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Abstract
Optical camera communication (OCC) has emerged as a promising technology for wireless communication owing to its enormous potential benefits. However, turbulence conditions can restrict the feasibility of OCC systems that employ multiple-input multiple-output (MIMO) techniques. In this work, we consider a vehicular MIMO-OCC system model in which the traffic light LEDs transmit data streams separately in parallel channels to the camera of a vehicle. We analyze the effect of turbulence on the inter-pixel interference (IPI) and bit error rate (BER) of the low speed vehicular MIMO-OCC system using the point spread function (PSF) of the optical channel. We introduce two performance metrics, i.e., digital number difference (DND) and the percentage of separable LEDs (PSLED) for strong and very strong turbulence conditions, respectively. The parameters of the camera and traffic light LEDs can affect the IPI caused by turbulence. It is shown that the DND of MIMO-OCC systems with larger focal length and LED size less affected by turbulence. In addition, smaller f-number and pixel size reduce the effect of turbulence on DND. It is also demonstrated that very strong turbulence can reduce PSLED for links longer than 20 m. Moreover, if turbulence reduces PSLED, increasing the LEDs’ intensity cannot efficiently reduce the BER in the MIMO-OCC system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The widespread use of light-emitting diode (LED) for illumination offers the opportunity to establish visible light communication (VLC) technology, which has received much attention due to providing illumination and communication simultaneously [1]. Optical camera communication (OCC) is a camera-based VLC technology that provides many advantages over conventional single-photodiode-based VLC systems, including the easier implementation of various services into smart devices [2], the ability to spatially separate incident light [3], and overcoming the interference problem [4]. VLC and OCC technologies are used in many applications such as digital signage [5], localization and navigation [6], internet of things [7,8], underwater communication [9,10], and vehicular communication [11]. Among them, vehicular communication plays an important role in intelligent transportation systems (ITSs) that include vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I), infrastructure-to-vehicle (I2V), and internet of vehicle (IoV) communications. OCC has been recognized as a promising technology for vehicular communication due to several features such as low cost, unlicensed spectrum, and safe for humans.
In vehicular OCC systems which employ multiple-input multiple-output (MIMO) techniques, LEDs in a light source send information separately in parallel channels to a receiver’s camera [12]. The camera’s pixels convert LED’s light into current or voltage, and then it is converted into the digital number (DN) using analog-to-digital converter (ADC). Since LEDs in the light sources are about a few millimeters apart, these parallel channels may disturb each other in MIMO-OCC systems. Some factors such as perspective, link distance, vehicle motion, lens blurring, bad weather conditions, and turbulence can cause adjacent channels to interfere with each other. As shown in Fig. 1(a), turbulence reduces the peak intensity and increases the beam waist of the received intensity at the receiver. Therefore, the pixels’ DN corresponding to the LED’s image on the camera’s image sensor (CIS) reduces, while DN of the neighboring pixels increases. This fact illustrates that turbulence increases the effect of each LED on the neighboring pixels, and thus inter-pixel interference (IPI) increases in the captured image (see Fig. 1(b) and Fig. 1(c)).
Fig. 1. Effect of turbulence on the IPI of the captured image. (a) Effect of turbulence on the intensity and pixels’ DN of one LED’s image. The pixels’ DN of $19$ LEDs’ images in the cases of (b) without turbulence and (c) with turbulence.
Some studies have focused on the optical signal propagation in outdoor environments, which are affected by atmospheric turbulence [13–17]. Experimental results in [13] showed that aperture averaging reduces the effect of atmospheric turbulence on the performance of the photodiode and camera-based VLC. The authors in [14] showed that turbulence cannot considerably degrade the performance of short-range camera-based VLC systems (i.e., about 7.5 m) in terms of peak signal-to-noise ratio. However, the influence of turbulence on the MIMO-OCC system has not been investigated in the Refs. [13] and [14]. Matus et al. in [15] used RGB LED strips to show that the influence of turbulence on short-range OCC systems (i.e., less than $5$ m) is negligible, while fog attenuation decreases the signal quality. The atmospheric turbulence channel was analyzed in [16] to quantify its influence on long-distance image sensor communication. Since the link distance in this article was several hundred meters long, the turbulence significantly affected this image sensor communication system. However, Ref. [16] has not investigated the effect of atmospheric turbulence on the MIMO-OCC system and LEDs’ interference. Reference [17] demonstrated that the scintillation caused by vehicle exhaust is very weak compared to the background noise in vehicular VLC applications.
