Mixed RF-VLC Relaying Systems for High-Speed Rail Communication

In the proposed system setup, a communication link is established between base station (BS) and end-user (UE) via a relay node mounted on the high-speed train (HST). The information is conveyed over backhaul RF links between BS and relay and over VLC links between relay and UE inside the train. It is assumed that RF links are encountered with dual shadowing due to slow-moving vehicles and pedestrians. Moreover, the relay node is not able to estimate the channel information perfectly due to HST mobility. Firstly, the statistical characteristics, such as the probability density function (PDF) and cumulative distribution function of the BS-relay link, are derived under imperfect channel information. Then, the system performance is examined by deriving the two key metrics, outage probability, and average bit error rate. Furthermore, we investigate the secrecy performance of the proposed system when the RF eavesdropper overhears the link between BS and relay and the VLC eavesdropper captures the information via the relay-UE link. To this end, the secure outage probability is derived into a closed form. Our results reveal that the proposed system setup can be adopted as a network architecture for existing as well as for future HST networks.


I. INTRODUCTION
E XPLOSIVE growth of high-speed trains (HSTs), as well as a huge increase in demand for seamless Internet-based services and other emerging applications (e.g., augmented reality (AR), virtual reality (VR), and video-on-demand), has made the continuous provisioning of high-capacity access networks HST an exciting topic of research. The main objective is to provide a high quality of service (QoS), such as low latency and high data rates, to a large number of passengers crowded on the HST at train speeds of up to 500 kmph, similar to low mobility users (or pedestrian users) [1]. On the other hand, the high speed of HST causes serious problems such as the Doppler effect and penetration loss due to train coaches, limiting the QoS of mobile users. Therefore, traditional techniques, including mobile relay-based network architecture [2], beamforming, multiple-input multiple-output (MIMO) [3], and optical wireless (UE) link, which is emerging as a key technology in 5G. The mobile relay architecture has several benefits over direct HST communication between the BS and the UE. The first benefit is broad coverage with enough power to support the multiple antenna relay for HST communication. Secondly, it provides the opportunity to utilize the indoor VLC frequency band to serve a large number of users in a static environment. Moreover, the mobile relay HST system provides the advantage of channel estimation at the mobile relay instead of UE, which reduces the Doppler shift effect and improves the system reliability.

A. Literature Overview
In the literature, several efforts have been devoted to study the mobile relay networks [20], [21], [22], [23], [24], [25]. In [20], [21], massive MIMO systems were studied. The firstorder Gauss-Markov Rayleigh fading channel model is used to characterize the channel aging in [20], whereas a single-cell urban scenario was considered [21]. The impact of mobility on system performance (i.e., average bit-error-rate (BER) and outage) was determined in [22] for both amplify and forward (AF) and decode-and-forward (DF) relaying by adopting an autoregressive model over a fast-fading channel. The authors in [23] adopted a random waypoint (RWP) mobility model to study the mobile wireless network over a generalized η-μ fading channel. In [24], the impact of channel state information (CSI) on mobile relay-based dual-hop RF-VLC network was studied in which the source-relay link was assumed as Nakagami-m distributed while VLC was used to model relay-UE link. Furthermore, the authors in [25], [26] proposed the random direction (RD) mobility model to investigate cooperative HST communication over Nakagami-m and generalized κ-μ fading channels, respectively.
On the other hand, a few studies suggest a cryptographic encryption mechanism to secure the transmission between the BS and the train [27], [28]. However, it has been proved that the eavesdroppers can overhear encrypted messages if chip processing is achieved and the codebook is acquired. Physical Layer Security (PLS) has recently surfaced as a promising approach for achieving perfect secrecy in wireless communication systems. By capitalizing on the stochastic nature and time-varying characteristics of wireless channels, PLS eliminates the need for a secret key, as supported by reference [29]. Rather than relying on traditional encryption techniques, the PLS can be applied to HST systems to improve the safety of train transportation by securing the train control messages.
PLS has been extensively analyzed in literature for various relay-based dual-hop communication systems such as mixed RF-FSO, FSO-RF, RF-VLC, and VLC-RF, as evidenced by numerous studies [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. Specifically, [30] and [31] proposed simultaneous wireless information and power transfer (SWIPT) based dualhop mixed RF-FSO and mixed FSO-RF systems, respectively, utilizing power splitting methods and investigating the impact of SWIPT parameters on secrecy performance. On the contrary, several studies [32], [33], [34], [35], [36], [37] have concentrated on analyzing the secrecy performance of RF-VLC and VLC-RF systems. In [32], a dual-hop mixed RF-VLC system was examined under two scenarios: RF-side and VLC-side eavesdropping attacks. Closed-form expressions for the secure outage probability (SOP) and strictly positive secrecy capacity (SPSC) were derived. For a dual-hop mixed VLC-RF system, [33] and [34] determined the SOP in scenarios involving multiple users and an Intelligent Reflecting Surface (IRS), respectively. The impact of light energy harvesting on system security, as indicated by the SOP, was assessed in [35] and [36], where closed-form expressions were provided. Lastly, reference [37] investigated the secrecy performance of a cooperative hybrid VLC-RF system, taking into account the random positions of the relay, legitimate receiver, and eavesdropper in different regions. Moreover, [38] and [39] proposed PLS approaches to secure the transmission link between the BS and HST by applying artificial noise and nonorthogonal multiple access (NOMA) techniques, respectively. However, none of the aforementioned studies address secrecy issues in HST communication that involve mixed RF and VLC links between the BS and UEs. It is noteworthy that transmission over VLC links plays a crucial role in green communication, as it can also be used for illumination purposes simultaneously.

