Moment-Based Analysis of Reg. Triangular QAM in Turbulent Atmospheric Channels with Pointing Errors incorporates Zero and Non-zero Boresight

In recent years, free-space optical (FSO) technology has gained fame in communication systems due to its high data rates and license-free feature. Triangular quadrature amplitude modulation (TQAM) is an efficient modulation scheme that uses even bits per symbol, and it has a low average symbol error rate (ASER) than square quadrature amplitude modulation (SQAM). In this paper, we theoretically investigate the performance of subcarrier triangular quadrature amplitude modulation (SC-TQAM) corrupted by atmospheric turbulence in the presence of pointing errors (P.E) in FSO communications. We have considered boresight displacement, P.E effect, and atmospheric turbulence. To represent the atmospheric turbulence we consider Log-normal, Rayleigh and Rician distributions which exhibit weak, moderate and strong weather effects. The P.E is employed using Rayleigh and Rician distribution that incorporates zero and non-zero boresight displacement respectively. The moment generating functions of these models have been derived. P.E is modeled using an optical beam radius and receiver aperture radius. The combination of these parameters will help enhance the average symbol error rate (ASER). Furthermore, we have derived an analytical expression that is used to develop numerical results. The ASER performance is observed against average received irradiance, optical beam radius, P.E standard deviation and receiver aperture radius. In the end, the ASER performance is evaluated against SNR and Monte Carlo simulations are performed which validates the theoretical results.


Introduction
Over the last two decades, FSO technology is readily earning fame as an efficient channel of transmitting data over short distances at high speed. FSO systems are presently developing as a replacement of fiber optics to overcome the huge bandwidth requirements. This is above all owing to exceptional attributes of the FSO technology, which comprise: almost endless bandwidth for acquiring very high accumulated capacity, cost-efficient, reduced intrusion, no licensing requirements, and highly secure link that is simple to design and implement [1,2]. FSO has already shown the superiority of high bandwidth utilization and zero electromagnetic Interference (EMI) [3].
The license-free feature of FSO systems has significant importance as the Radiofrequency spectrum is getting congested due to an increase in the number of users every day. Although FSO has considerable potential to fix the last mile problems, FSO communications is the least deployed technology among the others due to many problems. These problems are primarily induced due to weather conditions viz. snow, fog, and rain. Atmospheric turbulence effect on a clear day has a considerable effect on the performance of FSO systems. It is a line of sight (LOS) communication utilizing wavelengths of different frequencies viz. 850nm, 1300nm and 1550nm [4].
The phenomenon of atmospheric turbulence occurs when the refractive index of channel i.e. air or free space is altered due to change in pressure and temperature. Therefore, the turbulent channel which is extremely fickle, random and exposed to weather conditions causes scattering and absorption which leads to severe errors in received signal [5].
Besides atmospheric turbulence, P.E also affects optical links performance that adds more fading at the receiver's end. Pointing error is a phenomenon that is observed at the receiver due to building sway and transceiver misalignment, which causes loss in terms of power and induce fading in the received signal. As a result, transceiver misalignment causes a random shift in the LASER beam footprint along the vertical and horizontal axis. Bore-sight and jitters are two main components of pointing errors. The offset induced in the beam center at the receiver's aperture is known as jitter and displacement of beam footprint in X-Y plane on receiver's aperture is known as bore-sight [6].
Many probabilistic models have been suggested to incorporate the effect of atmospheric turbulence as weak, moderate and strong intensity fluctuations. The log-normal statistical model is used to describe weak-to-moderate, while gamma-gamma represents strong fading regime [7].
Over the past few years, a number. of generalized pointing error, models have been presented. In [8], a generalized P.E model is proposed that incorporates zero bore-sight and similar jitter in elevation and azimuth directions. In [9], a more realistic approach is considered that incorporates the effect of wind and thermal expansion as jitter variances in terrestrial FSO Links.
In FSO, the most used modulation scheme is on-off keying (OOK), due to its simplicity and less cost demanding features. However, opting for the right value of the adaptive threshold in OOK modulation is hard to achieve, due to fluctuation intensity in the signal [10]. As compare to OOK, pulse position modulation (PPM) has exceptional power and poor spectral efficiency [11].
A hexagonal constellation was presented that has a structure similar to honeycomb (HCSS) was presented, another constellation named as triangular shape signal set (TSSS) was proposed.
HCSS and TSSS both performs better than SQAM, but they have high receiver detection complexity [16,17]. In [18], triangular quadrature constellation was proposed, the constellation points were taken on the vertices of the regular polygon. TQAM, as compare to SQAM, is efficient in terms of performance and receiver detection complexity [18,19]. TQAM transmits even bits encoded data and due to its compact geometry, it performs better than SQAM [20]. In [21], an approximate SEP expression for TQAM was derived and also, In [22,23], an exact SEP expression was proposed and its performance evaluated over various fading channels in RF communication. It is noticed from the afore discussed literature, that to the best of authors' knowledge no research has been conducted on performance evaluation of TQAM over Freespace communication.
Hence, our motivation is to derive statistical models, probability density function (PDF), for TQAM which models the fading effect of the turbulent channel and pointing errors considering transmitter beam radius, jitter standard deviation, and photodetector aperture. Analytical expression for general order TQAM has been utilized, the effect of transmitter beam-width, jitter's standard deviation and photodetector aperture on ASER performance is observed. The mathematical ASER performance is evaluated and validated via Monte Carlo simulation.
Moreover, the impact of previously discussed factors e.g. jitter's standard deviation and transmitter beam radius 0 is used to optimize ASER performance.
This paper comprises of the following sections: In section 2 we will discuss system model, in section 3 fading channels are discussed, in section 4 statistical models are derived along with their moment generating functions, in section 5 expression for ASER is presented, in section 6 numerical results are discussed, and conclusion is given along with future recommendations in section 7.

