1. Introduction
The International Mobile Telecommunications 2020 (IMT-2020) specifications, developed by the third generation partnership project (3GPP) for new radio (NR) operations in the fifth generation (5G) spectrum, is expected to accomplish the following performance requirements: ultra-reliable and low latency communications (URLLC) in the user plane as low as 1 ms; massive machine type communications (mMTC) that support up to 1 million devices per square km; and enhanced mobile broadband (eMBB) with uplink and downlink speeds of up to 10 and 20 Gbits/s [
1,
2]. These technical requirements are needed for the high bandwidth demands of augmented reality (AR), virtual reality (VR), and mixed reality (MR) applications, as well as the seamless and optimal functionality of Massive Internet of Things (MIoT) and Vehicle-to-Everything (V2X) connections for the smooth running of emerging smart cities [
3,
4,
5].
Free space optical communication (FSOC), whether as a standalone or hybrid technology, is a promising complementary solution platform for 5G backhaul networks [
6]. FSOC systems convey bidirectional information at high data rates through the atmosphere between line-of-sight (LOS) optical transceivers. Their numerous advantages include: very high throughput, highly secure transmission, relatively low cost, and ease of deployment when compared to the rigours encountered in the installation of fibre-optic infrastructures, high resistance to signal eavesdropping, and low latency communication since the velocity of light in the atmosphere is about 40% faster than in the fibre-optic cable [
7,
8]. In spite of these advantages, the performance of FSOC systems is severely affected by atmospheric impairments.
Dense fog, haze, and snow storms are known to cause the aerosol scattering of optical signals and consequently degrade the availability of FSOC links [
9]. In clear weather, atmospheric turbulence or scintillation is the most significant cause of impairment in received signal quality [
10]. Atmospheric turbulence causes fluctuations in air temperature, pressure, density, and humidity, which results in rapid variations in the atmosphere’s refractive index. The impact of these changes leads to irradiance fluctuations of received information signals [
11]. Other turbulence effects on FSOC link performance are disruptions in the coherence of the laser beam and distortions in the optical wave front. Optical wave front distortions result in laser beam broadening, uneven beam energy redistribution within a cross-section of the laser, and beam wander [
12]. Improving the bit error rate (BER) performance of FSOC systems during these adverse weather situations is the major challenge in the design of FSOC links [
13].
In addition, the misalignment between FSOC transceivers cause pointing errors, which increase the performance degradation of FSOC links. These misalignments arise from either mechanical vibrations in the system as a result of wind or building movement or errors in the tracking system. The displacement of the laser beam along vertical (elevation) and horizontal (azimuth) directions, which are generally expected to be independent Gaussian random variables, result in pointing errors [
14,
15,
16,
17,
18,
19]. Beam width, boresight, and jitter are the three fundamental components of a pointing error. The beam width is the beam waist (radius computed at
e−2), while the jitter is the random offset of the beam centre at the detector plane produced by building motion, minor earthquakes, and dynamic wind loads. The boresight denotes the fixed displacement between the beam centre and the alignment point. It should be noted, however, that boresight displacements are of two kinds: inherent boresight displacement and additional boresight error. The first is related to the spacing between the detector’s receive apertures. This inherent boresight displacement corresponds to a fixed distance, namely the distance between each received aperture and its associated alignment point. The second is related to the boresight error caused by the building’s thermal expansion [
14,
15,
16,
17,
18,
19].
Conventionally, FSOC systems employ intensity modulation/direct detection (IM/DD) schemes. Most commercial FSOC links are based on the on-off keying (OOK) modulation schemes due to their low cost and simple implementation. However, FSOC systems employing OOK require adaptive thresholding, which is difficult to implement when combating irradiance fading, hence their sub-optimal performance over atmospheric turbulence channels [
11,
13,
20]. Binary phase-shift keying subcarrier intensity modulation (BPSK-SIM) FSOC systems have also been investigated extensively. In spite of their superior BER performance when compared with other coherent and non-coherent modulation schemes, BPSK-SIM FSOC links have poor power efficiency when compared to pulse position modulation (PPM) FSOC links [
20,
21]. FSOC systems employing sub-carrier intensity quadrature amplitude modulation (SIM-QAM) have also been investigated. SIM-QAM FSOC links are found to have better spectral efficiency compared to PPM FSOC links which exhibit poor bandwidth performance. SIM-QAM FSOC systems have great potential for future FSOC systems since they deliver a higher data rate without an increase in the required bandwidth due to their inherent attribute of transmitting more bits per symbol [
11,
22,
23,
24].
