Performance Analysis of Coherent Source SAC OCDMA in Free Space Optical Communication Systems

Abstract

In this paper, we investigate the performance of spectral amplitude coding optical code division multiple access (SAC OCDMA) systems under the effect of beat noise and turbulence. Three different multi-laser source configurations are considered in this analysis: shared multi-laser, separate multi-laser, and carefully controlled center frequency separate multi-laser. We demonstrate through Monte Carlo simulation that the gamma–gamma probability density function (pdf) cannot adequately approximate the measured intensity of overlapping lasers and that an empirical pdf is required. Results also show it is possible to achieve error-free transmission at a symmetrical data rate of 10 Gbps for all active users when only beat noise is taken into account by precisely controlling the center frequencies. However, only 30% of the active users can be supported when both beat noise and turbulence are considered.

1. Introduction

Free space optical (FSO) communications is a subset of optical wireless communication that has recently attracted increased attention due to its potential to replace microwave links in symmetrical and asymmetrical point-to-point communications. In addition to its unrestricted, extremely large (measured in terahertz) bandwidth and straightforward transmitter and receiver designs, FSO communication has become very popular. Figure 1 shows a simple FSO communications system.

The propagation of an optical carrier in the FSO channel is subject to a number of impairments. Photon absorption and Rayleigh scattering are the main sources of attenuation of the optical signal. However, a more severe problem that affects the optical carrier is turbulence [1,2,3,4,5]. Turbulence is the random variation of temperature in the atmosphere, which results in a variation in the refractive index of the air particles, which causes fluctuation in the intensity and phase of the optical carrier. Figure 2 shows an illustration of a turbulent channel effect on a received optical signal. It has been shown that the detected intensity of an optical signal passing through a turbulent channel can be modeled by the gamma–gamma probability density function (pdf) [3].

If FSO communications are to effectively compete with terrestrial communication systems, the enormously huge bandwidth must be utilized to its fullest extent. The use of wavelength division multiplexing (WDM) and polarization shift keying has been studied in [6] to achieve higher bandwidth. Additionally, in [7], a dual polarization ultra-dense WDM has been implemented to achieve higher data rates. Nawawi et al. proposed in [8] to use of a catenated orthogonal frequency division multiplexing (OFDM) modulation scheme with zero cross-correlation codes to improve the cardinality and performance. Harun et al. prosed to use a multiphoton quantum communication with multiple Beams [9] to reduce the quantum bit error rate (QBER). Bai et al. proposed in [10] the use of polarization multiplexing with direct optical code division multiple access (OCDMA) in an FSO channel, where a gamma–gamma pdf is applicable. Moghaddasi et al. studied in [11,12] the use of coherent sources in the OCDMA system to increase the data rates.

Spectral amplitude coding (SAC) OCDMA is a simple spectral-encoded OCDMA system that has been proposed for both optical fiber and FSO communications systems [13,14]. A broadband source is spectrally divided into spectral bins, and each user is assigned a specific number of bins to represent their individual code. In most codes, there is an overlapping between the spectral bins for the different users, thus leading to multi-access interference (MAI) [15]. The SAC OCDMA transmitter and receiver structures are shown in Figure 3a and Figure 3b, respectively.

A multi-laser source is fed into the encoder, where only a select few wavelengths (which stand in for the user’s individual code) pass, and the others are blocked. At the receiver, the received signal is split into two branches, the decoder branch, and the complementary decoder branch. The decoder has the same spectral response as the encoder, which implies that it is capable of passing the encoder’s produced wavelengths. The complementary decoder branch passes all other wavelengths because it has a complementary spectral response to the encoder. By subtracting these two signals from each other, only the desired user data will be present, and the MAI will be canceled [13].

The majority of the literature has focused on using light-emitting diodes (LED) and amplified spontaneous emission (ASE) sources in the design of SAC OCDMA systems because it requires a broadband optical source [3]. However, these sources perform weakly in the FSO environment; thus, the preferred light source is the coherent source [11]. Sources of coherent nature have been suggested for use in SAC OCDMA systems, and they have been shown to perform well under specific conditions [11,16,17,18].