B. Motivation and Contributions
Motivated by the aforementioned studies, the RF and VLC technologies are combined together to develop a relaying system suitable for HST communication. In the proposed system, a relay mounted on the HST serves as an intermediate node between the BS and onboard UEs and receives and transmits the information by exploiting DF relaying scheme. BS transmits signals over a strong line-of-sight (LoS) link with minimal signal scattering. Consequently, the BS-relay link can be represented by a double-shadowed Rician (DSR) fading distribution, which has been demonstrated to be highly effective in modelling strong line-of-sight (LoS) propagation with minimal scattering. Another reason for choosing the DSR fading distribution for the BS-relay link is that it can capture the effects of slow-moving vehicles and pedestrians on the LoS path. Moreover, the DSR distribution can be used to characterize the realistic channels in the frequency band of 2.45 GHz, which is a key technique for long-term evolution (LTE) [5], [6], [7], [8], [9]. It also describes different fading scenarios as special cases for which the values of fading parameters can be found in [31, Table II]. The relay has the capability of converting RF signals into optical signals and transferring the information over the VLC link, which is established between the relay and the UE. This means that the proposed system employs a dual-hop communication scheme where the relay receives an RF signal from the BS and converts it into an optical signal using the DF strategy, which is then transmitted to the onboard UE. The primary contribution of this paper is the introduction of a mixed RF-VLC system for HST communication. Specifically, the system performance under outdated CSI at the relay is examined, with a focus on   [25], [26], the impact of train mobility is described by the outdated CSI at the relay. The estimated channel information is used to derive statistical characteristics, such as the probability density function (PDF) of the received instantaneous SNR at the relay. The derived results facilitate obtaining the closed-form expressions for outage probability (OP) and average BER. Additionally, further insights are revealed by deriving the asymptotic outage at high large light-emitting diode (LED) power and BS power. 3) Different from [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], a relay-based mixed RF-VLC system for HSTs is studied in this work, in which the SOP is derived for end-to-end SNR of the BSrelay-UE link. The results in our work are presented for both assumptions of RF-side and VLC-side eavesdropping attacks. 4) We demonstrate the numerical results to reveal the effects of various system parameters on the performance of the proposed system.