Between April 2015 and February 2016, the First European South African Transmission ExpeRiment (FESTER) was conducted in False Bay, South Africa, to study the influence of atmospheric turbulence on wave propagation [
25,
26]. The experiment focused on measuring and modelling optical turbulence, electro-optical system performance, and imaging. Despite the fact that wind direction, wind speed, and the kinematic vertical sensible heat flux all have an effect on optical turbulence, thermal forces were found to have the greatest impact on it, with both exhibiting a direct relationship regardless of the seasons. Additionally, it was discovered that as friction velocity increases, optical turbulence increases. Onshore and offshore wind directions produced differences in the turbulence strength. With onshore conditions during the winter, the turbulence strength is extremely low. Spring brings an increase in the variability of turbulence strength. The highest refractive index structure parameter (
) values above 10
−14 m
−2/3 may be reached during the summer [
25,
26].
The
, which is also dependent on the root-mean-square (RMS) wind speed and altitude of a location, is used to characterize atmospheric turbulence as weak, moderate or strong at any point in time [
27,
28,
29,
30]. Most of the results obtained in literature [
6,
10,
11,
12,
13,
20,
23,
31,
32,
33] assume arbitrary
values or estimate them based on average wind speed measurements for a particular location. In some cases, worst case scenarios of atmospheric turbulence based on the maximum values of wind speed are investigated [
29,
33]. However, these measurements are based on data spanning less than 4 years. As a result, they cannot be accurately used to estimate the maximum attenuation due to turbulence-induced irradiance fading. In this paper, the focus is placed on the wind distributions based on data spanning over 8 years for the various locations of interest where FSOC links are to be deployed. This will allow for accurate estimation of the
, and consequently, correct calculations of the maximum attenuation due to turbulence, and the performance of various FSOC links during such periods.
Therefore, the key contributions of this work are as follows:
Computation of the scintillation profile for Gaussian beam FSOC signals in the nine cities under investigation based on the zero inner scale and infinite outer scale model and finite inner and finite outer scale model. To the best of our knowledge, the computation of the scintillation profile for Gaussian beam FSOC links transmitting at 1550 nm in the cities of interest, while considering periods not exceeded 50%, 99%, 99.9%, and 99.99% of the time have not been reported in open literature.
Aerosol scattering losses over various distances for FSOC links transmitting at 1550 nm with respect to events not exceeded 50%, 99%, 99.9%, and 99.99% of the time, for nine major locations in South Africa, are investigated.
Outage probabilities of Gaussian beam FSOC links based on the aforementioned scintillation models, while taking into account the effect of pointing errors for events not exceeding the previously mentioned time intervals, are presented for various locations of interest.
Analysis of the bit error rate (BER) performance for intensity modulation and direct detection (IM/DD) avalanche photodiode (APD) FSOC systems transmitting at 1550 nm and based on OOK, BPSK, square, and rectangular SIM-QAM schemes during weak, moderate, and strong atmospheric turbulence, with regards to average weather measurements and events not exceeding 99%, 99.9%, and 99.99% of the time are presented.
The rest of this paper is organized as follows:
Section 2 presents the ground wind speed distributions for nine cities in South Africa;
Section 3 presents and analyses the modified Rytov theory based on zero inner scale and infinite outer scale model and finite inner and finite outer scale model for Gaussian beam waves.
Section 4 presents aerosol scattering losses over various link distances for the nine cities under investigation. Weak, moderate, and strong atmospheric turbulence parameters during clear weather for the locations of interest based on the Lognormal and Gamma–gamma turbulence models are provided in
Section 5, while outage probability analysis of FSOC links with respect to the effect of pointing errors is presented in
Section 6. In
Section 7, the average BER analysis, taking in account pointing error effects for various FSOC systems in weak, moderate, and strong turbulence regimes is derived and the results are analysed, while conclusions are provided in
Section 8.
