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Abstract— The demand for access to broadband data services in high speed trains is increasing as

more people are travelling to and from work, which is not met by the existing radio frequency (RF)

technology. Therefore an alternative technology known as free space optics (FSO) could be readily

adopted that could overcome the bandwidth bottleneck problem. The paper presents a mathematical

model of an FSO link for ground-to-train communications link and analyses the system performance

in terms of the signal to noise ratio and the bit error rate (BER). We show that the simulated BER is in

good agreement with the predicted results for bit rates up to 50 Mbps. The link budget analysis for the

proposed system is also presented showing a link margin of 17.75 dB.

Index Terms—communications, free space optical, Gaussian beam, ground-to-train.

1. INTRODUCTION

Free-space optical (FSO) communications is an alternative wireless access technology to the existing radio

frequency (RF) wireless systems. FSO also known as the optical wireless communications (OWC) have

multiple advantages that can complement the existing RF links such as a huge unregulated spectrum,

immunity to electromagnetic interference, high security since optical beams do not penetrate opaque objects

and frequency reuse resulting in a high capacity per unit volume [1-5]. The FSO system is the preferred

option where there is limitation or restriction in the use of RF based systems in application including

hospitals, airplane, military where RF interferes with monitoring equipment. The FSO technology can be

utilised in both indoor and outdoor environment capable of supporting very high data rates up to Gigabit per

second. Various works [3, 6, 7] have reported indoor FSO communications with data rates beyond 1 Gb/s.

For outdoor systems, 100 Gb/s per channel link is reported in [8] whereas 1.28 Tb/s FSO link (32×40 Gb/s)

Modelling of Free Space Optical Link for

Ground-to-Train Communications Using a

Gaussian Source

Rupak Paudel, Zabih Ghassemlooy, Hoa Le-Minh,S. Rajbhandari

Optical Communications Research Group, NCRLab, Faculty of Engineering and Environment,

Northumbria University, Newcastle upon Tyne, UK

is reported in [1] using the dense wavelength division multiplexing (DWDM). FSO supports a number of

modulations, forward error correction coding schemes and protocols for a variety of applications, including

voice and multimedia services. FSO is also designed to support a large number of users, with multiple

connections per terminal, each with its own quality of service requirement. In indoor FSO systems mobility

offered is not as advanced as in the RF technology. The FSO technology is relatively young but mature and

its widespread deployment will in parts depend on (i) the cost of devices as compared to the RF technology,

and (ii) how the RF dominated industry will see FSO not as a threat but a complementary technology to RF

in areas where there is a need for a high speed link.

Due to the exponential growth of handheld devices such as smart-phones or tablets, there is a growing

demand for high speed internet connections in trains, ships, buses, etc. The existing infrastructure based on

the RF technology such as Wi-Fi/WiMAX used in trains are capable of delivering, theoretically, peak data

rates up to 54/75 Mbps, where in real scenario this could be lower than 10 Mbps at the best of times [9]. This

of course is [10] could

be improved by increasing the carrier frequency, thus by adopting the millimetre wave technology beyond

the 60 80 GHz band. In the future, the offered internet facilities are expected to be falling short of the

increasing demand for high-quality multimedia type services in train. One possible and viable solution to

address the demand for higher data rates is the FSO. A ground-to-train communications system using the

FSO technology is proposed in [11] where a tracking control algorithm is used to establish a stable

communication link between the mobile unit and the ground. In [12], an FSO system with a faster handover

mechanism (124 ms) is proposed to achieve a data rate in excess of 500 Mbps in high speed trains. Although,

practical based data are reported by [11, 12], a detailed mathematical modelling of the FSO ground-to-train

communication link, which is essential for system modelling and performance analysis has not been

addressed. In our previous work [13], we have reported a mathematical model for the FSO ground-to-train

system using a Lambertian source. In such systems, laser sources are the preferred option for higher power

requirements and longer coverage length and can be modelled as Gaussian. This paper outlines a

mathematical model for the FSO ground-to-train communications link and introduces a new expression for

the received optical power based on the proposed geometrical model. The proposed system would be a

complementary technology to RF based schemes providing higher data rates for the end users. In this

proposed system, intensity modulation with direct detection (IM/DD), which is the most popular method in

FSO, is incorporated due to its simplicity and low cost. The rest of the paper is organised as follows: Section

2 discusses the proposed system along with the mathematical model, Section 3 presents system design,

Section 4 presents results along with discussion and Section 5 concludes the paper.

