Measurements-Based Channel Models for Indoor LiFi Systems

The total data traffic is expected to become about 49 exabytes per month by 2021, while in 2016, it was approximately 7.24 exabytes per month [1]. With this drastic increase, the fifth generation (5G) networks and beyond must urgently provide high data rates, seamless connectivity, robust security and ultra-low latency communications [2, 3, 4]. In addition, with the emergence of the internet-of-things (IoT) networks, the number of connected devices to the internet is increasing dramatically [5, 6]. This fact implies not only a significant increase in data traffic, but also the emergence of some IoT services with crucial requirements. Such requirements include high data rates, high connection density, ultra reliable low latency communication (URLLC) and security. However, traditional radio-frequency (RF) networks, which are already crowded, are unable to satisfy these high demands [7]. Network densification [8, 9] has been proposed as a solution to increase the capacity and coverage of 5G networks. However, with the continuous dramatic growth in data traffic, researchers from both industry and academia are trying to explore new network architectures, new transmission techniques and new spectra to meet these demands.
Light-fidelity (LiFi) is a novel bidirectional, high speed and fully networked wireless communication technology, that uses visible light as the propagation medium in the downlink for the purposes of illumination and communication. It can use infrared in the uplink so that the illumination constraint of a room remains unaffected, and also to avoid interference with the visible light in the downlink [10]. LiFi offers a number of important benefits that have made it favorable for future technologies. These include the very large, unregulated bandwidth available in the visible light spectrum (more than 2600 times greater than the whole RF spectrum), high energy efficiency [11], the straightforward deployment that uses off-the-shelf light emitting diode (LED) and photodiode (PD) devices at the transmitter and receiver ends, respectively, and enhanced security as light does not penetrate through opaque objects [12]. However, one of the key shortcomings of the current research literature on LiFi is the lack of appropriate statistical channel models for system design and handover management purposes.

Some statistical channel models for stationary and uniformly distributed users were proposed in [13, 14, 15], where a fixed incidence angle was assumed in [13, 14] and a random incidence angle was assumed in [15]. However, accounting for mobility, which is an inherent feature of wireless networks, requires a more realistic and non-uniform model for users’ spatial distribution. Several mobility models, such as the random waypoint (RWP) model, have been proposed in the literature to characterize the spatial distribution of mobile users for indoor RF systems [16, 17]. However, these studies were limited to RF spectrum where statistical fading channel models were used. Recently, [18, 19] employed the RWP mobility model to characterize the signal-to-noise ratio (SNR) for indoor LiFi systems. In [18], the device orientation was assumed constant over time, which is not a realistic scenario, whereas in [19], the incidence angle of optical signals was assumed to be uniformly distributed, which is not a proper model for the incidence angle, since it does not account for the actual statistics of device orientation.
Device orientation can significantly affect the users’ throughput. The majority of studies on OWC assume that the device always faces vertically upward. This assumption may have been driven by the lack of having a proper model for orientation, and/or to make the analysis tractable. Such an assumption is only accurate for a limited number of devices (e.g., laptops with a LiFi dongle), while the majority of users use devices such as smartphones, and in real-life scenarios, users tend to hold their device in a way that feels most comfortable. Such orientation can affect the users’ throughput remarkably and it should be analyzed carefully. Even though a number of studies have considered the impact of random orientation in their analysis [20, 21, 22, 23, 24, 25, 26, 27], all these studies assume a predefined model for the random orientation of the receiver. However, little or no evidence is presented to justify the assumed models. Nevertheless, none of these studies have considered the actual statistics of device orientation and have mainly assumed uniform or Gaussian distribution with hypothetical moments for device orientation. Recently, and for the first time, experimental measurements were carried out to model the polar and azimuth angles of the user’s device in [28, 29, 30, 31]. It is shown that the polar angle can be modeled by either a truncated Laplace distribution for the case of stationary users or a truncated Gaussian distribution for the case of mobile users, while the azimuth angle follows a uniform distribution for both cases. Motivated by these results, the impact of the random receiver orientation on the SNR and the bit error rate (BER) was studied for indoor stationary LiFi users in [32].
Solutions to alleviate the impact of device random orientation on the received SNR and throughput were proposed in [33, 34, 35]. In [33], the impact of the random receiver orientation, user mobility and blockage on the SNR and the BER was studied for indoor mobile LiFi users. Then, simulations of BER performance for spatial modulation using a multi-directional receiver configuration with consideration of random device orientation was evaluated. In [34], other multiple-input multiple-output (MIMO) techniques in the presence of random orientation were studied. The authors in [35], proposed an omni-directional receiver which is not affected by the device random orientation. It is shown that the omni-directional receiver reduces the SNR fluctuations and improves the user throughput remarkably. All these studies emphasize the significance of incorporating the random spatial distribution of LiFi users along with the random orientation of LiFi devices into the analysis. However, proper statistical channel models for indoor LiFi systems that encompass both the random spatial distribution and the random device orientation of LiFi users were not derived in the literature, which is the focus of this work.