Most of the previous works use experimental methods to study the influence of turbulence on the MIMO-OCC systems with a link distance of less than $10$ m. To the best of our knowledge, there is no comprehensive study on the effect of turbulence on the IPI using point spread function (PSF) analysis. In this paper, we address this issue and propose PSF analysis to investigate the impact of turbulence on the IPI of MIMO-OCC systems in the long links (i.e., 10 m to 100 m). Since increasing the link distance causes each LED to fill fewer pixels and images of LEDs will be closer to each other in the CIS, the turbulence effect intensifies in longer links. Note that geometrical optics is not appropriate to compute the IPI of MIMO-OCC systems since the light intensity in geometric calculations is only modeled in a circle of a specific diameter. In practice, a portion of the light intensity leaks out of this area, causing interference between the LEDs. A comparison between geometric and PSF analysis is given in Section 4.1.
1.1 Contribution
In this work, it is assumed that a traffic light allocates different data to each LED. Then, the LEDs transmit information separately to a vehicle’s camera in the low speed vehicular MIMO-OCC environment, e.g., when cars stop behind the red traffic light or move slowly in traffic jams that can last for many minutes in densely populated cities. The main contributions of this paper are summarized as follows:
† To derive the effect of turbulence on the IPI and the bit error rate (BER) of the system, we analyze the PSF of the MIMO-OCC system and obtain the received LEDs’ light intensity in the presence and absence of turbulence. Then, the LEDs’ light intensity is used to calculate the digital number of pixels, which is the ADC output.
† We introduce two performance metrics namely digital number difference (DND) and the percentage of separable LEDs (PSLED) for strong and very strong turbulence conditions, respectively. These metrics help us to evaluate the effect of turbulence on our scheme.
† We derive the effect of strong and very strong turbulence on the IPI for various link distances, traffic light settings, and camera parameters such as focal length (f), $f$-number (FN), and pixel size.
The rest of the paper is organized as follows. In Section 2, the system model is described. Then, the PSF of the OCC-based vehicular system with and without turbulence is analyzed. The transmitter and receiver structures are described in Section 3. Section 4. provides simulation results of our theoretical analysis. Finally, in Section 5, an overview of the results and conclusions are drawn.
2. Vehicular OCC system model and PSF analysis
In this work, we consider a vehicular optical camera communication scenario, in which an OCC-based traffic light transmits various information to a vehicle’s camera (see Fig. 2). The traffic light consists of 61 LEDs and each of them simultaneously transmits independent data streams. We analyze the PSF of the MIMO-OCC system to obtain the received LEDs’ light intensity at the CIS. For our analysis, we need to acquire the positions of the traffic light LEDs based on the optical axis of the camera lens. In the first step of the PSF calculation, we assume that the optical axis of the camera lens and traffic light LEDs are parallel to the centerline of the street. Then, we choose coordinate system $1$ ($CS_1$ in Fig. 2(a)) in which the camera lens and the center of traffic light LEDs are located on its $Y~\textrm {axis}$ in position $(0, y_1, 0)$, and $XZ$ plane in position $(x_1, 0, z_1)$, respectively. As shown in Fig. 2(b), if the optical axis of the camera lens rotates $\alpha$ degrees around the $Y \textrm {axis}$ on the $XY$ plane, then we choose coordinate system 2 ($CS_2$) in which the traffic light LEDs and the camera lens will be in the positions $(x_2, 0, z_2)$ and $(0, y_2, 0)$, respectively. The relationship between the parameters of $CS_1$ and $CS_2$ is shown in part $1$ of Fig. 2(d). If the optical axis of the camera lens rotates $\theta$ degrees around the $Y^{\prime } \textrm {axis}$ on the $Y^{\prime }Z^{\prime }$ plane, as shown in Fig. 2(c), then the appropriate coordinate system would be as $CS_3$, where the traffic light LEDs and the camera lens are located in $(x_3, 0, z_3)$ and $(0, y_3, 0)$, respectively. Part $2$ of Fig. 2(d) shows the relationship between the parameters of $CS_2$ and $CS_3$. Without loss of generality, we can assume that traffic light and camera lens are in parallel planes, otherwise, we can calculate an equivalent coordinate system like $CS_3$. Figure 3 shows the front view of $CS_3$, which is more suitable for PSF analysis. As shown in this figure, the light of a LED in the source plane $u_o$ propagates through the atmosphere and passes through the lens plane at $u_l$. Then, the lens focuses the light onto the CIS at the plane $u_i$. To compute the intensity of a traffic light LED at the point $(x_i, z_i)$ of the image plane $u_i$, we obtain the PSF of our optical system denoted by $h(x_i, z_i; x_o, z_o)$. This function represents the impulse response of an imaging system to the point source, which affects the quality and spatial resolution of the imaging system. We first assume that the LED is a point source with the position $(x_o, z_o)$ in the source plane $u_o$, as depicted in Fig. 3, and then generalize the results to an extended source. Since the point source emits light in all directions with spherical wavefronts, the PSF in the position $(x_l, z_l)$ of the lens can be written as [18]
Fig. 2. The architecture of an OCC-based vehicular communication system. (a) $CS_1$, (b) $CS_2$, (c) $CS_3$, and (d) relationship between $CS_1$, $CS_2$, and $CS_3$.