2. SYSTEM MODELLING

A typical ground-to-train FSO communication link is shown in Fig. 1. The link consists of optical

transceivers (Tx/Rx) positioned on the roof of the train and base stations (BSs) positioned alongside the train

track. Each BS emits a narrow width optical beam fully covering the entire train. BSs are only active when

the train is within its transmission range and the optical footprint, otherwise they are in the off mode to save

(a)

(b)

Fig. 1(a): A typical ground-to-train communications system and (b). Front view of the proposed system.

energy. Exchange of information from BS to the service provider will take place via the fibre-optic backbone

network, which is normally laid down along the rail tack as in UK.

The proposed geometrical model of the downlink communications for an over-ground train for a straight

track is depicted in Fig. 2. Usually for a straight track, the maximum span between the power overhead lines

(gantries) is 75 m [14], which is used as the spacing between the BSs in this analysis. Of course the BS

separation can be extended to few hundred metres for a longer track. In this scenario, one could either

increase the transmit power at the BS within the eye-safe limit, which is further explained in section 3.3 or

move to a longer wavelengths of 1300 nm and 1550 nm where eye safety is not a major issue compared to

lower wavelengths.

In this section, we will model the available received optical power distributed along the track length L, see

Fig. 2. The BS is positioned a metre away from the track, and is adjustable. Let the half angle divergence be

1/2 (i.e., = 21/2). The BS transmitter (A), which is at the same height as the optical Tx/Rx (i.e. ~ 4 m above

the ground level), could be tilted by an angle 1/2+ represented by along the horizontal plane so that it

points towards the optical Rx.

CD

(d2) is the horizontal separation distance between BS and the shortest

coverage point C, which .

AD

(d1) is fixed at 1 m (BS separation from the track), is the coverage angle at the longest point B and is the

coverage angle at the shortest point C. Using a simple geometry, and for ACD and ABD

respectively, we have

and

.The estimated transmitter beam divergence angle can be

(a)

(b)

Fig. 2 (a). Proposed system geometrical modelling and (b). System geometry.

written as:

Hence, based on the position of the BS and the effective coverage length, the beam divergence angle can

be approximated. The optical beam radius of a Gaussian beam is given as [15]:

where wo is the beam waist of the laser source at the transmitter, z is the axis of propagation and is the

operating wavelength of the optical source.

In Fig. 2, let

AC

be denoted by x (= ). The OHB (see Fig. 2(b)) can be written as (1/2 + ) and

represented by . The following analysis is performed in order to derive a general equation for the power

received along the track. Here, the length along the axis of propagation,

AO

(z) would be related to length

CB, which is the effective coverage length L so that the power along L could be determined. From Fig. 2(b),

z = AH + HO and the length AH can be written as AG + GH. Similarly, length HO can be written as HO = (L

- CH) cos. Hence z can be given as. Using basic geometry, z can be written

as:

Rewriting the beam radius from (2) in terms of the effective coverage length along the track using (3) as:

OHB, denoting OB as the offset r from the axis AO at the point B, we get

Hence, the radial offset from the axis of propagation AHO orthogonal to the axis can be

written as:

The received power at the receiver along the z axis for a Gaussian beam is given by [15]:

where Ptx is the total transmit power from BS and Acoll is the collection area at the receiver. Using (4), (5) and

(6), Ptx along the track

CB

, when the BS is positioned at a distance d1 from the track based on the Gaussian

beam profile, is given as:

Once the average optical transmitter power, the collection area of the receiver, beam divergence, the tilting

angle of the receiver at the longest coverage point, the BS position and the effective coverage length are

determined, the received power can be evaluated using (7). The angular parameters for the model and the

system model parameters used for the simulation are given in Tables 1 and 2 respectively.

3. SYSTEM DESIGN

This section describes the optical wireless link for the proposed system, which consists of an optical

transmitter and a receiver and a wireless communications channel. Also the eye safety analysis is discussed

for the average transmitter power and the link budget analysis is performed for the system.