II. SYSTEM AND CHANNEL MODEL
As shown in Fig. 1, a system setup is developed for HST communication based on a mixed RF-VLC relaying system in which a DF relay R receives signals from BS S via an RF link and forwards it to the UE D over the VLC link. R is equipped with a LED that converts electrical signals into optical signals. It is assumed that two adversaries, i.e., E1 and E2, are present on the RF link and VLC link, respectively, where E1 can hear the RF link, whereas E2 is capable of overhearing optical signals and positioned close to UE. BS and E1 are static, while relay, UE, and E2 are present on the train and moving with HST speed. It is assumed that BS and UE cannot communicate over a direct path due to the obstacles, and hence, BS sends information to UE through a relay mounted on HST. R receives signals from BS through an RF link which is decoded and then forwarded to the UE via a VLC link. R is capable of converting perfectly 2.45 GHz signals into nanometer Wave signals. We assume that BS-relay and BS-E1 links experience the quasi-static DSR fading channel. Notably, E1 and E2 are not present when performance metrics are derived, while secrecy performance is investigated in the presence of E1 and E2.

A. BS-Relay and BS-E1 Channel Models
where P s is the transmit power at the BS, h SR is denoting the estimated DSR fading coefficients of BS-relay link and z SR denotes zero-mean complex valued Gaussian noise with variance σ 2 SR . Since the relay is dynamic and moving with the HST at the same speed, it cannot estimate the channel perfectly. Therefore, the estimated channel gain is outdated and correlated with precise channel gain (i.e., h SR ) of BS-relay link, which can be written as where ε SR is zero-mean Gaussian random variable with unit variance and is uncorrelated with h SR . ρ SR is the correlation coefficient between h SR and h SR and can be characterized by using Jakes' autocorrelation model as, where J 0 (•) is the zeroth order Bessel's function of first kind, T s denotes the transmitted symbol duration, f D = (f c v/c) describes the Doppler shift frequency where f c is the carrier frequency and c represents the speed of light. Note that if the train is static, ρ SR = 1 otherwise, 0 < ρ SR < 1.
The instantaneous SNR of the BS-relay link can be expressed asγ whereγ SR is the average SNR and defined asγ SR = P s σ 2

SR
. The PDF and cumulative probability distribution (CDF) ofγ SR are derived in the following Lemma.
Lemma 1: The PDF and CDF of γ SR are obtained as where m s denotes inverse Nakagami-m parameter, m d denotes Nakagami-m parameter, k is Rician parameter, r is the mean-square value of r, Ω =r 2(1−ρ 2 SR )γ SR , is Proof: Please see Appendix A. Remark 1: It can be observed from (6) that OP of BS-relay link occurs zero floor, which is equal to 1 − ξ at γ SR → ∞. This is because ξ is independent of γ SR . In other words, h SR is always positive for practical scenarios while ε SR can be negative or positive.
Remark 2: It is worth pointing out that the two-fold infinite summation in (5) and (6) are convergent and can be truncated only for a few terms of (p, q). This can be followed from the theorem that if ∞ q=0 a pq converges for q ∈ N to a non-negative number b p and ∞ p=0 b p converges for p ∈ N, then ∞ p=0 ∞ q=0 |a pq | also converges for a finite number of (p, q). One can apply the d'Alembert's ratio test to testify to the convergence of infinite series representation of (5) and (6). Therefore, we require only a finite number of terms to achieve the high-precision sum results. For instance, the test for the convergence of the series in (6) is available in Appendix B.
Since E1 is static and hears the transmission between BS and relay, it can estimate perfect channel parameters. Therefore, the received signal at E1 can be written as [36] where z SE1 denotes zero-mean complex valued Gaussian noise with variance σ 2 SE1 . From (5), the instantaneous SNR of the BS-E1 link can be written as whereγ SE1 is the average SNR of the BS-E1 link and can be defined asγ SE1 = P s σ 2