To evaluate the IPI of our MIMO-OCC system, we select a traffic light and calculate the DN of each pixel of CIS in the next section.
3. Transmitter and receiver of MIMO-OCC system
As shown in Fig. 4(a), a traffic light with 61 LEDs is arranged in rings with radii of $d_{LED}$, $2d_{LED}$, $3d_{LED}$, and $4d_{LED}$ related to the central LED. Note that similar traffic lights are available at [27]. Since our MIMO-OCC system uses the on-off keying (OOK) modulation technique, each LED of the traffic light turns on and off separately to transmit bits 1 and 0, respectively. The optical signal of traffic light LEDs propagates through the atmosphere and is received by a camera in the receiver. Thanks to the spatial separation of the lens, the camera can distinguish different LEDs, as shown in Fig. 4(b). The CIS converts the received optical signal into an electric voltage. Then, the ADC converts the voltage of each pixel to a digital number, DN, proportional to the magnitude of the voltage.
Fig. 4. (a) The traffic light with $61$ LEDs and (b) the CIS with images of traffic light LEDs. Both X and Y axes correspond to pixels.
The receiver processes the data sent from each LED based on the DNs in pixels. Initially, the receiver determines the region of interest (ROI) in the first frame that contains the image of the traffic light, as depicted with a blue circle in Fig. 4(b). Various methods are used to determine the ROI, including image processing [28], deep learning algorithms, and convolutional neural networks [29]. The size, position, and threshold level of LEDs in the captured images are then determined. Typically, blocks of frames are provided, with a few preamble frames at the beginning and payload frames at the end [30]. The preamble frames are used to extract the necessary information about ROI, threshold level, and LEDs [31]. This information allows the receiver to identify the on or off state of the LEDs and to extract the sent bits 1 and 0, respectively. In order to make a bit decision and calculate the BER of the MIMO-OCC system, the receiver only processes the ROI and compares the DN of pixels in the image of each LED against a threshold level.
Let assume DNL stands for the average DN of pixels inside the images of 61 LEDs (see Fig. 5(a)), and DNI stands for the average DN of pixels between the images of 61 LEDs. The letters “L” and “I” in the acronyms DNL and DNI are derived from the words LEDs and IPI, respectively. The blue circles in Fig. 5(a) represent the area of LEDs in geometric calculations. According to geometric calculations, the image of a LED with a diameter $r_{LED}$ and center $(x_o, z_o)$ is a circle with a diameter $r_{img}=M r_{LED}$ and center $(M x_o, M z_o)$ in the image sensor. Since the LEDs in traffic lights are near to each other ($d_{LED}$ is about 1 to 2 cm), their images interact. As a result, even in the absence of noise and background light, the DNI has a non-zero value. The effect of LEDs on adjacent pixels is called IPI. If all 61 LEDs of the transmitter are separable in the received image, they can convey data independently in each frame. As a result, the data rate is $61\times FR$ (bits per second), where FR denotes the frame rate.
Fig. 5. (a) The DNL and DNI, and (b) the PSLED in the images of traffic light LEDs. Both X and Y axes correspond to pixels.
When the IPI rises, DNI increases, while DNL decreases. As a result, increasing the IPI lowers the quality and clarity of the LED images. The image quality in this paper is represented by the parameter digital number difference, $DND=DNL-DNI$. As the IPI increases, the DND decreases, and the images of the LEDs become a blur and lose sharpness. Since the IPI has a stronger effect on the LEDs in the middle of the traffic light image, they start overlapping as the IPI rises (see Fig. 5(b)). In this situation, the image of the LEDs in the overlapping area grows so significantly that it differs greatly from the region of LEDs indicated by the geometric relations, making it impossible to calculate DNI in this area. In this case, to evaluate the effect of turbulence on the MIMO-OCC link, we define the parameter percentage of separable LEDs (PSLED), as shown in Fig. 5(b). Since overlapping LEDs cannot convey data independently, decreasing PSLED causes the BER of the MIMO-OCC system to increase (Section 4.3). The parameters PSLED and DND are employed in subsection 4.2 to evaluate the effect of turbulence on the IPI of the MIMO-OCC system.
