot α ) , A p = ( sin α − j cos α ) / N , W = exp ( − j 2 π / N ) , and m , n = 0 , 1 , ⋯ , N − 1 . Here, Δ t represents the sampling interval in the time domain, and Δ u denotes the sampling interval in the pth-order fractional Fourier domain, satisfying the relationship Δ u = 2 π | sin α | / ( N Δ t ) . Additionally, I N × N and J N × N refer to the identity matrix and the exchange matrix of size N × N , respectively.
The inverse DFRFT (IDFRFT) can be expressed in matrix form as
x = F − p X p
where F − p = F p H , and
F − p ( n , m ) = ρ 2 − n 2 W − m n ρ 1 − m 2 A p * , p ≠ 2 n I N × N , p = 4 n J N × N , p = 4 n + 2
In IM/DD VLC systems, it is essential for OFDM signals to be positive real numbers. To achieve this, we modify the structure of the FRFT algorithm to ensure that the output signals are real and implement unipolarity by adding a DC bias to the transmitter. In FT-OFDM schemes, Hermitian conjugate symmetry is often applied to signals before the FT to guarantee that the transformed output signals remain real numbers. However, since Hermitian conjugate symmetry is specific to the FT, a new form of the FRFT is required. In [35], a method was proposed to derive a discrete fractional cosine (sine) transform variant that maintains real-valuedness and decentralization, along with most of the properties needed for FRFT matrices. We have adopted the RPFRFT method introduced in [35,36] to meet the signal requirements in our FRFT-OFDM scheme.
Let F p be a complex-valued DFRFT matrix with size N / 2 , where N is even, and let x = [ x 1 , x 2 , ⋯ , x N ] T be a real signal of length N. We construct a complex vector x ^ = [ x 1 + j x N / 2 + 1 , x 2 + j x N / 2 + 2 , ⋯ , x N / 2 + j x N ] T of length N / 2 from x [37]. By performing a DFRFT on x ^ , we obtain a complex vector,
y ^ = F p x ^ = [ Re ( F p ) + j Im ( F p ) ] · [ Re ( x ^ ) + j Im ( x ^ ) ] = [ Re ( F p ) · Re ( x ^ ) − Im ( F p ) · Im ( x ^ ) ] + j [ Re ( F p ) · Im ( x ^ ) + Im ( F p ) · Re ( x ^ ) ]
Next, we separate the real and imaginary parts of the complex vector y ^ .
y = Re ( F p ) · Re ( x ^ ) − Im ( F p ) · Im ( x ^ ) Re ( F p ) · Im ( x ^ ) + Im ( F p ) · Re ( x ^ ) = Re ( F p ) − Im ( F p ) Im ( F p ) Re ( F p ) · Re ( x ^ ) Im ( x ^ ) = R p x
R p is the RPFRFT matrix given by
R p = Re ( F p ) − Im ( F p ) Im ( F p ) Re ( F p )
And, the inverse RPFRFT (RPIFRFT) matrix can be expressed as
R − p = Re ( F − p ) − Im ( F − p ) Im ( F − p ) Re ( F − p )