3.1. Transmitter

The transmitter comprises of a laser source and a laser driver, which can be modulated using the most

common modulation format non return-to-zero (NRZ) on-off keying (OOK). The transmitter parameters

TABLE 2

SYSTEM MODEL PARAMETERS

Symbol

PARAMETERS

Value

Operating wavelength

850 nm

d1

Vertical position of BS

1 m

d2

Horizontal BS position

15 m

Adet

Photodetector area

7 mm2

Ptx

BS optical transmit power

15 mW

Sr

Receiver sensitivity

-36 dBm @

10 Mbps

R

Responsivity

0.59 A/W

Rcoll

Radius of the optical

concentrator

25 mm

f

Focal length of the lens

50 mm

n

Refractive index of the

telescope

1.5

L

Coverage length

75 m

D

Source diameter

5 mm

Pn

Noise power

10 µW

such as the beam divergence and the average transmit power for the system is estimated based on the

proposed geometrical model. The beam divergence is estimated from the positions of d1 and d2 and L as

given by (1). The transmit power at the BS can be derived from (6) by replacing the received power with the

receiver sensitivity and for L = 75 m.

3.2. Receiver

The receiver positioned on the roof of the train will be tilted at an angle . This is also the angle made by

the propagation axis with the train track so that the FOV of the receiver at both the longest and shortest

points B and C, respectively will be within the beam divergence of the transmitter. The receiver incorporates

an optical concentrator, a photodiode and receiver electronics. The concentrator collects and focuses the

incoming light onto the photodiode whereas the receiver electronics is used to recover the signal. The

concentrator gain at the receiver can be evaluated as [16]:

where n is the refractive index of an optical concentrator and c is the half-angle FOV of the receiver after

the lens. The FOV of the receiver using the optical concentrator is given by [17]:

where Acoll is the effective light collection area of the receiver and Adet is the area of the photodetector. The

proposed system parameters are tabulated in Table 1. The noise source at the detector is the combination of

the shot noise, the thermal noise [18] and the background noise [19]. The typical value for the background

radiation used is 10 µW [20, 21]. The total noise variance can be written as:

where

,

and

is the variance due to shot noise, thermal noise and the background noise

TABLE 1

ANGULAR PARAMETERS

Symbol

Parameters

Calculated

[deg.]

Beam divergence

3.20

c

Receiver FOV

5.15

Coverage angle at B

0.65

Coverage angle at C

3.85

Tx/Rx tilting angle

2.25

respectively. The total noise present at the detector can be modelled as the additive white Gaussian noise

(AWGN). The signal-to-noise ratio (SNR) at the receiver is hence given by [16]:

The BER for OOK-NRZ is then evaluated as:

where

3.3. Eye safety

In order for the system to operate in public places, the utilised optical source should conform to the

international eye safety standards [22]. The Acceptable Emission Limit (AEL) is determined by the angular

subtense angle and the operating wavelength. The angular subtense of the apparent source should be

calculated in order to classify the laser source used based on Fig. 3, which depends on the source diameter

as:

where rmeasure ( = 100 mm) is the measuring distance and D is the source diameter. D is assumed to be 5 mm,

therefore the source can be classified as an extended source since the angular subtense can be calculated as

50 mrad as given by (13min max of 1.5 mrad and 100 mrad, respectively. With

Ptx = 15 mW, which is below the AEL limit of 20 mW for an exposure time of 100s for extended source

when the operating wavelength is 850 nm, the system proposed conforms to the eye safety standards. If

higher power transmission is required, we can move to higher wavelengths around 1300 nm where the AEL

limit increases by a factor of 20 as compared to that of 850 nm. In this work, 850 nm wavelength is adopted

as this is the most commonly used window for optical communications where the components available are

the cheapest [23].

Fig. 3. Angular subtense measurement.

3.4. Link budget analysis

The link budget analysis is performed in order to evaluate the system link margin after taking into account

losses associated with the system. The losses considered are the atmospheric loss due to weather conditions,

the geometrical loss Lgeom, the pointing loss Lpt, the transmitter loss Ltx and the receiver loss Lrx. The

attenua [24]. The link visibility is derived

from the fog attenuation using the Kim model to reflect the attenuation in dB/km as [2]:

where V q is the size distribution of

scattering particles and is given by [25]:

(15)

Using (14) for a moderate fog, i.e visibility of 500 m, fog attenuation is calculated as 34 dB/km. The other

loss in the system is due to the spreading of the transmitted beam as it propagates through the atmosphere

known as the Geometrical loss. Geometrical loss can be approximated by the following expression for a

uniform transmitter power distribution [26]:

where dr is the receiver aperture diameter, and dt is the transmitter aperture diameter. Typical

transmitter/receiver loss is considered to be 3 dB. The link budget equation can be written as:

TABLE 3

LINK BUDGET

Parameters

Value

Transmitter

power

losses (Ltx)

pointing loss (Lpt)

Channel losses(Lfog+Lgeom)

Receiver losses (Lrx)

Receiver telescope gain

Receiver sensitivity

11.80 dBm

-3 dB

-5 dB

-43 dB

-3 dB

24 dB

-36 dBm

Link margin for weather conditions

17.75 dB

(17)

where M is the link margin of the system, Sr is the receiver sensitivity of the photodetector, and Grx is the

gain of the optical concentrator. Table 3 shows the link budget analysis for this system with a link margin of

nearly 18 dB after considering all different losses in the system.