4. Simulation results
In this section, we compare the results of PSF analysis given in Section 2 with the geometrical analysis to calculate the IPI in CIS. In addition, a comprehensive simulation study is conducted to evaluate the effect of turbulence and various parameters of the car camera and traffic light (e.g., $f$, $f$-number ($FN$), pixel size ($l_{pix}$), $d_{LED}$, and LED’s diameter ($r_{LED}$)) on the IPI and BER of the OCC-based vehicular communication. Firstly, the collected intensity of LEDs in the CIS for cases without and with turbulence (i.e., (12)) is simulated in Matlab software. Then, the DN of each pixel is calculated based on (14). It is assumed that the vehicle’s velocity is small enough such that the location of LEDs in the CIS is fixed during one exposure time ($t_{exp}=0.1 ms$). For example, vehicles at intersections stay stationary or move at low speeds in traffic, which is feasible in real environments. This assumption has already been used in the literatures such as [32–34]. The simulation parameters are listed in Tables 1 and 2, where $QE$, $A_{SN}$, $A_{SF}$, and $D_R$ represent quantum efficiency, sense node conversion gain, linear source follower gain, and the average dark current rate, respectively. Moreover, parameters $\sigma _{DSNU}$ , $\sigma _{PRNU}$, $\sigma _{SF}$, and $\sigma _{reset}$ denote the standard deviations of dark signal nonuniformity noise, photoresponse nonuniformity noise, source follower noise, and reset noise, respectively [24]. It is assumed that the positions of the traffic light and car camera lens are set in $(5, 5, 0)~\textrm {m}$ and $(0, 0, y_3)~\textrm {m}$ in coordinate system $1$ (i.e., $CS_1$ in Fig. 2(a)), respectively.
Table 2. The reference simulation parameters.
4.1 PSF analysis versus geometric analysis
The image of the LED is shown as a circle with diameter $r_{img}=M r_{LED}$ and center $(M x_o, M z_o)$ in geometric calculations, where $r_{LED}$ and $(x_o, z_o)$ stand for the diameter and center of a LED, respectively. In this circle, the light intensity is uniform and equals to $\frac {P_{image}}{ M^2 A_{LED}}$, where $A_{LED}$ is the area of a LED (see Eq. (15) in Appendix A). The intensity of the received light outside this circle is zero in geometric calculations (Eq. (24) in [36]). Figure 6(a) shows the discrepancy between geometric and PSF analysis for a LED, where $C_n^2=0~\textrm {m}^{-2/3}$ represents PSF analysis without the presence of turbulence. In this simulation, thevalue of $\frac {P_{LED}}{str}$ in Eq. (15) for geometric calculations is $1~\textrm {cd}$ and $|u_o(x_o,z_o)|^2$ in Eq. (8) for PSF analysis equals to $\frac {1~\textrm {cd}}{A_{LED}}$. In Fig. 6(a), the received power from a LED is equal to $3.86\times 10^{-8}~\textrm {lm}$ and $3.84\times 10^{-8}~\textrm {lm}$ in geometric and PSF calculations, respectively, which are very close. However, in the PSF analysis, unlike geometric calculations, the light intensity at the margins of the LED image drops smoothly. Additionally, increasing turbulence spreads the LED’s light on the surface of the CIS, according to the PSF analysis, which makes it appropriate for IPI calculation. Figure 6(b) depicts the image of two LEDs with $d_{LED}=20~\textrm {mm}$. The images of two LEDs are isolated in the geometric calculation, i.e., they have no effect on each other or the pixels between them. On the other hand, the PSF analysis reveals the impact of each LED on nearby LEDs as well as the entire CIS surface. The PSF analysis also demonstrates that as turbulence grows, the LED’s effect on the other LEDs increases, and as a result, the IPI rises.
Fig. 6. The received intensity of (a) one LED, and (b) two LEDs for $y_3=40~\textrm {m}$ in the CIS.
The DN of pixels in the CIS for images of two LEDs is shown in Fig. 7. The blue circles represent LEDs images in geometric calculations. As seen from Fig. 7, turbulence elevates the IPI by increasing DNI and lowering DNL. Thus, the clarity of the LEDs’ images degrades. In addition, as the number of LEDs increases, the IPI intensifies. The next subsection examines the effect of turbulence on IPI.
Fig. 7. Pixels’ DN of two LEDs images in the CIS for (a) absence of turbulence, and (b) turbulence with $C_n^2=4\times 10^{-11}~\textrm {m}^{-2/3}$ for $y_3=40~\textrm {m}$. Both X and Y axes correspond to pixels.
4.2 Effect of turbulence on IPI
Since the parallel channels of LEDs in MIMO-OCC systems are very close to each other, turbulence increases IPI in these schemes (see Fig. 7). Generally, turbulence is classified into four regimes [37]:
1-weak: ($C_n^2\leq 10^{-14}~\textrm {m}^{-2/3}$),
2-moderate: ($10^{-14}< C_n^2\leq 10^{-13}~\textrm {m}^{-2/3}$),
3-strong: ($10^{-13}< C_n^2\leq 10^{-12}~\textrm {m}^{-2/3}$),
4-very strong: ($10^{-12}< C_n^2\leq 10^{-10}~\textrm {m}^{-2/3}$).