2.2. System Model

We have designed an indoor VLC and VLP system based on FRFT-OFDM, as shown in Figure 1. The input serial bit streams are processed using QAM to produce the output X [ u ] . To ensure that the transformed output signal is real, we apply the RPFRFT method described in Section 2.1 to separate the real and imaginary parts of X [ u ] . Subsequently, we perform serial-to-parallel (S/P) conversion on the real and imaginary parts independently. At this stage, the real part of X [ u ] corresponds to subcarriers 1 , ⋯ , N / 2 , while the imaginary part corresponds to subcarriers N / 2 + 1 , ⋯ , N . This can be represented as X c [ u ] = [ Re ( X [ u ] ) , Im ( X [ u ] ) ] T . The separated signals undergo the RPFRFT, followed by a parallel-to-serial (P/S) conversion to produce x c [ n ] . The real-valued output x c [ n ] obtained at this stage can be expressed as
x c [ n ] = R − p X c [ u ]
Due to the chirp periodicity of the signal, a chirp cyclic prefix (CP) is required at the transmitter instead of the conventional CP. By selecting the optimal FRFT order, the chirp CP effectively mitigates inter-symbol interference (ISI). However, the chirp CP has a significant drawback: it necessitates a feedback channel between the transmitter and receiver to relay the optimal FRFT order, which complicates the overall system. To simplify the system, we have adopted the approach from [38] and utilized zero padding (ZP) in place of the chirp CP. This adjustment results in an output x z p [ n ] that incorporates ZP while simultaneously reducing the transmit power required by the system.
Due to the limited dynamic range of LEDs, it is necessary to scale the signal x z p [ n ] before adding a DC bias to ensure the signal falls within an appropriate range. We have
x i n [ n ] = ζ I m a x x z p [ n ] ( ζ + 1 ) · max { | x z p [ n ] | }
where I m a x is the maximum current desired after scaling and biasing, and ζ is the modulation index, which is given as
ζ = I m a x − I DC I DC
where I DC represents the DC-bias current.
Various studies have attempted to model LED nonlinearities, employing approaches such as the Volterra series [39] and the Wiener [40], Hammerstein [41], and Wiener– Hammerstein [42,43] models. The authors of [41] surveyed nonlinear effects in VLC and discussed several classical models based on polynomial expansions. Among these, the Volterra series provides the most comprehensive description of nonlinearities with memory [44]. However, it requires a substantial amount of training data and incurs significant computational complexity. To mitigate these challenges, the Hammerstein and Wiener models have been simplified based on the Volterra series by assuming separable memory effects over nonlinearity [45,46,47].
In [48], the author demonstrated that the Hammerstein model is more effective than the Wiener model for describing LED nonlinearities. We have adopted this model, which consists of a memoryless nonlinear module followed by a linear time-invariant (LTI) system (see Figure 2).
The LED nonlinear module is represented by a third-order polynomial [49], while the LTI module is modeled as a first-order low-pass filter [12]. The output x o u t of the intensity-modulated LED is expressed as
x o u t ( t ) = f NL ( x i n ( t ) ) ⊗ h LED ( t )
where f NL and h LED are given by
f NL ( x ) = 0.15 , x < 0.1 1.2897 x 3 − 3.2738 x 2 + 3.3919 x − 0.1577 , 0.1 ≤ x < 1 1.25 , x ≥ 1
h LED ( t ) = e − 2 π B m t
where B m is the modulation bandwidth, and ⊗ denotes the time-domain convolution.
This paper considers a VLC and VLP system. As illustrated in Figure 3, four LEDs are mounted on the ceiling of the room. They are symmetrically positioned relative to the center of the ceiling and serve both communication and positioning functions. To enhance spectrum utilization, we multiplex the signals from the four LEDs in the fractional Fourier domain, with the subcarriers corresponding to LED 1 – LED 4 illustrated in Figure 4. The receiver, equipped with a PD, is mounted on the observation plane to receive light signals from the LEDs. In this scenario, the indoor wireless optical communication VLC channel comprises both line-of-sight (LOS) and diffuse or non-line-of-sight (NLOS) paths, which can be expressed as
h VLC ( t ) = h LOS ( t ) + h NLOS ( t )
Here, we consider only the LOS component.
For the LOS path, the impulse response is given by
h LOS ( t ) = A d 2 R ( ϕ ) cos ( θ ) δ ( t − d c )
where ϕ and θ denote the radiation angle of the transmitter and the incidence angle of the receiver, respectively. A represents the surface area of the PD, d is the LOS path length, c is the speed of light, and R ( ϕ ) is the radiation intensity of the light source. Considering the optical characteristics of the LED, the angular distribution of the light source’s radiation intensity follows the Lambertian radiation intensity model, characterized by the following distribution:
R ( ϕ ) = m + 1 2 π cos m ( ϕ )
where m denotes the Lambertian emission order, which is related to the semiangle at half power ϕ 1 / 2 of the LED light emission and is defined as
m = − ln ( 2 ) ln ( cos ( ϕ 1 / 2 ) )
For the VLP module, the optical power P r x received by the PD at the receiver can be used to calculate the channel gain G. The channel gain can be expressed as
G = P r x P t x
where P t x represents the transmitter’s signal power. It is assumed that only the LOS link component in the VLC channel is considered, and the normal vector of the PD at the receiver is perpendicular to the horizontal plane. When the incidence angle of the PD is smaller than its field of view (FOV), the channel gain can be expressed as
G = A d 2 R ( ϕ ) cos ( θ ) = ( m + 1 ) A 2 π d 2 cos m ( ϕ ) cos ( θ )
From this, we can derive the estimated distance d e s t from the LED to the PD as
d e s t = ( m + 1 ) A P t x 2 π P r x cos m ( ϕ ) cos ( θ )
We calculate the estimated distance from each LED to the PD individually. With the specific positions of each LED and the corresponding estimated distances to the PD, we can determine the PD’s position at the receiver using the least squares method.
For the VLC system, the received optical signal is converted back into the electrical format using an optical Rx, which is a PD and a trans-impedance amplifier (TIA). The signal at the receiver can be expressed as
y ( t ) = G TIA R PD ( x o u t ( t ) ⊗ h VLC ( t ) + n ( t ) )
where n ( t ) represents the additive white Gaussian noise (AWGN) generated by the PD, R PD denotes the response of the PD, and G TIA is the gain of the TIA. The output y ( t ) is then processed by applying the RPFRFT following analog-to-digital (A/D) conversion and ZP removal. This output is merged into a complex bit stream, and the final data are obtained through QAM demodulation. The resulting data are compared bit by bit with the original data from the transmitter to calculate the bit error rate, thereby validating the effectiveness of the FRFT within this system configuration.