3.5 Train aerodynamics and turbulence effect

The train moving at a high speed creates aerodynamic forces around the train. These forces are

influenced by three factors namely, train speed, distance from the train and the train geometry [27]. At low

train speeds, there is a significant velocity variation around the train height with high turbulence intensity.

The turbulence intensity is low at high train speeds with more uniform velocity profile [28]. The moving

train creates a boundary layer along the train length resulting in the airflow in the direction of the train and a

wake behind it. As shown in Fig. 4, although the movement of the train creates pressure peaks at front and

back ends of the train, there is a uniform pressure along the length of the train carriage [27]. Train moving at

high speed induces wind in its surrounding [27]. At a speed of around 200 km/h, it would generate wind

speeds of around 15 m/s [29]. According to [30], the link performance improves with the wind speed. To

illustrate this, we use the method adopted in [30] where the mean <SNR> as a function of SNR with no

turbulence SNR0 is given by [31]:

where is the scintillation index for various wind speed for a Gaussian beam as given by [30]:

Fig. 4. Air flow around the train (Courtesy of [27])

for Kolmogorov spectrum,

denotes the longitudinal component of the scintillation index,

is a non-Kolmogorov Rytov variance for plane wave,

and are large scale and small scale

parameters,

, denotes an effective pointing error,

is long

term spot size caused by large scale induced beam wander, W is the free-space beam spot radius and r is the

radial distance from the optical axis. Fig. 5 shows the system performance as a function of the wind speed

based on (18) and (19) where <SNR> (gain) increases for increasing wind speed. Positioning the transceiver

in the middle part of the train roof, where the pressure is almost constant and as a result the refractive index

is almost constant would make the FSO link less susceptible to the turbulence effect. Although when the

train is stationary, the scintillation index variation due to the pressure and temperature would be high, but for

a train moving at a constant speed the turbulence effect can be ignored due to the constant pressure and

temperature along the length of the train. The link margin of 17.75 dB would ensure that the system is

functioning at all conditions.

Fig. 5. Mean SNR as a function of SNR with no turbulence for various wind speeds (Adapted

from [30])

0 5 10 15 20 25 30

0

2

4

6

SNR0 (dB)

<SNR> (dB)

No turbulence

v = 20 m/s

v = 11 m/s

v = 7 m/s

4. RESULTS AND DISCUSSION

The numerical analysis of the proposed system is performed using MATLAB®. The communications

channel for this terrestrial FSO link is assumed to be AWGN channel in our simulation. In order to estimate

the appropriate beam divergence the horizontal BS position d2 is varied from 5 m to 25 m and the beam

divergence values for a range of d2 are shown in Fig. 6. For d2 below 15 m, due to the wide beam divergence

of over 5° and 10° at d2 of 10 m and 5 m respectively, the required transmitted power is over 20 mW and 50

mW, respectively as most of the transmit power is wasted. This is due to a large amount of power, which is

outside the train-BS communications area as the beam profile is circular. When d2 >15 m, although the

required transmitted power is low (<15 mW) but the received power fades away quickly since the BS

position is further away from the shortest coverage point C. The power profile for different values of d2 is

plotted and compared in Fig. 7 based on (7), which suggests that the power profile appears to be more

uniform for d2 values over 15 m. Since, the desired BS separation distance from the shortest coverage point

C along the track is small, the horizontal BS position value is chosen to be 15 m. Using (1), the beam

divergence for the track coverage length of 75 m, fixing d1 at 1 m and d2 =15 m, would be 3.20° as is evident

from Fig. 6.