Against the above background, we investigate in this paper the channel statistics of indoor LiFi systems. Novel realistic and measurement-based channel models for indoor LiFi systems are proposed, and the proposed models encompass the random motion and the random device orientation of LiFi users. Precisely, the statistics of the line-of-sight (LOS) channel gain are derived for stationary and mobile LiFi users with random device orientation, using the measurements-based models of device orientation derived in [28]. For stationary LiFi users, the model of randomly located user is employed to characterize the spatial distribution of the LiFi user, and the truncated Laplace distribution is used to model the device orientation. For mobile LiFi users, the RWP mobility model is used to characterize the spatial distribution of the user and the truncated Gaussian distribution is used to model the device orientation. In light of the above discussion, we may summarize the paper contributions as follows.

• •

  For stationary LiFi users, two channel models are proposed, namely the modified truncated Laplace (MTL) model and the modified Beta (MB) model. For mobile LiFi users, also two channel models are proposed, namely the sum of modified truncated Gaussian (SMTG) model and the sum of modified Beta (SMB) model. The accuracy of the derived models is then validated using the Kolmogorov-Smirnov distance (KSD) criterion.

• •

  The BER performance of LiFi systems is investigated for both cases of stationary and mobile users using the derived statistical channel models. We show that the random orientation and the random spatial distribution of LiFi users could have strong effect on the error performance of LiFi systems.

• •

  We propose a novel design of indoor LiFi systems that can alleviate the effects of random device orientation and random spatial distribution of LiFi users. We show that the proposed design is able to guarantee good error performance for LiFi systems under the realistic behaviour of LiFi users.

• •

  The proposed statistical LiFi channel models are of great significance. In fact, any LiFi transceiver design, to be efficient, it needs to incorporate the channel model into the design. Therefore, having realistic channel models will help in designing realistic LiFi transceivers.

The rest of the paper is organized as follows. The system model is presented in Section II. Section III presents the exact statistics of the LOS channel gain. In Sections IV, statistical channel models for stationary and mobile LiFi users are proposed. Finally, the paper is concluded in Section V and future research directions are highlighted.
The notations adopted throughout the paper are summarized in Table I. In addition, for every random variable $X$, $f_{X}$ and $F_{X}$ denote the probability density function (PDF) and the cumulative distribution function (CDF) of $X$, respectively. The function $\delta(\cdot)$ denotes the Dirac delta function. The function $\mathcal{U}_{[a,b]}\left(\cdot\right)$ denotes the the unit step function within $[a,b]$, i.e., for all $x\in\mathbb{R}$, $\mathcal{U}_{[a,b]}\left(x\right)=1$ if $x\in[a,b]$, and 0 otherwise.

From a statistical point of view, the instantaneous location and the instantaneous orientation of the LiFi receiver are independent. Thus, the couples of random variables $\left(r,\alpha\right)$ and $\left(\Omega,\theta\right)$ are independent. In addition, based on the results of [38, 18], the random variables $r$ and $\alpha$ are independent, since $r$ defines the polar distance and $\alpha$ defines the polar angle. On the other hand, based on the results of [28], the angle of direction $\Omega$ and the elevation angle $\theta$ are also statistically independent. Therefore, the random variables $r$, $\alpha$, $\Omega$ and $\theta$ are independent. In addition, for both cases of stationary and mobile LiFi users, the random variables $\alpha$ and $\Omega$ are uniformly distributed within $[0,2\pi]$ [38, 18, 28]. However, this is not the case for the polar distance $r$ and the elevation angle $\theta$. In fact, as we will show in the following, the statistics of $r$ and $\theta$ depend on whether the LiFi receiver is stationary or mobile.