SYSTEM MODEL AND MODULATION
A system using general order TQAM is considered and, the source generates the data which is pre-modulated to radio RF can be written as [24].
( ) = J I, n cos(2 ) + J Q, n sin(2 ), 0 ≤ ≤ Where J I, n and J Q, n are the amplitudes of in-phase and quadrature components respectively, is the carrier frequency and is the symbol period. The SC-TQAM is used to derive the LASER diode circuit as the SC-TQAM varies and generates a modulated signal in terms of irradiance.
The generalized transmitted signal is written as where 0 is the average irradiance power, is the modulation index and ( ) is the transmitted signal. The SC-TQAM signal is used to modulate the intensity of LASER, with proper biasing to avoid any chopping, the LASER beam is aligned with the receiver using a collimator (telescope) and transmitted over the channel. The received signal is expressed as where is the scaling factor of channel and it temper's the signal due to the presence of atmospheric turbulence and pointing errors. The 0 is DC noise and it can be filtered by using a band-pass filter. The LASER beam distorted by atmospheric turbulence Figure 1 Basic block diagram of FSO communication system is collected at the receiver's aperture, focused on the photo-detector (PD) generates an electrical signal. The signal recovered at photo-detector can be written as where is the responsitivity of photo-detector and ( ) can be modeled as white Gaussian noise (AWGN). The generalized expression for equation (4) can be written as where is the received signal, is the transmitted symbol, is the scaling factor of channel used and is receiver noise which is modeled as white Gaussian noise (AWGN) having a spectral density 0 . Assuming the scaling factor of fading channel is accurately known to receiver, the electrical signal is TQAM demodulated and implementing equalization can be written as where is the recovered signal, which is compared with transmitted signal , symbol error rate where is the probability of symbol error, is total no. of symbols received in error and represents the total no. of symbols transmitted.