Assuming a typical Rx sensitivity of -36 dBm at 10 Mbps, and a radius of receiver optics of 25 mm, the

required transmit power at the BS is taken to be 11.8 dB (15 mW) as in Table 3. The receiver FOV is

estimated based on the size of the receiver optics as given in (9). Based on the parameters considered, the

optimum receiver semi-FOV for a 7 mm2 photodetector would be 5.15° (half angle). For this link, a margin

Fig. 6. Beam divergence for varying d2.

of ~18 dB is used, which is evident from the link budget analysis. The transmitter power could be increased

to the AEL limit as discussed above to ensure 100 % link availability.

The SNR along the track for various bit rates are shown in Fig. 8. The simulation is performed up to a data

rate of 100 Mbps due to the limitation in the bandwidth (50 MHz) of the measured laser impulse response,

which is adopted for simulation. As can be seen, to achieve a SNR of 13.6 dB the effective coverage length

for the data rate of 10 Mbps is 75 m. However, the observed SNR within the link margin offers an additional

SNR of 10 dB i.e., 23.6 dB at 10 Mbps. Increasing the bit rate to 100 Mbps does not affect the theoretical

coverage length of 75 m. The SNR drops from 23.6 dB at 10 Mbps to 13.6 dB at 100 Mbps since the noise

bandwidth increases at higher bit rates for a constant transmit power.

Fig.7. Power profile variation for varying d2.

Fig. 8. SNR variation along the track length CB.

The BER performance of the ground-to-train system is evaluated based on the angular parameters in Table

1 and system model parameters in Table 2. The BER curve along the train track for various bit rates is

plotted in Figs. 9(a) and 9(b) for beam divergence angles of 3.2° and 4°, respectively. The plot for a 4° beam

angle (commercially available typical value) is used to compare the achievable coverage length with the

proposed beam angle .For an AWGN channel, the simulated BER curve shows a relatively good agreement

with the predicted curve using (12) at 50 Mbps for both beam divergence angles

(a)

(b)

Fig. 9. Bit Error Performance along the track for beam divergence of (a) 3.2° and (b) 4°.

30 45 60 75 90 105 120

6

5

4

3

2

1

0

Length along the track (m)

- log10 (BER)

Simulation 10 Mbps

Theory 10 Mbps

Simulation 50 Mbps

Theory 50 Mbps

Simulation 100 Mbps

Theory 100 Mbps

30 45 60 75 90 105 120

6

5

4

3

2

1

0

Length along the track (m)

-log10 (BER)

Simulation 10 Mbps

Theory 10 Mbps

Simulation 50 Mbps

Theory 50 Mbps

Simulation 100 Mbps

Theory 100 Mbps

For 100 Mbps case, there is mismatch between the

predicted and the simulated results. This is due to the rise and fall times of the impulse response of the

system, which is not ideal. The bandwidth of the laser limits the transmitted bit rate beyond 100 Mbps. The

bit error performance of 10-6 is achieved for a track length of 75 m at 50 Mbps for a beam divergence of 3.2°.

The coverage length drops to 68 m for a beam divergence of 4° for the same transmit power. For bit rates

beyond 50 Mbps, the coverage length decreases for the desired BER due to the increase in the noise

bandwidth of the system. The coverage length of 75 m with given beam divergence is to be used as a

reference for the performance analysis of the system. However, moving to longer wavelengths could increase

the transmitted power by up to 50 times, thereby increasing the effective coverage length along the track

over few hundred metres. Thus, the number of BSs alongside the track could be significantly reduced.

Hence, high bandwidth availability of FSO in excess of THz and license free operation [32] would encourage

internet service providers to adopt this technology for ground-to-train communications.

5. CONCLUSIONS

A mathematical modelling for ground-to-train FSO communications link is proposed using Gaussian beam

theory. Receiver power equation for a Gaussian source is derived based on the geometrical position of the

BS from the track. The analytical and simulated BER performance of the proposed system is carried out

showing a good agreement to each other for data rates up to 50 Mbps. With the optimum parameters, it is

possible to have beam coverage for track length of 75 m for data rates up to 50 Mbps. Also, link budget

analysis for the proposed system is presented showing a link margin of 17.75 dB for worse weather

conditions. The paper also pointed out the proper positioning of the receiver on the train based on the

aerodynamics of the train. Hence, FSO technology with the proposed system modelling can be an alternative

to provide a high bandwidth broadband access to high speed trains.

ACKNOWLEDGMENT

R. Paudel thanks the Faculty of Engineering and Environment Northumbria University for financially

supporting this research. This work was supported in part by the EU FP7 Cost Actions of IC0802 and

IC1101.

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