B. VLC Channel Model
Inside the HST, the signals received at the UE and E2 are given as where t ∈ {RD, RE2}, x is the non-negative emitted intensity by the LED, which is upper bounded as E[x ] = P opt due to safety concerns, P opt is the transmitted optical power and z t denotes mean zero complex valued Gaussian noise with variance The end users (i.e., D and E2) are illuminated by the LED source at illumination angle ψ t and irradiance angle θ t and positioned at a height L from the receiving plane, as shown in Fig. 1. The location of the end users (i.e., D and E2) can be determined with the help of Euclidean distance d t , angle ϕ t and radius r t on the polar coordinate plane.
In indoor VLC subsystem, the channel gain of the tth link can be written from [24] as where Λ denotes the physical area of the photodetector, T represents the optical filter gain, g is the photodetector's concentrator gain, which is related to the field of view (FOV) of the receiver Ψ and refractive index of the lens ζ as Moreover, it is postulated that the surface plane of the photodetector is oriented parallel to the ground plane, then θ t = ψ t , d t = r 2 t + L 2 , and cos(θ t ) = L √ r 2 t +L 2 . Then, (12) can be expressed as where . The position of the tth user can be characterized using uniform distribution, therefore, PDF of r t is expressed as f r t (r) = 2r r 2 ω . Thus, the PDF of h t can be derived with the help of change of variable method from (13) as and h t max = χ t /L ω+3 . The PDF and CDF of the corresponding SNR can be obtained from (14) as follows where where η is the electrical to optical conversion efficiency, and B denotes the baseband modulation bandwidth.

III. PERFORMANCE ANALYSIS
In this section, we investigate the performance of the proposed dual-hop mixed RF-VLC HST system when no eavesdropper is present. It is assumed that DF relaying scheme is utilized, the end-to-end SNR of the relay-UE link, referred to as γ eq , can be written as

A. Outage Probability (OP)
Generally, the OP is defined as the probability of an event in which instantaneous SNR γ eq is smaller than the predetermined threshold SNR γ th . Mathematically, OP can be expressed as Theorem 1: The OP for the dual-hop mixed RF-VLC HST system can be obtained from (18) with some mathematical manipulation as γ th ∈ C 4 (19) Proof: By using the (6) and (15), the terms in (18) can be evaluated as following.
Using [38, Eq. (2.01.1)], we can achieve where The proof is completed by substituting (20) and (22) into (19). Remark 3: For high transmitted BS power i.e., P s → ∞, it is concluded thatγ SR → ∞. From (19), it becomes evident that in this scenario, the instantaneous SNR at D is equivalent to that of the VLC link, i.e., γ eq ≈ γ RD . This implies outage performance of the proposed system is independent of the RF backhaul link and can be controlled by the relay-UE link in a high-power regime. The asymptotic OP can be written as Remark 4: At a high LED power regime, i.e., P opt → ∞, only the BS-relay link remains dominating as the instantaneous SNR of the VLC link reaches its high value, i.e., γ RD → ∞. From (19), it is concluded that γ eq ≈ γ SR . The OP for P opt → ∞ is determined as Similar to Remark 3, the asymptotic OP depends on the RF subsystem of the proposed HST system and becomes independent of VLC fronthaul.

B. Average Bit-Error-Rate (BER)
In the DF relay-based systems, the binary modulated data is decoded at the relay and then transferred to the destination after re-encoding. The received data, either at the relay or the destination, is error-free and erroneous. In this case, the decoding at the relay and the destination are independent of each other. Considering the decoding errors at the relay, we derive the average BER expression for the proposed system when binary phase-shift keying (BPSK) or differential BPSK (DBPSK) is applied. The average BER of a wireless system can be written in terms of the CDF of γ f from [24,Eq. (47)] as where f ∈ (SR, RD). The different values of (a, b) parameters are used to define the modulation schemes. For example,(a, b) = (0.5, 1) for BPSK and (a, b) = (1, 1) for DBPSK.  (26) where P SR e and P RD e denote the average BER of the RF and VLC subsystems, respectively, and are derived in (27) and (28).

Remark 5:
One can obtain the average BER for a mixed RF-VLC HST system considering perfect relaying or error-free decoding at the relay by subtitling the CDF of γ eq into (25). The CDF of γ eq can be written as where F γ SR (γ) and F γ RD (γ) are given in (6) and (16), respectively.
Remark 6: Similar to OP, the asymptotic expressions for average BER can be obtained for P s → ∞ and P opt → ∞ in (29) and (30), respectively.