2. Wind Speed Distribution
Figure 1 shows the map of South Africa depicting the average wind speed (m/s) at 100 m above ground level for selected cities investigated in this work. The average wind speed data used in plotting
Figure 1 spans January 2008 until December 2017 and was sourced from [
34]. The data in
Figure 1 is very similar to the average measurement values presented in Table I of [
10,
32]. Wind speed data from January 2010 until June 2018 was also acquired from the South Africa Weather Service (SAWS) for major locations in each of the nine provinces of South Africa. The data was collected hourly for the 8½ year period. The locations of interest investigated in this work are: Bloemfontein, Cape Town, Durban, Johannesburg, Kimberley, Mafikeng, Mbombela, Polokwane, and Port Elizabeth. The data provided by the SAWS, which was collected from various weather stations placed a few meters above the ground, was statistically processed and used for all our computations in this work.
Figure 2a–c, alongside all other analysis done in this work, are based on the measurement data obtained from the SAWS.
Figure 2a shows the CDF of wind speed for various cities in South Africa, while
Figure 2b presents the PDF of wind speed for the same locations. In
Figure 2a, the coastal cities of Port Elizabeth and Cape Town have the highest probabilities of occurrence of high wind velocities compared to other cities in South Africa. The cities of Bloemfontein, Mbombela, and Polokwane have the highest likelihood of occurrence of low wind speeds. In
Figure 2b, the probability of occurrence of wind speeds of 1 m/s in the cities of Mbombela and Polokwane is ~0.3, while the cities of Cape Town and Port Elizabeth have the lowest likelihood of occurrence (less than 0.15) of low wind velocities when compared with other cities in South Africa.
Figure 2c shows the wind speed exceedance against the percentage of time for the various locations of interest.
Figure 2c validates the results in
Figure 2a,b.
Wind velocities greater than 4 m/s occur ~60% of the time in the cities of Port Elizabeth and Cape Town, while in Polokwane, Mbombela, and Bloemfontein, wind speeds higher than 4 m/s occur less than 25% of the time.
The
in m
−2/3 based on the Hufnagel–Andrews–Phillips (HAP) model is presented in [
35,
36,
37] as:
where
is the scaling factor,
is the reference height of the ground above sea level in metres,
is the height of the first FSOC transceiver above the ground in metres,
is the altitude from the reference height
to the height of the other (second) FSOC transceiver above the ground in metres, and
is the root-mean-square (RMS) wind speed in m/s. The RMS wind speed in Equation (1) is calculated using the Bufton wind model, which is given in [
35,
36,
38,
39] as:
where
is the beam slew rate associated with a satellite moving with respect to an observer on the ground in rad/s and
is the ground wind speed in m/s.
The climate of South Africa is considered to be highly variable, both spatially and temporally. Spatial variations in elevation across the country contribute significantly to this variability. According to the Council for Scientific and Industrial Research’s (CSIR) Köppen–Geiger climate classification for South Africa, the country is predominantly semi-arid, with influences from temperate and tropical zones [
40,
41,
42]. A large part of the geographical space of South Africa and Namibia is characterized by arid and hot climates, with clear skies and low annual rainfall [
28,
29]. Due to the similarities in the two countries’ climatic patterns, the
based on the HAP model are expected to adequately estimate the atmospheric turbulence losses encountered by FSOC links deployed in various cities of South Africa considered in this work.
Other values where
= 1,
= 0.1 mrad/s,
= 10 m, and
= 15 m were used in computing the
throughout this work. The altitude measurements above sea level are given in
Table 1, and the ground wind speed data from SAWS when inserted into Equations (1) and (2) are used for determining the
of the various locations shown in
Figure 3.