From the literature, it has been shown that the primary sources of noise in SAC OCDMA systems are phase-induced intensity noise when the incoherent light source (LED and ASE) is used and beat noise when coherent light sources are used [14,19,20].

In this paper, we investigate the performance of coherent source FSO SAC OCDMA system under the effect of both beat noise and turbulence noise. In our study, we consider three different coherent light sources, a centralized shared multiple laser source, a uniformly distributed multi-laser source, and a controlled multi-laser source. Figure 4 shows bin occupation for the three sources.

2. The Probability Density Function of Intensity

In order to assess the performance of the coherent light source FSO SAC OCDMA system, the pdf of the received signal must be examined in relation to both beat noise and turbulence. The overlapping of the different users’ bins occurs in the spectral domain because SAC OCDMA is a spectrally encoded OCDMA technique and due to the lack of an analytical expression for the pdf for a number of overlapping laser diodes with the same center wavelength and random phase and polarization angles, Monte Carlo simulation will be used [21,22].

2.1. The Characteristic Functions and Probability Density Functions for Different Sources

2.1.1. Shared Multi Laser Source

Here, we assume that all users share a common multi-laser source; based on this assumption, for each spectral bin, all the overlapping users will have the same central frequency. However, since the different users are all located at different places from the centralized source, the polarization and phase angles will be random. A visualization of the distribution of the frequency of the laser within a spectral bin is given in Figure 4a.

To obtain the pdf, we start by obtaining the characteristic function; we follow the same steps as in [17]. The electric field can be expressed as

$$E_{u,i}=\left[\begin{array}{c}\sin ({\varphi }_{u,i})\exp ({\gamma }_{u,i})\\ \cos ({\varphi }_{u,i})\exp ({\gamma }_{u,i})\end{array}\right]\exp (j{\omega }_{u}t)$$

(1)

where u is the intended user, φ_u,i, γ_u,i, and ω_u are the polarization, phase, and angular frequency, respectively, i is a specific realization of the angles, and t is time. It is assumed that the distribution of all the angles over the period [0, 2π] is uniform.

In the presence of L overlapping lasers, the detected intensity for specific polarization and phase angles can be given by

$$I_{L,i}=\left({\displaystyle \sum _{u=1}^L{E}_{u,i}}\right)\left({\displaystyle \sum _{u=1}^L{E}_{u,i}^{\ast }}\right)$$

(2)

where E* denotes the conjugate of the received signal. We begin by computing the cumulative distribution function (cdf) by creating 10⁷ distinct realizations of the polarization and phase angles in order to obtain the empirical pdf:

$$F_B(x|L)=\frac{1}{{10}^7}{\displaystyle \sum _{i=1}^{{10}^7}{1}_{\{I_{L,i}\le x\}}}$$

(3)

where the value of 1_{$I_{L,i}\le x$_} is equal to “1” if the inequality is satisfied and to “0” in all other cases.

Through differentiation, the pdf may be generated from the cdf:

$$f_B(x|L)=\frac{d(F(x|L))}{dx}$$

(4)

and by taking the Fourier transform of the pdf, we can obtain the characteristic function (cf):

$${\mathsf{\Omega }}_B(\omega |L)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x|L)e^{-j\omega x}}dx$$

(5)

Equation (5) only takes into consideration the beat signal generated by the multiple lasers. In order accurately model the FSO channel, we need to include the effect of turbulence.