IV. PHYSICAL LAYER SECURITY ANALYSIS
In this section, we derive the SOP by considering the presence of active eavesdroppers on BS-relay link and relay-UE link, respectively. For this scenario, the security is compromised if the instantaneous secrecy capacity i.e., C o is less than the predefined constant rate i.e., R o and otherwise perfect secrecy is achieved. As the proposed system is utilizing the DF relaying protocol, the capacity of the system is dominated by the worst hop and expressed as where C SR and C RD present the capacity of BS-relay and relay-UE links, respectively. One of the common and useful secrecy performance metric SOP can be defined as the probability of C o ≤ R o and is expressed as [30] Under DF scheme, both links (i.e., BS-relay and relay-UE) are independent of each other. Therefore, (32) can be rewritten by using basic probability theory as where

A. Eavesdropping on BS-Relay Link
In this case, it is assumed that a static eavesdropper E1 is present on the BS-relay link and can hear the secure transmission through RF link, then the instantaneous capacity of the BS-relay link can be written as Similarly, the capacity of the second hop can be written from [32] as Theorem 3: The lower bound on SOP for DF mixed RF-VLC HST system can be obtained under the RF eavesdropping attacks as given in (45).
Proof: Substituting (36) into (34), we have (38) can also be represented in terms of predetermined threshold γ th as R o = 1 2 log 2 (1 + γ th ). (38) further can be expressed in terms of PDF and CDF as (39) Since (39) cannot be obtained in exact closed-form, the upper bound of (39) can be derived by utilizing a similar method presented in [32] as follows: By substituting (6) and (9) where Θ = . Now, substituting (37) into (35), we have From (15) and (42), we have Using [38, (2.01.1)], (43) can be expressed as where Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Finally, the lower bound on SOP is deduced in (45) by substituting (41) into (44) in (33) as where 2 , m s SE1 , 0 . Remark 6: It can easily be found from (45) that smaller values of R o will lead to smaller SOP while the larger values of R o will result in a larger SOP. This is because, at larger R o , the possibility of outage occurrence on the relay-UE link will be increased. Therefore, an optimal R o can be obtained to minimize the SOP and maximize the secrecy throughput by utilizing the root-finding method [43].
Remark 7: Since only the BS-relay link is under eavesdropping attack, the secrecy performance is significantly affected by the variation in Doppler shiftf D T s or the speed of HST. The larger f D T s leads to higher estimation errors at the relay and hence results in low SNR at the relay. The effects of f D T s can be mitigated by varying the values of parameters controlling the relay-UE links, such as FOV and height of LED from the UE L only when 2 2R o − 1 ∈ C 3 otherwise, the secrecy will be compromised with the larger f D T s .
Remark 8: Considering the case of P s → ∞, (41) becomes independent of γ SR i.e., Pr(C SR ≥ R o ) ≤ ξ. Therefore, the asymptotic expression for lower bound on SOP can be written from (45) as 46) Similar to (24) and (30), it can be shown that lower bound on SOP for P opt → ∞ is given by

B. Eavesdropping on BS-UE Link
This is the case when eavesdropper E2 is present on the train and wiretapping the relay-UE link. This system setup is similar to the scenario of the indoor VLC system as presented in [24], [32], [37]. In this case, the capacity of the relay-UE link can be expressed as [32] and the capacity of the BS-relay link is given as Theorem 4: By considering the presence of an eavesdropping attack on the indoor VLC link inside HST, the lower bound of the SOP is derived in (58).
Proof: On substituting (48) into (35), we have In indoor VLC, it is reasonable to assume that γ RD >> 1 and γ SE1 >> 1. Therefore, the upper bound on (50) is given by Since γ RD and γ SE1 are independent of each other, the solution of (51) depends on γ th . Therefore, we define the four different scenarios in Fig. 2 to calculate the integral regions of Pr(γ RD < γ th γ SE2 ). Based on the considered scenarios, (51) is derived as follows: where With the help of [42, Eq. (2.111.1)], the integrals in (53) and (54) are deduced as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Now, by substituting (49) into (34), we obtain By substituting (52) and (57) into (33), then the lower bound on SOP can be written as Remark 9: From (58), it can be seen that SOPs with lower σ 2 RE2 or larger σ 2 RD are higher. This is because a lower value of σ 2 RE2 results in a higher channel gain h RE2 , and a larger σ 2