In links smaller than $100$ m, turbulence is usually in the weak and medium regimes. However, since our MIMO-OCC system has been employed in a vehicular environment, we could observe more severe turbulence due to the road’s surface on hot days and vehicle exhausts [13]. For example, Refs. [14,15] experimentally investigate the effect of very strong turbulence ($10^{-12}< C_n^2\leq 10^{-10}~\textrm {m}^{-2/3}$) on a vehicular MIMO-OCC system for $7$ and $5$ m long links, respectively. However, we apply PSF analysis to infer the influence of turbulence on IPI in MIMO-OCC systems for $y_3\leq 100~\textrm {m}$ analytically. Simulation results show that the influence of the weak and moderate turbulence ($C_n^2\leq 10^{-13}~\textrm {m}^{-2/3}$) on MIMO-OCC systems is small for $y_3\leq 100~\textrm {m}$. Therefore, in this section, we investigate the effect of turbulence on the IPI of MIMO-OCC systems in two parts: (1) effect of strong turbulence (i.e., $10^{-13}\leq C_n^2\leq 10^{-12}~\textrm {m}^{-2/3}$) on IPI; and (2) effect of very strong turbulence ( $10^{-12}\leq C_n^2\leq 10^{-10}~\textrm {m}^{-2/3}$) on IPI. In reality, we will receive an image similar to Fig. 5 in the receiver’s camera. If the LEDs cannot be separable in the received image, as shown in Fig. 5(b), we check the PSLED parameter; otherwise, as in Fig. 5(a), we examine the DND parameter. Since the simulation results show that very strong turbulence usually causes the LEDs to overlap, the PSLED parameter is used to show its influence on IPI in subsection 4.2.2. However, the strong turbulence typically cannot cause the LEDs to overlap; hence the PSLED’s value is $100\%$. Then, in this study, we use the parameter DND to investigate the effect of strong turbulence on IPI of the MIMO-OCC system, as explained in subsection 4.2.1.
4.2.1 Strong turbulence regime
As demonstrated in Fig. 7, strong turbulence causes the DNs of LEDs images (i.e., DNL in Fig. 5) to decrease while the DNs of surrounding pixels (i.e., DNI in Fig. 5) increase. Therefore, strong turbulence reduces the clarity of LEDs’ images and DND, which indicates an increase in IPI of MIMO-OCC systems. Since LEDs in the MIMO-OCC system send data separately, reducing the clarity of LEDs’ images affects the system performance. Figures 8 and 9 show DND in the CIS for various turbulence, camera, and traffic light parameters, where $C_n^2=0~\textrm {m}^{-2/3}$ represents the absence of turbulence. The more the turbulence affects the OCC link, the more DND decreases accordingly. As a baseline, we run a reference simulation using the settings listed in Table 2, where $d'_{LED}$ is the distance between LEDs ($d'_{LED}=d_{LED}-r_{LED}$). Then, we change a parameter in Table 2 and compare DND variations to the reference simulation.
Fig. 8. The comparison of the DND of reference simulation with LEDs’ parameters of (a) green light (i.e., $G$) and (b) $d'_{LED}=5~\textrm {mm}$ (i.e., $d_2$) and $r_{LED}=20~\textrm {mm}$ (i.e., $r_2$).
Fig. 9. The comparison of the DND of reference simulation with camera parameters of (a) $FN=4$ ($\textit {i.e.}, FN_2$) and $f=35~\textrm {mm}$ ($\textit {i.e.}, f_2$), and (b) $l_{pix}=1.85~\mu \textrm {m}$ (i.e., $l_2$), $\alpha =10^\circ$ (i.e., $\alpha _2$) and $\theta =10^\circ$ ($\textit {i.e.}, \theta _2$).
We first change the parameter $\lambda$ to $505~\textrm {nm}$ (green light). Figure 8(a) demonstrates that the DND of green light is higher than that of red light, allowing it to withstand more severe turbulence. Moreover, the higher DND value of green light indicates that green light has sharper LEDs’ images than red light. However, as the turbulence increases, the green light’s DND value declines faster than the red light’s. Therefore, the destructive effect of turbulence on the IPI of green light is greater than red light. As shown in Fig. 8(a), the DND value drops as the link length grows, indicating that the images of the LEDs has a lower clarity on longer links. In addition, the slope of $y_3=100~\textrm {m}$ curves is steeper than the $y_3=20~\textrm {m}$ curves, since the longer link is more impacted by turbulence. For example, increasing the turbulence to $C_n^2=10^{-12}~\textrm {m}^{-2/3}$ reduces DND of red light by $0.50\%$ and $25.65\%$ for $y_3=20~\textrm {m}$ and $y_3=100~\textrm {m}$, respectively.