3. Simulation Results and Discussion

In the previous section, we outlined the component architectures of the VLC and VLP systems and presented the relevant mathematical formulas. In this section, we will simulate the performance of the FRFT-OFDM scheme, compare it to that of the FT-OFDM scheme, and draw conclusions from the results. We assume that all LEDs are mounted on the ceiling, with LED 1 – LED 4 positioned at (−1, −1, 3), (−1, 1, 3), (1, 1, 3), and (1, −1, 3), respectively. These LEDs are driven by OFDM signals using QAM. The specific parameters of the entire system are detailed in Table 1. To simplify the calculations, we assume that there are no obstacles between the transmitter and receiver and only consider the LOS path. Notably, we set the PD’s FOV to 60 ° to minimize the impact of NLOS on the BER and ensure robust overall system communication performance. Consequently, the PD will not be placed too close to the surrounding walls; positioning it near the walls may result in some direct light from the LEDs being undetectable, thus adversely affecting system performance. For accurate positioning throughout the room, a larger FOV for the PD would be necessary.
We first assume that the receiver’s PD is located at (1, 1, 0.8). From Section 2.1 and Equation (2), we know that the FRFT is a periodic function. Specifically, when the transform order p spans a range of 2, the transform can cover the entire time–frequency domain, transitioning from the time domain to the frequency domain. Based on this, we selected a range of 0.01 ≤ p ≤ 1.99 for traversal and used 4-QAM to determine the optimal transform order p in this environment. We transmitted 20,000 FRFT-OFDM signals, each containing 64 bits, resulting in 12,80,000 bits sent for each p calculation. Figure 5 shows the relationship between the BER performance and the transform order p at fixed E b / N 0 levels of 20 dB and 30 dB. Notably, the BER trends are similar for different p values with both E b / N 0 settings. From the trend of the BER curves in Figure 5, it is evident that the transform order p corresponding to relatively low BER values is concentrated around p = 1 . This suggests that the nonlinear effects of the LED and the limited bandwidth have a minimal impact on the overall BER of the VLC system. In other words, the BER corresponding to the optimal order p does not differ significantly from the BER at p = 1 . Specifically, the lowest BER observed at E b / N 0 = 30 dB is 4.1 × 10 − 5 for p = 1.04 , while the BER for p = 1 is approximately 2.6 × 10 − 4 . Thus, the optimal order p in this scenario is determined to be 1.04.
Figure 6 presents a comparison of BER performance between the FRFT-OFDM scheme at the optimal order p and the FT-OFDM scheme across various modulation modes. As illustrated in the figure, both systems exhibit similar BER performance when E b / N 0 ≤ 8 dB, regardless of the modulation scheme used. However, in higher- E b / N 0 environments, the FRFT-OFDM scheme shows significant improvements over the FT-OFDM scheme. For instance, with 4-QAM at E b / N 0 = 30 dB, the BERs for the FRFT-OFDM scheme and FT-OFDM scheme are 4.1 × 10 − 5 and 2.6 × 10 − 4 , respectively. In the case of 16-QAM, although both systems experience degraded BER performance due to the increased number of modulation bits, the FRFT-OFDM scheme still demonstrates a notable advantage in BER performance compared to the FT-OFDM scheme in higher- E b / N 0 scenarios.
In the positioning system, we evaluated the localization error for both the FRFT-OFDM scheme and the FT-OFDM scheme at the optimal order p = 1.04 with the receiver positioned at (1, 1, 0.8). As shown in Figure 7, the FRFT-OFDM scheme demonstrates a significant reduction in localization error compared to the FT-OFDM scheme in higher- E b / N 0 environments. Notably, when E b / N 0 ≥ 20 dB, the decline in localization errors for both the FRFT-OFDM scheme and the FT-OFDM scheme becomes more gradual as E b / N 0 increases, leading to a negligible effect of E b / N 0 on localization error beyond a certain point. Specifically, at E b / N 0 = 20 dB, the localization errors are 6.6 cm for the FRFT-OFDM scheme and 8 cm for the FT-OFDM scheme. This results in a 1.4 cm improvement in positioning accuracy for the FRFT-OFDM scheme over its FT counterpart.
In addition, we investigated the impact of varying the number of lights on localization error at a fixed receiving location. The four-LED positioning scheme was modified to use three LEDs positioned symmetrically around the ceiling’s center: (0, 1, 3), ( − 3 / 2 , −1/2, 3), and ( 3 / 2 , −1/2, 3). As shown in Figure 7, at the optimal order, the three-LED FRFT-OFDM scheme shows minimal differences compared to the four-LED scheme. This indicates that, when the number of LEDs is sufficient, further increasing the number of LEDs has a limited impact on the positioning accuracy of the VLP system, as each LED’s position estimation for the receiver is primarily influenced by its position relative to the receiver. Additionally, as shown in Figure 7, both the three-LED and four-LED FRFT-OFDM schemes perform significantly better than their corresponding FT-OFDM schemes. Both the three-LED and four-LED FRFT-OFDM schemes achieve optimization over 1 cm compared to the FT-OFDM scheme when E b / N 0 > 20 dB.
To investigate the positioning accuracy of the FRFT-OFDM scheme at various locations in the room and to compare it with the FT-OFDM scheme, we selected a PD with a FOV of 70 ° . This ensures that the angle of incidence on the PD remains within its FOV, regardless of the receiver’s position within the observation plane, allowing it to capture all direct light emitted by the LEDs. The relationship between localization error and the PD’s location is depicted in Figure 8. As the receiver approaches a wall, the localization error increases significantly. This is because the LED is a Lambertian transmitter, and its radiation characteristics are typically angle-dependent. Due to the effect of the semiangle at half power, when the receiver is near the wall, the angle between the receiver and the more distant LED increases, leading to a significant reduction in the power received from that LED and, consequently, a higher localization error. As shown in Figure 8, the localization error can reach up to 17 cm in the four corners of the room. Conversely, when the receiver is positioned at the center of the room, the localization error approaches 0, which is attributed to the symmetrical arrangement of the positioning LEDs relative to the center of the ceiling. As shown in Figure 8, a circular area with a radius of 2.8 m from the room’s center ensures that the localization error remains below 12 cm, while a smaller circle with a radius of 2 m guarantees that the error is less than 8 cm.
Figure 9 illustrates the improvement in positioning accuracy of the FRFT-OFDM scheme with the optimal order p compared to the FT-OFDM scheme for various indoor locations. It is evident that the FRFT-OFDM scheme achieves varying degrees of improvement in localization error across nearly all positions. Table 2 presents the localization errors of the FRFT-OFDM scheme at specific locations, along with the improvement in positioning accuracy compared to the FT-OFDM scheme. Combining this with Figure 8, we observe that when the receiver is very close to the center of the room, the increased power from direct LED reception leads to better positioning accuracy for the FT-OFDM scheme, with minimal improvement from the FRFT-OFDM scheme in this scenario. When the receiver is positioned close to the wall, the power of the received direct light is significantly reduced, resulting in a marked increase in localization error for both the FRFT-OFDM and FT-OFDM schemes. However, even in this scenario, the positioning accuracy of the FRFT-OFDM scheme remains notably superior to that of the FT-OFDM scheme. Overall, the FRFT-OFDM scheme demonstrates strong positioning performance in most scenarios, achieving localization errors of less than 10 cm and improvements of over 1 cm compared to the FT-OFDM scheme.