4. Aerosol Scattering Losses
The aerosol scattering coefficient in dB/km, as defined by the Kim and Ijaz models, is given in [
42] as:
where
is the meteorological visibility in km,
is the maximum spectrum wavelength of the solar band, and
is the particle size distribution parameter. In the Kim model,
is expressed in terms of visibility as [
48]:
while
is expressed in terms of wavelength in the Ijaz model as [
49,
50]:
For visibility measurements less than 1 km, the Ijaz model is used to calculate scattering losses, while the Kim model is used to calculate the specific attenuation associated with visibility values greater than or equal to 1 km. The two models are used in estimating scattering losses encountered by the transmission wavelength of 1550 nm. The visibility data used in computing the aerosol scattering losses over different distances in
Figure 4a–d was obtained from the SAWS for the nine major South African cities of interest investigated in this work. The data was collected three times daily (8:00 a.m., 2:00 p.m., and 8:00 p.m.) over an 8½ year period (January 2010 until June 2018). Over a link distance of 1 km, FSOC links transmitting at 1550 nm in Mbombela would encounter scattering losses of ~0.34, 23, 94, and 188 dB based on the periods not exceeded 50%, 99%, 99.9%, and 99.99% of the time, respectively.
Similarly, scattering losses of ~0.15, 0.50, 1.54, and 4.29 dB would be encountered by the same FSOC links over a distance of 1 km in the city of Mafikeng, as shown in
Figure 4a–d, respectively.
5. Intensity Distribution
In this section, the statistical analysis of the irradiance fluctuations and the channel characteristics for the weak and moderate to strong turbulence regimes are carried out using the Lognormal and Gamma–gamma turbulence distributions, respectively. The PDF of the lognormal distribution is given in [
14,
15,
17,
18] as:
While the PDF of the Gamma–gamma turbulence distribution can be expressed as [
17,
39,
51,
52]:
where
is the normalized irradiance,
represents the Gamma function, and
is the modified Bessel function of the second kind and of order
.
is the effective number of large-scale turbulence eddies. It is defined as [
43]:
where
is the normalized large-scale (refractive) variance.
is the effective number of small-scale turbulence eddies. It is given as [
43]:
where
is the normalized small-scale (diffractive) variance. Using equation (07.34.03.0605.01) in [
53], where:
Equation (29) can be rewritten as:
where
is the Meijer G function, which is well defined in [
54]. Integrating
in Equation (33) gives the CDF of
. This is derived by using equation (07.34.21.0003.01) in [
53]. Thus, we have:
In this work, pointing errors represent the misalignment between the transmitter and receiver caused by the laser beam being displaced horizontally or vertically, i.e., a two-dimensional configuration is being considered. The transmitter and receiver planes are assumed to be parallel, and the laser beam is perpendicular to the receiver area. The pointing error parameter,
, is defined as the ratio between the equivalent beam waist or radius at the receiver (
) and the standard deviation of the jitter or pointing error at the receiver (
). It can be expressed as [
6,
16,
55,
56]:
The beam waist
of a Gaussian beam, which is the radius calculated at
, determines the value of the parameter,
at distance,
.
is given as [
6,
16,
18,
55,
56,
57]:
where
,
is the error function and parameter
is expressed as [
6,
16,
55,
56]:
where
represents the radius of a circular detector aperture. At distance
, the fraction of the collected power is represented by parameter
. It is expressed as [
6,
16,
55,
56]:
where
is the complementary error function.
Therefore, the PDF of the Lognormal distribution, considering the effect of pointing errors, is derived in [
17,
18,
58] as:
where
and
Additionally, the PDF of the Gamma–gamma distribution model, taking into account the effects of misalignment, is derived in [
6,
16,
56] as:
After some mathematical manipulations, the PDF can be further simplified as [
6,
56]:
The expression for the CDF of the Gamma–gamma distribution model, considering pointing error effects, is derived in [
6,
56] as:
For commercial FSOC links employing the use of intensity modulation/direct detection (IM/DD) schemes and avalanche photodiode (APD) detectors, the instantaneous signal-to-noise ratio (SNR) at the receiver is defined as [
42,
56,
59,
60]:
where
is the responsivity,
is the APD gain,
is the total noise at the APD receiver, and
is the average optical power detected at the receiver.
is well defined in Equation (9) of [
42].
The average SNR at the receiver is defined as [
42,
56]:
The total noise at the APD receiver comprises the thermal and shot noise. It is given as [
42,
51,
59,
60]:
where
is the temperature of the receiver,
is the Boltzmann constant,
is the bit rate,
is the noise figure of the amplifier,
is the APD load resistance,
is the electron charge, and
is excess noise factor. The excess noise factor is expressed as [
42,
51,
59,
60]:
where
is the ionization factor.