Because the influence of turbulence on longer links is more pronounced, we explore the efficacy of the other parameters in Table 2 on IPI at $y_3=60~\textrm {m}$ and $y_3=100~\textrm {m}$. Fig. 8(b) depicts the effect of LED size and inter LED distance (i.e., $d'_{LED}$) on the DND parameter. The DND value is lower for smaller $d'_{LED}$ and $r_{LED}$, which reduces the clarity of LEDs’ images. Additionally, the slope of DND curves is steeper for smaller $r_{LED}$ and $d'_{LED}$, which indicates that the turbulence has a more damaging impact on the IPI of MIMO-OCC systems with smaller $r_{LED}$ and $d'_{LED}$.
Figure 9(a) investigates the effect of $FN$ and $f$ on the IPI caused by turbulence. When $FN$ decreases while $f$ remains constant or when $f$ raises while $FN$ remains constant, the camera’s aperture widens. As a result, the amount of light incident on the CIS increases, and DND values grow. As a result, lowering $FN$ or raising $f$ makes LED separation easier and reduces the camera’s sensitivity to noise and turbulence. However, the slope of the DND curves are steeper for MIMO-OCC systems with lower $FN$ and higher $f$, demonstrating that turbulence has a stronger effect on them. Figure 9(b) depicts the effect of $l_{pix}$, $\alpha$, and $\theta$ on the DND. The amount of light collected by a pixel rises as the pixel size grows. As a result, the DND value for $l_{pix}=1.85~\mu \textrm {m}$ are higher than those for $l_{pix}=1~\mu \textrm {m}$. Therefore, the pixels with larger sizes can tolerate more strong turbulence. However, it should be noted that increasing the pixel size reduces the image resolution. In this case, the images of the LEDs were no longer separable at $y_3=100~\textrm {m}$, and the corresponding diagram is not drawn in Fig. 9(b). As shown in Fig. 9(b), the rotation of $\alpha$ and $\theta$ by $10$ degrees does not affect the IPI caused by turbulence. At $y_3=20~\textrm {m}$, increasing $\alpha$ and $\theta$ by $10$ degrees brings LEDs closer to the optical axis of the camera lens (i.e., $x_3\simeq ~1.5\textrm {m}$ and $z_3\simeq ~1.5\textrm {m}$). As a result, the values of DND for the rotated camera increase at $y_3=20~\textrm {m}$.
4.2.2 Very strong turbulence regime
Very strong turbulence can cause an excessive amount of LEDs’ light to disperse, so that the LEDs’ images in the captured image overlap each other. Therefore, the parameters DNL and DNI cannot be calculated for a very strong turbulence regime. Thus, in this subsection, we study the effect of turbulence on the percentage of separable LEDs and PSLED. For instance, Fig. 10(a) shows that the camera with parameters $f=18~\textrm {mm}$, $FN=2$, and $l_{pix}=1.85~\mu \textrm {m}$ can separate all LEDs of the traffic light with parameters $r_{LED}=10~\textrm {mm}$ and $d'_{LED}=2~\textrm {mm}$ in the absence of turbulence and $y_3=30~\textrm {m}$. To better investigate the effect of turbulence on the IPI of MIMO-OCC system, no noise is added to this figure. However, when the turbulence is so strong, i.e., $C_n^2=5\times 10^{-11}~\textrm {m}^{-2/3}$, some of the LEDs will no longer be separable (see Fig. 10(b)). Therefore, overlapping LEDs cannot transmit independent data. As the turbulence intensifies to $C_n^2=10^{-10}~\textrm {m}^{-2/3}$, the number of separable LEDs considerably reduces, as depicted in Fig. 10(c).
Fig. 10. Effect of very strong turbulence on the IPI at $y_3=30~\textrm {m}$. The LEDs’ images in (a) absence of turbulence, (b) $C_n^2=5\times 10^{-11}~\textrm {m}^{-2/3}$, and (c) $C_n^2=10^{-10}~\textrm {m}^{-2/3}$. Both X and Y axes correspond to pixels.