4. Conclusions

This paper addresses the challenges of indoor VLC and VLP, which are affected by LED nonlinearities and limited modulation bandwidth. We introduce an OFDM scheme based on the FRFT. By discretizing the real and imaginary parts of the modulated signal, the system is applied to both optical communication and positioning. The performance of this system is then compared to that of the FT-OFDM scheme. In the FRFT-OFDM scheme, it is necessary to recalculate the optimal FRFT order p under the current environmental conditions, which is used as the basis for subsequent communication or positioning. Our results demonstrate that the FRFT-OFDM scheme, when using the optimal order p, achieves superior performance in both communication and positioning compared to the FT-OFDM scheme. This is reflected in improvements in both the BER and positioning accuracy. Specifically, for communication, the FRFT-OFDM scheme provides over a 6-dB E b / N 0 gain at a BER of 3 × 10 − 4 compared to the FT-OFDM scheme. For positioning, the FRFT-OFDM scheme significantly outperforms the FT-OFDM scheme in most positions throughout the room, maintaining a low localization error. Simulation results indicate that the FRFT-OFDM scheme offers promising applications and valuable research potential. Currently, research on VLC and VLP systems has reached a relatively mature stage. However, the integration of the FRFT with VLC or VLP systems remains underexplored. This may be attributed to the need for further research and optimization of discrete FRFT algorithms and their supporting techniques within these systems. In the future, key research directions include exploring different coding methods in FRFT-OFDM schemes, designing channel equalization techniques in the fractional domain, and selecting appropriate positioning methods tailored to various environments.