The PDF of SNRs for weak atmospheric turbulence using the Lognormal distribution model with pointing errors is derived by substituting Equation (45) into Equation (39), and is given below as [
18]:
Applying the relation in [
18,
31] where:
The PDF of SNRs for moderate to strong atmospheric turbulence using the Gamma–gamma model with pointing errors, as derived in [
55,
56], is given as:
The
is used in characterizing the atmospheric turbulence strength due to the effect of scintillation. In a weak turbulence regime,
, and the Lognormal distribution model is employed. For moderate to strong fluctuations,
, and the Gamma–gamma turbulence distribution is used [
38,
44]. In certain instances where
but
or
, then the Gamma–gamma distribution is employed. In other situations, where
and
or
, some computations involving the Gamma–gamma distribution would produce undefined results [
43]. Thus, in this work, when
and
or
, the Lognormal distribution is used.
6. Outage Probability Analysis
Outage probability is a critical performance indicator that defines the likelihood of the instantaneous SNR going below the threshold SNR. Once this occurs, the link’s communication will fail. It is expressed below as [
45,
61]:
where
is the threshold SNR and
is the CDF of the instantaneous SNR.
From Equation (45), the normalized irradiance at the receiver can be expressed as [
6,
57]:
Therefore, substituting for
in Equation (44) presents the expression for estimating the outage probability of the FSO link over the turbulent atmospheric channel while considering the effect of pointing errors [
6,
56]:
Figure 5a–d are based on the numerical values in
Table 2,
Table 3,
Table 4 and
Table 5, respectively. These figures are generated using the parameters in
Table 6, while computing with Equations (1)–(3), (15), (30), (31), (35), (46) and (55). The average receiver SNRs needed to achieve various outage probabilities over different link distances while considering the effect of pointing errors in the presence of turbulent eddies are presented in
Figure 5a–d for the locations of interest. In these figures, the average receiver SNRs required to attain different outage probabilities are quite similar. It is also evident from these figures that the impact of the normalized jitter standard deviation on the outage probability is significant (when comparing
Figure 5c,d where
to
Figure 5a,b where
). This implies that the lower the value of
, the better the overall system performance. Additionally, the higher the value of
, the better the outage probability performance of the FSOC links. In the presence of finite inner and outer scales of turbulence, that is, where
lo = 0.005 m and
Lo = 10 m, the outage probabilities of the FSOC links are quite similar to when these turbulent eddies have sizes of zero and infinity in the Kolmogorov model with an infinitely large inertial range.
8. Conclusions
In this paper, analysis of atmospheric turbulence effects on terrestrial SISO FSOC links based on the RMS and ground wind speeds prevalent in various cities of South Africa are presented. Wind speed data provided by the SAWS were statistically processed, and the corresponding CDF, PDF, and percentage of time plots are shown for each location of interest. The
based on RMS wind speeds during clear and sunny weather are computed. The scintillation indices not exceeded 50%, 99%, 99.9%, and 99.99% of the time, based on the zero inner scale and infinite outer scale model and finite inner and finite outer scale model are calculated. Aerosol scattering losses based on visibilities not exceeded 50%, 99%, 99.9%, and 99.99% of the time for the various cities of South Africa are shown. Outage probability and BER analysis, taking into account the effect of pointing errors over weak and moderate to strong atmospheric turbulence channels, were then carried out for OOK, BPSK, and SIM-QAM SISO FSOC links deployed at the different locations of interest. All through
Figure 5,
Figure 6,
Figure 7,
Figure 8 and
Figure 9, the SISO FSOC links deployed in all the locations of interest have similar outage probability and BER performances based on the zero inner scale and infinite outer scale model and finite inner and finite outer scale model. This is because the values of
in all the investigated cities are approximately equivalent over all the time intervals (
Table 2,
Table 3,
Table 4 and
Table 5) considered in this work. As part of future work, all the analytical results in this work would be verified experimentally. The
based on important meteorological parameters such as temperature, pressure, and the structure parameter for temperature, as well as three-dimensional pointing errors effects, will also be investigated for FSOC links deployed in the locations of interest.