Fading Channels
Free-space optical (FSO) system is a technology that combines two different types of communication i.e. optical fiber and wireless communication. The optical beam is transmitted in the atmosphere (Channel), to recover transmitted signal at the receiver, a mathematical model is required to be formulated. This light signal experience attenuation due to the turbulent nature of the channel. The particles present in the atmosphere act as a barrier; reflect, absorb, refract, attenuate, disperse and diffract the optical beam [24]. The phenomena of dispersion happen when a signal is transmitted, it reaches the receiver at different time intervals through multiple paths. If the terminals of transceiver are relatively in motion it is dispersion in frequency. Otherwise, if transmitter and receiver are stationary it is dispersion in time [25]. The phenomena of fading can be classified as follows: Shadowing and multi-path fading. Shadowing referred to fading induced due to reflection, diffraction and absorption from small stationary and non-stationary obstacles e.g. buildings, vehicles, and trees. These obstacles cause small variations in mean intensity of signal amplitude. In shadowing, the coherence time is larger than symbol time period > .
The effect of shadowing can be completely removed by employing specialized systems. In contrary to that, the multipath fading or small-scale fading is occurred when signal reaches at receiver at different times reflecting through obstacles or reaches directly from the transmitter.
These delayed multipath adds up at receiver and behaves constructively or destructively.
the subsequent section, to study the effect of different fading conditions; Log-Normal as shadowing; Rayleigh and Rician as multipath fading has been employed.

A. Log-Normal Shadowing
Log-Normal Shadowing is utilized when the particles present in the air are almost frozen. It is a normal practice to model path loss using Log-Normal distribution in multi path fading. When a signal propagates through channel, it gets corrupted due to reflection, refraction and diffraction.
When all possible losses are added to together, according to central limit theorem it tends to follow Log-Normal distribution [26]. The random variable can be modeled as Log-Normal distribution. In FSO communication, Log-Normal is used to illustrate the weak atmospheric fading effect. The PDF of Log-Normal distribution is expressed as where is the signal intensity level, σ I 2 = 0.31 7 6 2 11 6 is log amplitude variance, is mean received irradiance, the relation of log-amplitude variance with Raytov variance is 2 ≈ 2 , 2 is Raytov variance. = 2 is the optical wave no. and is the wavelength.
In a communication system, the symbol error rate is a key parameter to analyze the performance of the modulation scheme in any channel. Now, we will derive a moment generating function (MGF) that will be utilized to investigate symbol error performance. The generalized mathematical expression of (MGF) is given as [25] ( , ) = 2 ∫ ( , ) ( ) substituting the 2 in equation (10), we get to reduces equation (12) to close form expression, we will use Gauss Hermite Quadrature approximation stated in [24], which is stated as where is zeros and is the weight factors. considering ( , ) = ( ) and replacing − 2 = in equation (13). Furthermore, alternate form of Q-function is utilized which is also known as Craig's form ( = 2 ) expressed as [27,28] Now substituting Craig's form in equation (12), new expression becomes now substituting = 2 2 2 in equation (15), we get (16) now substituting in equation (11) in equation (16) ( , ) = 2 Now, substituting 0 = 0 2 2 2 as different types of noise limiting conditions, now reducing equation (17) using Craig's form, we get the final expression as The mathematical expression shown above will be used in subsequent section. It will be utilized to evaluate the symbol error rate for M-TQAM in Log-Normal Channel.

B. Rayleigh Fading Channel
Rayleigh fading channel in strong turbulence regime incorporates stronger fluctuations in the amplitude of received signal irradiance. Its application is considered when a number of scatterers are present in the atmosphere, which scatters the transmitted signal. It involves no direct line of sight communication between transmitter and receiver. Normally, the Rayleigh fading channel is utilized when the scatterings are independent, severe in effect and the optical link spans a few kilometers. These independent scatterings are modeled as Gaussian. Rayleigh Fading is observed when the radius ( ) of particles present in the channel is smaller in size then wavelength ( ) of LASER beam ( < ). The Rayleigh distribution is modeled using negative exponential distribution and its probability density function can be written as [29].
where is the signal intensity level and is mean received irradiance. Moreover, in saturation regime the scintillation index of strong turbulence is almost unity and it is also known as fully developed speckle regime. The Rayleigh channel is modeled using Gaussian random with zero mean. Thus, SNR per symbol is central chi-square distributed [25].
In order to derive the close form expression for to evaluate the symbol error rate performance of M-TQAM in Rayleigh channel. Now, substituting equation (19) and equation (15) in equation (9), we get substituting ( ) = 2 2 2 and by approximating 2 to in equation (20) can be written as Now taking − common from the exponential term we get By solving integral with variable we get After applying the limits and simplifying the above expression we get where the of Rayleigh Fading channel is written as After solving the integral in equation (24) and applying limit we get The expression written above will be utilized in subsequent section for evaluating symbol error rate of M-TQAM in Rayleigh fading channel.