Simulation results in Fig. 11 show that increasing the turbulence from $C_n^2=10^{-12}$ to $C_n^2=10^{-10}$ for links longer than $20~\textrm {m}$ can reduce the number of separable LEDs. As expected, the results of Fig. 11 illustrate that the effect of turbulence on the overlap of LEDs increases at the longer links. Additionally, this figure shows that by increasing $d'_{LED}$ and decreasing $l_{pix}$, the PSLED increases at the longer $y_3$. Therefore, the implementation of a MIMO-OCC system is accomplished for longer $y_3$. For example, the PSLED of $FN_1\&f_1$, $FN_2\&f_1$, and $FN_1\&f_2$ for $y_3=60~\textrm {m}$ and $C_n^2=10^{-11}~\textrm {m}^{-2/3}$ is approximately $65\%$, $0\%$, and $0\%$, respectively, as depicted in Fig. 11(a). However, Fig. 11(b) shows that they are all $100\%$ for $y_3=60~\textrm {m}$ and $C_n^2=10^{-11}~\textrm {m}^{-2/3}$. Moreover, the PSLED increases in the larger $f$ and smaller $FN$. For instance, the PSLED of $FN_1\&f_1$, $FN_2\&f_1$, and $FN_1\&f_2$ for $y_3=100~\textrm {m}$ and $C_n^2=6\times 10^{-12}~\textrm {m}^{-2/3}$ is approximately $70\%$, $0\%$, and $90\%$, respectively, as illustrated in Fig. 11(b). Since the decreasing $f$ and the increasing $FN$ reduce the amount of incident light in the CIS, the quality of the LEDs’ image decrease. For this reason, PSLED reduces in the captured image for lower $f$ and higher $FN$. In addition, $60~\textrm {m}$ curves in Fig. 11(a) show that increasing $f$ can increase PSLED in long $y_3$ as well.
Fig. 11. PSLED versus various $C_n^2$ for $r_{LED}=10~\textrm {mm}$, $\lambda =625~\textrm {nm}$, and different $y_3$, $f$ and $FN$. PSLED for (a) $l_{pix}=1.85~\mu \textrm {m}$ and $d'_{LED}=2~\textrm {mm}$, and (b) $l_{pix}=1~\mu \textrm {m}$ and $d'_{LED}=10~\textrm {mm}$.
4.3 Effect of turbulence on BER
In this section, we investigate the effect of turbulence on the BER of the MIMO-OCC system. It is assumed that the traffic light has features $r_{LED}=10~\textrm {mm}$ and $d'_{LED}=20~\textrm {mm}$, and the camera features are $f=18~\textrm {mm}$, $FN=2$, $FR=1000~\textrm {fps}$, and $l_{pix}=1.85~\mu \textrm {m}$. In addition, we suppose the location of the traffic light and the car camera lens in coordinate system $1$ (i.e., $CS_1$ in Fig. 2(a)) are $(5, 5, 0)~\textrm {m}$ and $(0, 0, y_3)~\textrm {m}$, respectively. The angles $\alpha$ and $\theta$ are set to zero. To perform BER measurements, the frames are transmitted block by block. In this simulation, we generate $1000$ frames’ blocks in Matlab software that each frame of them consists of 61 bits of data for 61 LEDs of the traffic light. The LEDs of the traffic light are randomly turned on and off during sending data. The receiver determines the ROI location and threshold level of the frames in a block. Then, on the receiver side, the DN of pixels in the frames is compared with the threshold level to determine the bit decision. The on and off states of LEDs in the OOK modulation correspond to symbols one and zero, respectively. The detected one and zero in the receiver are compared with the sent one and zero in the transmitter to determine BER.
The effect of turbulence on BER for $y_3=10~\textrm {m}$ and $y_3=50~\textrm {m}$ with a data rate of $61000~\textrm {bits} / \textrm {s}$ is illustrated in Fig. 12. As seen in Fig. 11(a), the parameter PSLED remains at $100\%$ even when the turbulence increases to $C_n^2=10^{-10}~\textrm {m}^{-2/3}$ for $y_3<20~\textrm {m}$. Therefore, increasing the turbulence only causes the parameter DND to decrease at $y_3=10~\textrm {m}$. Figure 12(a) illustrates that the BER curve for $y_3=10~\textrm {m}$ grows as the turbulence increases. However, BER curves dramatically decline as LED power increases. Therefore, although the turbulence increases the BER of the MIMO-OCC system, it can be compensated by increasing the LEDs’ intensity. In contrast to $y_3=10~\textrm {m}$, the very strong turbulence in $y_3=50~\textrm {m}$ causes LEDs to overlap and PSLED to decrease. The PSLED for $y_3=50~\textrm {m}$ is $91/8\%$ and $88/5\%$ for $C_n^2=5\times 10^{-12}~\textrm {m}^{-2/3}$ and $C_n^2=7\times 10^{-12}~\textrm {m}^{-2/3}$, respectively. As demonstrated in Fig. 12(b), since the very strong turbulence reduces PSLED for $y_3=50~\textrm {m}$, the BER of the MIMO-OCC system with the data rate of $61000~\textrm {bits} / \textrm {s}$ increases. As a result, raising the intensity of the LEDs cannot rectify the turbulence impact. In cases where the LEDs’ intensity is low, the BER occurs due to low signal to noise ratio (SNR). Therefore, the existing result in Fig. 12(b) demonstrates that raising LEDs’ intensity can reduce BER in low signal power scenarios. However, at higher LEDs’ power, the BER that occurs due to overlapped LEDs is not compensated by increasing the LEDs’ intensity and, therefore, the BER curves decrease very smoothly. Note that the most intense turbulence we considered for $y_3=10~\textrm {m}$ in Fig. 12 is $C_n^2=10^{-10}~\textrm {m}^{-2/3}$, which is much greater than the most intense turbulence in $y_3=50~\textrm {m}$ that is $C_n^2=7\times 10^{-12}~\textrm {m}^{-2/3}$. This fact indicates that as $y_3$ increases, the turbulence more affects the BER curves.