The pdf of the irradiance or the detected intensity (I) is assumed to have a gamma-gamma distribution [11]:

$$f_T\left(I\right)=\frac{2{\left(\alpha \beta \right)}^{\frac{\left(\alpha +\beta \right)}{2}}}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}{I}^{\left[\frac{\left(\alpha +\beta \right)}{2}-1\right]}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta I}\right)$$

(6)

where Γ( ) and K_a( ) are the gamma function and the modified Bessel function of the second kind, respectively. The variables α and β are the gamma distribution coefficients and are given by [3]:

$$\alpha ={\left\{\exp \left[\frac{0.49{\sigma }_R^2}{{\left(1+1.1{\sigma }_R^{12/5}\right)}^{7/6}}-1\right]\right\}}^{-1}$$

(7)

$$\beta ={\left\{\exp \left[\frac{0.51{\sigma }_R^2}{{\left(1+0.69{\sigma }_R^{12/5}\right)}^{5/6}}-1\right]\right\}}^{-1}$$

(8)

where σ²_R is the Rytov variance, which is used to determine the turbulence strength. We can obtain cf from the pdf of the turbulence by

$${\mathsf{\Omega }}_T(\omega )={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{f}_T(I)e^{-j\omega x}}dx$$

(9)

To combine both beat noise and turbulence, we multiply the respective cf.

$${\mathsf{\Omega }}_I(\omega |L)={\mathsf{\Omega }}_B(\omega |L)\times {\mathsf{\Omega }}_T(\omega )$$

(10)

Thus, the overall pdf can be obtained by

$$f(x|L)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\mathsf{\Omega }(\omega |L)e^{j\omega x}}d\omega$$

(11)

2.1.2. Uniformly Distributed Central Frequency Source

Here, it is assumed that each user has a separate source composed of multiple lasers. Due to manufacturing flaws, the central frequencies for overlapping lasers will not be exactly aligned, but they will all be within the bin spectral width (B_o). If the gap between two or more lasers’ central frequencies is smaller than or equal to the electrical bandwidth (B_e), these lasers will beat together. This is equivalent to dividing the optical bandwidth (B_o) into r subdivisions, each of which is equal to the electrical bandwidth (B_e). Beating between lasers can only occur if the lasers fall within the same subdivision. A visual representation of this configuration is given in Figure 4b.

Thus, the cf for any given spectral bin is [21]:

$${\mathsf{\Omega }}_B(\omega |L)={\displaystyle \sum _{\left\{m\right\}}\frac{\left|D_m\right|}{r^L}{\mathsf{\Omega }}_B(\omega |L,m)}$$

(12)

where n and A_n are the weight vector and occupancy vector, respectively.

2.1.3. Controlled Central Frequency Source

It is assumed in this case that each user has a separate multi-laser source and that the center frequencies of these sources are controllable. For any SAC OCDMA code, there is a maximum number of overlapping lasers K (maximum cross-correlation) that fall within any spectral bin. This spectral bin bandwidth (B_o) is divided into K subdivisions, and each of the lasers is assigned to a subdivision. We presume that the subdivision’s center is where the laser’s central frequency is located. However, to account for the controllability of the lasers’ accuracy, a Gaussian approximation is considered for the central frequency position.

The cf for controlled lasers is given by [21]:

$${\mathsf{\Omega }}_B(\omega |L)={\displaystyle \sum _{b=0}^{L/2}P(b|L)\cdot {\mathsf{\Omega }}_B{(\omega |2)}^b\cdot {\mathsf{\Omega }}_B{(\omega |1)}^{L-2b}}$$

(13)

where b represents the total number of beating pairs and P(b|L) represents the conditional probability that these pairs are beating when there are L active lasers.

2.2. Bit Error Rate (BER)

The first step in computing the bit error rate (BER) is to determine the number of lasers present at bin x in the decoder branch, denoted by the letters d_x, and bin y in the complementary decoder branch, denoted by the letters cd_y. Secondly, using the computed occupation numbers (d_x and cd_y), the characteristic function is calculated for each bin. To retrieve the desired user’s signal and cancel the MAI, the detected intensity of the complementary decoder branch is subtracted from the detected intensity of the decoder branch. The overall characteristic function of the balanced detector is computed by multiplying the characteristic functions of the bins in the decoder branch by the characteristic functions of the bins in the complementary decoder branch (due to their independence).