RD
leads to a smaller channel gain h RD . This can also be observed from (52) that the integral range of Pr(γ RD < γ th γ SE2 ) is enhanced for lower σ 2 RE2 or larger σ 2 RD . Remark 10: The SOP in (58) is enlarged for larger values of γ th and if γ th ≥ γ RDmax γ RE2 min , the SOP reaches a maximum value of 1and hence information can never be transmitted securely.
Remark 11: One can derive the SOP for the case of simultaneous eavesdropping attacks on RF and VLC links by using the similar approach adopted to achieve (45) and (58). In this case, the instantaneous capacity of the corresponding links can be expressed as given in (36) and (48). Then, by substituting (41) and (51) into (33), the lower bound on SOP can easily be obtained.

V. NUMERICAL RESULTS AND DISCUSSIONS
This section presents the analytically evaluated results for the proposed dual-hop mixed RF-VLC high-speed train (HST) system with varying system parameters, such as Doppler shift and RF and VLC fading parameters, and different system configurations. Simulation results are provided for verifying the correctness of the derived results. Firstly, we examine the effect of Doppler shift and fading parameters on the proposed system's performance, followed by an analysis of its secrecy performance in the presence of eavesdroppers. The system parameters utilized in the analysis are drawn from various sources, including [15], [23], [36], [38], and are listed in Table I.

A. Characteristics Performance of Mixed RF-VLC HST System
The outage probability of the considered system derived in (19) as a function of transmitted power at the BS under varying Doppler shift is plotted in Fig. 3. We consider four different values of Doppler shift as {0.2, 0.1, 0.08, 0}, which correspond to different train speeds {400, 200, 150, 0} in kmph. This means that higher train speeds provide larger f D T S as they are related to each other as given in (3). The result clearly shows that outage is enhanced with increasing f D T S . This is because estimation of the fading channel at the relay is accurate at relatively smaller f D T S . On the other hand, larger f D T S leads to higher channel estimation errors, and hence the performance degrades significantly. The effect of f D T S can be mitigated on the cost of increasing transmitted power at BS, as shown in the result. However, the outage probability decreases to an error floor on high transmitted power due to channel estimation errors and dominance of the VLC link, which is derived in (23). Our results are consistent with the results in [3], [16], [40], [41], as the  effects of the Doppler shift on system performance in our study are similar to these studies.
Similar behavior of outage performance under varying Doppler shifts is presented in Fig. 4, which demonstrates the outage performance as a function of optical power. With increasing P opt , the decrement in an outage probability is noticed. Similar to Fig. 3, an error floor is noticed for high values of optical power. Hence, further increments in optical power will not result in an improvement in system performance. Fig. 5 depicts the outage probability for various RF fading conditions. One can see that severe fading conditions (i.e., low m s and m d ) will result in a smaller OP, meaning better system performance, since higher m s or m d lead to more diversity gains. The reason for this is that scattered waves or LoS signals are perturbed and heavily shadowed at the low values of m s or m d . This condition may arise when the number of slow-moving vehicles or pedestrians is increased in the vicinity of BS or when BS is installed in a crowded area. It is noted that stronger scattered waves provide larger SNR at the relay and are beneficial for the enhancement of the system performance. The trend of this result is also supported by [31] and [46]. Fig. 6 illustrates the dependence of average BER on the distance between LED and receiving plane, considering different  Doppler shifts. It can be observed that average BER is enlarged for higher L. This is because the larger L represents a longer path of optical signal propagation, resulting in a reduction in received power at the UE. Moreover, when the UE is closer to the relay, the BS-relay link conditions have a stronger impact on the average BER at low transmitted power at BS. Hence, a crossover point can be observed in the low transmitted power regime, while in the high transmitted power regime, the closer the UE to the relay, the less the average BER is achieved. Additionally, due to the larger channel estimation errors at the higher speed of the train, the system performance has become worse. Additionally, it is worth noting that the presence of channel estimation errors can cause the performance of the BER to converge to a discernible error floor. Fig. 7 shows the secrecy performance under BS-relay link eavesdropping for different values of average SNR of BS-E1 channel as a function of transmitted power at BS. As expected, the security of the proposed system is enhanced significantly with increasing P s or decreasingγ SE1 , as testified in [30]. The reason is that the SNR at the relay is improved with increasing P s , while the SNR at the E1 is reduced with decreasingγ SE1 due to the higher noises in the BS-E1 link. Thus, from (36), the difference between the channel capacities of the BS-relay and BS-E1 links is increased, and hence SOP is reduced which can be seen from (33). Another interesting observation that can be noted from the result is that a smaller Doppler shift has an insignificant impact on SOP in the low P s regime, while SOP is enhanced with an increasing Doppler shift in the high P s regime. This is because the difference between channel capacities of the BS-relay and BS-E1 links is very small, with lower P s resulting in a larger SOP. Fig. 8 depicts the secrecy performance under relay-UE link eavesdropping for different eavesdroppers' noise and predetermined secrecy rate as a function of transmitted power at BS. We can see that the secrecy performance is improved with the increase in σ 2 RE2 , which coincides with Remark 9. This is because a larger σ 2 RE2 will result in a smaller SNR at the E2, and hence the difference between the channel capacities of relay-UE and relay-E2 links is increased, which can be calculated from (48). This implies that the larger the noise present in the relay-E2 link leads to lower SOP as SOP depends on the difference between the channel capacities of the relay-UE and relay-E2 links. Moreover, the secrecy performance can be improved by controlling the predetermined secrecy rate R o , as one can easily find from (33) that a larger predetermined secrecy rate increases the chances of eavesdropping. At larger R o , the BS has no choice other than to increase its transmission power to raise the channel capacity of the legitimate link, which may compromise the security of the system. Additionally, the impact of the Doppler shift remains similar to the previous case because channel estimation errors at the relay are forwarded the same to both UE and E2.