Fig. 12. Effect of turbulence on BER in the MIMO-OCC system with f-number of $2$ and focal length of $18~\textrm {mm}$ for a) $y_3=10~\textrm {m}$ and b) $y_3=50~\textrm {m}$.
The BER, DND, and PSLED values of the MIMO-OCC system for $y_3=40~\textrm {m}$ are shown in Table 3 for various $C_n^2$. The results are computed under the assumption that $d'_{LED}=2~\textrm {mm}$, $l_{pix}=1.85~\mu \textrm {m}$, and LED’s intensity $=300~\textrm {mcd}$. Table 2 includes a list of the other parameters. Since the value of PSLED for up to $C_n^2=10^{-11}~\textrm {m}^{-2/3}$ is equal to $100\%$, the value of DND has been calculated for $C_n^2\leq 10^{-11}~\textrm {m}^{-2/3}$. Based on the outcomes stated in Table 3, Figs. 13(a) and (b) display the BER versus PSLED and DND, respectively. For $\textrm {PSLED}=100\%$, as shown in Fig. 13(a), the BER will continue to decline with the reduction of $C_n^2$, and as a result, the BER curve in this part will be vertical. To better clarify the reduction in BER caused by $C_n^2$ reduction, the BER in this part is shown in terms of DND in Fig. 13(b). Table 3 and Fig. 13 demonstrate that when $C_n^2$ increases from $0$ to $10^{-11}~(\textrm {m}^{-2/3})$, the DND value drops slowly, resulting in a gradual increase in the BER value. However, in the case of $C_n^2$ greater than $10^{-11}~(\textrm {m}^{-2/3})$, as the value of PSLED quickly declines, the value of BER likewise increases rapidly and achieves its maximum value.
Fig. 13. The BER of the MIMO-OCC system versus (a) PSLED and (b) DND, for $y_3=40~\textrm {m}$ and LED’s intensity of $300~\textrm {mcd}$.
Table 3. The BER, DND and PSLED values of the MIMO-OCC system in various $C_n^2$ for $y_3=40~\textrm {m}$ and LED’s intensity of $300~\textrm {mcd}$.
5. Conclusion
In this paper, we investigated the effect of strong and very strong turbulence regimes on vehicular MIMO-OCC systems by applying the PSF of the system that affects the quality and spatial resolution of the imaging system. We studied the effect of the strong turbulence on the IPI of MIMO-OCC systems in various parameters of camera and traffic light. The turbulence less affects the IPI of MIMO-OCC systems with larger $f$ and $r_{LED}$ and smaller $FN$ and $l_{pix}$. The results showed that the very strong turbulence could cause adjacent LEDs’ images to overlap, and then the PSLED reduced in the CIS for the long distance. Additionally, we showed that as the IPI increases, the BER of the MIMO-OCC system increases as well. If turbulence caused the LEDs’ overlap in the CIS, increasing the LEDs’ intensity could not compensate the effect of turbulence on the BER. Although the experimental implementation of this system are important, they are not included in the current paper due to space constraints. They would be taken into account in our future works.
6. Appendix A
To calculate the factor $K_h$ in Eq. (8), we compare a total LED’s received power using geometric calculations with the received power in Eq. (8). In the first step, we calculate the received power on the image plane using geometric equations. We assume that a LED with total power $p_{LED}$ shines through a lens at distance $\sqrt {y_{3}^2+x_o^2+z_o^2}$. The lens collects light and transmits it to the image plane. The received power at the image plane is equal to the received power at the lens surface $p_{image}=p_{lens}$, in which
where $str$ is the steradian of LED’s radiation, $A_{lens}=\pi R^2$ denotes the area of the lens and $\frac {p_{LED}}{str}$ is the intensity of LED light.In the next step, we derive the received power on the image plane using Eq. (8). For simplicity, we assume the image size is infinite. Then, the received power is as follow:
Funding
R&D Center of Mobile Telecommunication Company of Iran (MCI) (RD-51-9911-0021).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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