As stated in Section II, the LiFi receiver can be located anywhere inside the outer cell with radius $R_{\rm e}$. However, it is connected to the desired AP if it is located inside the LiFi attocell, i.e., if $r\in[0,R]$. In other words, in order to have a communication link between the desired AP and the LiFi receiver, the only admitted values of the polar distance $r$ should be within the range $[0,R]$. Due to this, we constrain the range of $r$ to be $[0,R]$, and therefore, the exact PDF of the polar distance $r$ becomes $\tilde{f}_{r}(r)=\frac{f_{r}(r)}{F_{r}(R)-F_{r}(0)}\mathcal{U}_{\left[0,R\right]}(r)$, where $F_{r}$ denotes the CDF of $r$. Consequently, the PDF of the distance $d=\sqrt{r^{2}+(h_{a}-h_{u})^{2}}$ is given by

where $d_{\min}=h_{a}-h_{u}$ and $d_{\max}=\sqrt{R^{2}+(h_{a}-h_{u})^{2}}$. On the other hand, consider the random variable $\cos\left(\Omega-\alpha\right)$ appearing in (6). Since $\Omega$ and $\alpha$ are independent and uniformly distribution within $[0,2\pi]$ and using the PDF transformation of random variables, $\cos\left(\Omega-\alpha\right)$ follows the arcsine distribution within the range $[-1,1]$. Thus, the PDF and CDF of $\cos\left(\Omega-\alpha\right)$ are expressed, respectively, as

Based on this, the exact PDF of the channel gain $H$ is given in the following theorem.

An approximate expression of the PDF of the LOS channel gain $H$ for the case of stationary LiFi users is given in the following theorem.

An approximate expression of the PDF of the LOS channel gain $H$ for the case of mobile LiFi users is given in the following theorem.

It is important to highlight here that, for $j=1,2,3$, the functions $h\mapsto\frac{1}{h^{\nu_{j}}}$ and $h\mapsto g_{j}(h)$ express respectively the effects of user mobility and the random orientation of the UE on the LOS channel gain $H$. At this point, the missing part is the functions $g_{j}$, for $j=1,2,3$, that provide the best approximation for the PDF of the LOS channel gain $f_{H}$. In the following, we provide two expressions for each function $g_{j}$ for $j=1,2,3$.

A detailed algorithm for implementing the proposed statistical channel models for indoor LiFi systems is presented in Algorithm 1.

For stationary LiFi users, Figs. 4 and 5 present the theoretical, simulated and approximated PDF and CDF of the LOS channel gain $H$, when a radius of the attocell of $R=1$m and $R=2.5$m, respectively. For both cases, two different values for the field of view of the UE were considered, which are $\Psi_{c}=90^{\circ}$ and $60^{\circ}$.

These figures show that the proposed MTL and MB models offer good approximation for the distribution of the LOS channel gain $H$. Analytically, in order to evaluate the goodness of the proposed MTL and MB models, we use the well-known Kolmogorov-Smirnov distance (KSD) [39]. In fact, the KSD measures the absolute distance between two distinct CDFs $F_{1}$ and $F_{2}$ [39], i.e.,

Obviously, smaller values of KSD correspond to more similarity between distributions. In our case, the KSD of the MTL and MB models are shown in Table III.

As it can be seen in this table, the maximum KSD value for the MTL and MB models are 0.0669 and 0.0444, respectively, which demonstrates the good approximation offered by the MTL and MB models.
For mobile LiFi users, Figs. 6 and 7 present the theoretical, simulated and approximated PDF and CDF of the LOS channel gain $H$, when the radius of the attocell is $R=1$m and $R=2.5$m, respectively. For both cases, two different values for the field of view of the UE were considered, which are $\Psi_{c}=90^{\circ}$ and $60^{\circ}$.