Author Contributions

Conceptualization, W.L. and Z.W.; software, W.L.; validation, W.L., Z.W. and J.Y.; writing—original draft preparation, W.L.; writing—review and editing, Z.W. and J.Y.; supervision, Z.W. and J.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Natural Science Foundation of China under Grant 61835003 and Grant 62005194.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Block diagram of RPFRFT-based OFDM system.
Figure 1. Block diagram of RPFRFT-based OFDM system.
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Figure 2. Hammerstein model.
Figure 2. Hammerstein model.
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Figure 3. VLC and VLP link with LOS and NLOS paths.
Figure 3. VLC and VLP link with LOS and NLOS paths.
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Figure 4. Subcarrier allocation in the fractional Fourier domain for 4-LED VLC and VLP systems.
Figure 4. Subcarrier allocation in the fractional Fourier domain for 4-LED VLC and VLP systems.
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Figure 5. BER performance comparison at various FRFT orders p.
Figure 5. BER performance comparison at various FRFT orders p.
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Figure 6. BER performance against E b / N 0 for FRFT-OFDM and FT-OFDM schemes using 4-QAM and 16-QAM.
Figure 6. BER performance against E b / N 0 for FRFT-OFDM and FT-OFDM schemes using 4-QAM and 16-QAM.
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Figure 7. Localization errors against E b / N 0 for FRFT-OFDM and FT-OFDM schemes with 3- or 4-LED positioning.
Figure 7. Localization errors against E b / N 0 for FRFT-OFDM and FT-OFDM schemes with 3- or 4-LED positioning.
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Figure 8. Localization errors at various positions throughout the room under the optimal order p of the FRFT.
Figure 8. Localization errors at various positions throughout the room under the optimal order p of the FRFT.
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Figure 9. Improvement in positioning accuracy at various positions throughout the room under the optimal order p of the FRFT compared to the FT.
Figure 9. Improvement in positioning accuracy at various positions throughout the room under the optimal order p of the FRFT compared to the FT.
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Table 1. The system parameters of the simulation setup.
Table 1. The system parameters of the simulation setup.
ParametersValue
Modulation typeQAM
Total number of SCs128
Number of data SCs64
Number of ZP samples16
Maximum value of current, I m a x 1 A
DC-bias current, I DC 0.55 A
Number of light-emitting diode (LEDs)4
Room size 6 × 6 × 3 m 3
Height of receiver0.8 m
Modulating bandwidth of LED, B m 4.5 MHz
Total bandwidth of OFDM, B5 MHz
Semiangle at half power, ϕ 1 / 2 70 °
PD’s FOV, θ c 60 °
Active area of PD, A 1 cm 2
Responsivity of PD, R PD 0.6 A/W
TIA gain, G TIA 50 dB
Table 2. Localization errors of the FRFT-OFDM scheme at specific locations and the improvement in positioning accuracy compared to the FT-OFDM scheme.
Table 2. Localization errors of the FRFT-OFDM scheme at specific locations and the improvement in positioning accuracy compared to the FT-OFDM scheme.
Specific LocationsLocalization ErrorsPositioning Accuracy Improvement
(0.5, 0.5, 0.8)3.09 cm0.74 cm
(1, 1, 0.8)6.07 cm1.46 cm
(1.5, 1.5, 0.8)9.12 cm2.18 cm
(2, 2, 0.8)12.14 cm2.93 cm
(2.5, 2.5, 0.8)15.19 cm3.63 cm
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Li, W.; Wang, Z.; Yu, J. Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems. Photonics 2024, 11, 1147. https://doi.org/10.3390/photonics11121147

AMA Style

Li W, Wang Z, Yu J. Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems. Photonics. 2024; 11(12):1147. https://doi.org/10.3390/photonics11121147

Chicago/Turabian Style

Li, Wenyang, Zixiong Wang, and Jinlong Yu. 2024. "Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems" Photonics 11, no. 12: 1147. https://doi.org/10.3390/photonics11121147

APA Style

Li, W., Wang, Z., & Yu, J. (2024). Performance Improvement by FRFT-OFDM for Visible Light Communication and Positioning Systems. Photonics, 11(12), 1147. https://doi.org/10.3390/photonics11121147

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