C. Rician Fading Channel
In this channel signal propagates through multipaths and reaches the receiver at different time intervals. There is one direct line of sight component and the rest of components are delayed random reflections of the transmitted signal. The channel with specular and non-specular components can be mathematically formulated by Rician distribution function which is as follows [24,25] where is the signal intensity, 0 is mean received irradiance, 0 refers to zeroth order modified Bessel function of first kind and 2 is K-factor of Rician distribution.It is noteworthy that Rician distribution is modeled using Gaussian random variables having non zero mean. Therefore, the SNR per symbol is non-central chi-square distributed.
Now, we will develop a close form mathematical expression to evaluate the symbol error rate in Rician Fading channel for M-TQAM. To develop the expression, we will substitute equation (14) in equation (9) we get In equation (28), the term written in square brackets can be solved using Laplace transform. The generalized expression for Laplace transform is written as Now, rewriting the equation (28) in terms of Laplace transform we get The mathematical expression written within square brackets in equation (28) is directly taken from [25] and written as where = 2 that is Rician K-factor, by substituting the above expression in equation (30) we get the final expression

Atmospheric turbulence model with pointing errors
In this section we will derive the composite channel model recalling the method stated in [4].
The channel gain $H$ is characterized by three factors viz. atmospheric turbulence , atmospheric attenuation and pointing errors . When optical signal passes through turbulent channel it gets distorted due to rain, fog and dust. The channel gain can be expressed as [8] = × × where, is a deteministic quantity exhibits no randomness, it is expressed as where, is attenuation coefficient and is the link distance.

A. Pointing Error Model
The fundamental reasons that introduces pointing error in optical signal is either due to misalignment of transceiver or vibrations of the buildings. Assuming, the Gaussian beam as beam waist radius reaches at distance received on photo-detector having a radius of . The expression for power collected at = 0 is denoted by 0 . The pointing error random variable can be written as [8] The PDF of random variable is expressed as [30] ( )] 2 is at radial displacement = 0 and = √ 2 . The parameter = 2 is the ratio of equivalent beam radius and jitter standard deviation . $ 2 is the equivalent beam radius written as where = √ 2 (1 + ( ( 2 ) −2 )) with 0 radius of transmitted beam at = 0, = 2 2 −2 + 1 and the parameter Where the is known as intensity of pointing error effect.
Rearranging equation (35) and substituting the value of in equation (40), the PDF for Rician pointing error with non-zero bore-sight can be written as

D. Atmospheric Turbulence Model
The random variable ( ) can be modeled as Log-Normal distribution. In FSO communication, Log-Normal is used to illustrate the weak atmospheric fading effect. The PDF of Log-Normal distribution is expressed as where, 2 = 0.31 7 6 2 11 6 is log amplitude variance, the relation of log-amplitude variance with Raytov variance is 2 ≈ 2 4 , 2 is Raytov variance. = 2 is the optical wave no. and is the wavelength.

E. Generalized Composite Channel Model
The density function of equation (33) can be expressed as [8] ( , ) = ∫ / ( ) ( ) where, ( , ) is composite channel PDF with parameters $H$ channel gain and is beam radius at distance , / is conditional that can be expressed as now substituting equation (51)in equation (9), it becomes now applying Craig's formula in equation (52) using equation (14), new equation becomes The above closed form mathematical expression is to evaluate the ASER for Log-Normal with Rician P.E model.

ASER Analysis
In this section, we will derive ASER expression of TQAM that is corrupted by turbulent channel in the presence of P.E with zero and non-zero boresight. The work shown below has not been found in the literature previously.
where the = √2 is normalized minimum distance between neighboring symbols and is defined as = Where = 1,2, … ,6. Table 1 System parameters and constants.