These figures show that the proposed SMTG and SMB models offer good approximation for the distribution of the LOS channel gain $H$. Furthermore, using the KSD metric, Table IV presents the KSD of the SMTG and the SMB models, where it shows that their maximum KSD values are 0.0238 and 0.0054, respectively. This result demonstrates the good approximation offered by the MTL and MB models.

Fig. 8 presents the average BER performance of the on-off keying (OOK) modulation versus the transmitted optical power $P_{\rm opt}$ for stationary and mobile users for the values of the attocell radius of $R=1$m and $2.5$m and for the values of the field of view $\Psi_{c}=90^{\circ}$ and $\Psi_{c}=60^{\circ}$. Considering stationary users, this figure shows that the BER results of the MTL and MB models match perfectly the simulated BER for both cases when $\left(R,\Psi_{c}\right)=\left(2.5\text{m},90^{\circ}\right)$ and when $\Psi_{c}=60^{\circ}$. However, for the case when $\left(R,\Psi_{c}\right)=\left(1\text{m},90^{\circ}\right)$, we remark that the average BER results of the MB model match the simulated BER better than the ones of the MTL model. This can be also seen from the values of the KSD in Table III, where we can see that the KSD of the MB model is lower than the one of the MTL model when $\left(R,\Psi_{c}\right)=\left(1\text{m},90^{\circ}\right)$. In other words, when $\left(R,\Psi_{c}\right)=\left(1\text{m},90^{\circ}\right)$, the MB model offers better accuracy than the MTL model. This is mainly due to the assumptions made for both models. In fact, when the radius of the attocell $R$ is small and by referring to (5), the random variable $\cos\left(\Omega-\alpha\right)$ is dominant in $\cos\left(\psi\right)$. Hence, assuming that the distribution of the random orientation of the UE can be approximated by a Beta distribution makes more sense. For mobile users, the same figure shows that the BER results of the SMTG and the SMB models match perfectly the simulated BER for both cases when $\left(R,\Psi_{c}\right)=\left(2.5\text{m},90^{\circ}\right)$ and when $\Psi_{c}=60^{\circ}$. However, for the case when $\left(R,\Psi_{c}\right)=\left(1\text{m},90^{\circ}\right)$, we remark that the BER results of the SMB model matches the simulated BER better than the ones of the SMTG model. Similar to the case of stationary users, the SMB model offers better accuracy than the SMTG model when $\left(R,\Psi_{c}\right)=\left(1\text{m},90^{\circ}\right)$ due to the assumptions made for both models.
Fig. 8 shows also two important facts about the BER performance of LiFi users. First, it can be seen that the BER performance degrades heavily when either the radius of the attocell $R$ increases or the field of view of the LiFi receiver decreases. Second, the BER saturates as the transmitted optical power increases. These two facts can be explained by the following corollary.

The result of Corollary 1 shows that, even when the transmitted optical power $P_{\rm opt}$ is high, the BER is stagnating at $\frac{F_{\cos(\psi)}(\cos(\Psi_{c}))}{2}$. This result is directly related to the cases when the AP is out of the FOV of the LiFi receiver. On the other hand, based on its expression in (19), $F_{\cos(\psi)}(\cos(\Psi_{c}))$ is a function of the attocell radius $R$ and the field of view of the receiver $\Psi_{c}$. Therefore, since $d_{\max}$ increases as $R$ increases, then $F_{\cos(\psi)}(\cos(\Psi_{c}))$ is an increasing function in $R$. In addition, since $x\mapsto F_{\cos(\psi)}(x)$ is a CDF, it is an increasing function, and due to the fact that $x\mapsto\cos(x)$ is a decreasing function within $[0,\pi/2]$, then $F_{\cos(\psi)}(\cos(\Psi_{c}))$ increases as $\Psi_{c}$ decreases. The aforementioned reasons explain the bad BER performance of the LiFi system when either the radius of the attocell $R$ increases or the field of view $\Psi_{c}$ decreases.
From a practical point of view, the above performance can be explained as follows. Recall that