VI. CONCLUSION
In this work, RF and VLC technologies are combined together to support communication between a BS and a UE present on the HST, where it is assumed that the BS-relay links are subject to DSR fading while the onboard passengers are served with VLC links. The impact of the speed of HST is considered in terms of imperfect CSI at the relay, where the correlation between estimated channel gain and precise channel gain is characterized with the help of Jake's model. Particularly, a performance framework is presented to investigate the effects of the Doppler shift on HST communication. First, the statistical characteristics, including PDF and CDF, for the proposed system were derived. Subsequently, we proceeded to derive the key performance metrics, including the outage and average BER, along with the corresponding closed-form expressions for the secrecy metric, i.e., SOP, accompanied by their respective asymptotic expressions. This study highlights the advantages of using the proposed mixed RF-VLC system for dealing with future challenges in HST communication, including spectrum scarcity and eavesdropping attacks.

APPENDIX A
The PDF of h SR is given by [31] f h SR (r) SR ε SR and Z = h SR , one can easily obtain the PDFs of X and Y as Based on convolution theorem, the PDF of Z can be written as Representing exp(−x) into Meijer's G function from [36, (8.4 Substituting (A.6) into (A.4), we have where G pq = G 1,2 2,1 (1+k SR )2(1−ρ 2 SR ) (m s SR −1) r 2 ρ 2 SR −m s SR + 1, −q/2 + 1 p + 1 .
In practical fading models, the channel gain coefficients must be positive. Keeping this in mind, we can rewrite (A.7) as With the help of [38, Eq. (3.326. 2)] and using ∞ 0 f h SR (z)dz = 1, ξ can be obtained as Finally, using change of variable based on (4), we obtain the PDF in (5) and then the CDF is obtained in (6) The ithand jth terms in the series expansion of CDF of DSR distribution are Therefore,