Numerical results and discussion
In this section, we will evaluate and compare the already derived analytical expression for M-     Table 2 and 3).
The symbol error rate of 10 −3 using 16-TQAM in Log-Normal, Rayleigh and Rician requires irradiance intensity of −71, 48 and 37 respectively.   When the transmitted optical signal travels through the Free-space reaches the receiver, it gets tempered. That received signal is recovered compared with the transmitted set of symbols. The ratio of received symbols in error to total transmitted is calculated. This ratio gives an estimated probability of symbols in error for that value of SNR. In Figure 5 symbol error rate of 16-TQAM in Log-Normal, Rician, and Rayleigh fading has been plotted as a function of SNR, in Table 4     Rayleigh as pointing error and Rician pointing error respectively. As the weather effects 2 and jitter's standard deviation increases, the ASER also increases. Moreover, as the value of transmitter beam radius approaches as closer to 0 ≈ 0.022 the ASER decreases and eventually at 0 ≈ 0.022 ASER is minimum. The ASER performance for given conditions is therefore optimized and 0 ≈ 0.022 is an optimal value. In Figure 6, due to Rayleigh pointing error model is employed the ASER value is smaller than Rician pointing error model at 0 ≈ 0.022 due to the presence of boresight displacement = 0.02 in Rician pointing model ( Figure 7). The difference in the values of ASER can be seen in Table 5 at 0 ≈ 0.022 .  In Figure 8 and 9, the ASER is evaluated against the transmitter beam radius for Rayleigh pointing error model and Rician pointing error model respectively, for different values of receiver aperture radius and weather conditions. The effect of aperture radius on the performance of optical communication is significance. From the figures, it can be seen that as the receiver's aperture size increases the ASER gets improved. The aperture radius at 0.042m and 0.052 gives significantly poor ASER performance than at 0.062 . Moreover, at an optical beam radius $w_0\approx0.022~m$ the ASER gives the minimum values, that is the optimal value of optical beam radius in all weather conditions.   can be seen that as the value of jitter's standard deviation increases, ASER also increases. The composite effect of weather conditions and an increase in jitter's standard deviation make ASER increase and system performance becomes poor. It is worth observing that the effect of P.E is more visible when the atmospheric turbulence is strong. Practically, the effect of P.E standard deviation cannot be minimized. By selecting the right width of the optical beamwidth the performance of the system can be optimized. But P.E effect cannot be reduced by only adjusting the optical beamwidth. In Figure 11, due to the employing of the Rician pointing error model, the ASER is slightly higher than the Rayleigh P.E model.     1000 . in Figure 15, the Rician P.E model is employed for the same parameters as mentioned before. As we know, due to the presence of boresight displacement = 0.01 in Rician pointing error model, it gives slightly higher values of ASER. In Figure 14 and 15, the falling curve with optimal beam requires small aperture and it performs better than others in terms of ASER. Also, it can be observed that by increasing the size of receivers aperture the ASER performance gets improved.   Figure 16 and 17, the rayleigh, and Rician P.E models have been plotted respectively. ASER of these models has been observed against receiver aperture radius for various values of P.E standard deviation = 0.05, = 0.07, = 0.1$ in meters, weather condition 2 = 1 × 10 −15 , = 30 and link length = 1000 . we can not control the jitter standard deviation . however, the effect of P.E standard deviation can be reduced by using the right size of the receiver aperture. As the aperture radius size increases the ASER decreases drastically.

Conclusion
In this article, we have investigated the performance of SC-TQAM over Log-Normal, Rayleigh and Rician fading channels. We have also investigated Log-Normal distribution with weak, moderate and strong atmospheric turbulence employed P.E effect with zero and non-zero boresight. The moment generating function of Log-Normal with Rayleigh and Rician P.E models has also been derived. The expression for ASER is derived and analyzed against the LASER average received irradiance, optical beam width, P.E standard deviation, receiver side aperture, and SNR. The selection of these parameters is critically important to achieve better performance in weak to strong turbulence conditions accordingly. Moreover, ASER performance is observed against SNR and Monte Carlo simulation is performed that validates the theoretical results, at higher values of SNR simulation and theoretical results show a close agreement.