Thus, the cf when the intended user is sending data “0” and data “1”, respectively:

$$\begin{array}{ll}{\mathsf{\Omega }}_0(\omega |U,T,m)=[{\mathsf{\Omega }}_I(\omega |& d_1)\dots {\mathsf{\Omega }}_I(\omega |d_x)]\\ & \times \left[{\mathsf{\Omega }}_I(-\omega /a|c{d}_1)\dots {\mathsf{\Omega }}_I(-\omega /a|c{d}_y)\right]\end{array}$$

(14)

and

$$\begin{array}{ll}{\mathsf{\Omega }}_1(\omega |U,T,m)=[{\mathsf{\Omega }}_I(\omega |& d_1+1)\dots {\mathsf{\Omega }}_I(\omega |d_x+1)]\\ & \times \left[{\mathsf{\Omega }}_I(-\omega /a|c{d}_1)\dots {\mathsf{\Omega }}_I(-\omega /a|c{d}_y)\right]\end{array}$$

(15)

where U is the desired user, T is a data vector, which elements are “0” or “1” based on what the different users are sending, and m is the number of interferers.

Equations (14) and (15) give the cf for a single realization of T. The occupation numbers (d_x and cd_y) for the different frequency bins will vary based on what the different users are sending. Thus, we have to take into consideration all the different combinations of the vector T.

Assuming that all the combinations of T are equally probable, then the cf for sending data “0” will be

$${\mathsf{\Omega }}_0(\omega |U,m)=E_T\left\{{\mathsf{\Omega }}_0(\omega |U,T,m)\right\}$$

(16)

where E_T is the expected value. Ω₁ can be derived in a similar way.

Assuming a binomial distribution for the data generation for all users (L), the cf for data “0” can be expressed by

$${\mathsf{\Omega }}_0\left(\omega |U,L-1\right)={\displaystyle \sum _{l=0}^{L-1}{\left(\frac{1}{2}\right)}^{L-1}\frac{\left(L-1\right)!}{l!\left(L-1-l\right)!}}{\mathsf{\Omega }}_0(\omega |U,m)$$

(17)

A similar equation can be derived for Ω₁.

After obtaining the pdfs for data “0” and “1” from the Fourier transform of the respective cfs, the BER can be expressed as

$$BER(U,L-1)=\frac{1}{2}\left({\displaystyle \underset{\gamma }{\overset{\infty }{\int }}{f}_0(x|U,L-1)dx}+{\displaystyle \underset{-\infty }{\overset{\gamma }{\int }}{f}_1(x|U,L-1)dx}\right)$$

(18)

where γ is the threshold value and its calculated numerically.

3. Results and Discussions

For our simulation, a balanced incomplete design block (BIBD) code with a weight of 6 (number of ones “1” in each code) and a cardinality (number of users) of 31 is used. This BIBD code was chosen because it has a fixed cross-correlation of 1, and the code length is equal to the number of users, which can provide a lower performance limit. The bin spectral width is 0.97 nm, and the system is evaluated at 1.25 Gbps and 10 Gbps, which determines the electrical bandwidth B_e.

3.1. The Probability Density Function for the Received FSO SAC OCDMA Signal

Figure 5 displays the gamma-gamma pdf model of the effect of turbulence on the FSO signal for three levels of turbulence strength: weak (σ²_R = 0.3), moderate (σ²_R = 0.8), and strong (σ²_R = 1.5). The figure shows that moderate and strong turbulence severely distorts the detected pdf.

Using Monte Carlo simulation, Figure 6 depicts the impact of beat noise and turbulence on the pdf of two overlapping lasers for various turbulence intensities. Figure 6 clearly illustrates how the pdf’s shape differs from the gamma–gamma pdf in Figure 5. As a result, an empirical model must be utilized as the gamma–gamma pdf cannot be used to mimic the effects of beat noise and turbulence. Additionally, Figure 7 shows that an empirical pdf is preferable unless there are 16 or more overlapping lasers, in which case a gamma pdf can be utilized as a good approximation.

3.2. The Bit Error Rate Versus the Number of Active Users

The following figures show the BER versus the number of active users for the three source configurations. Figure 8 shows the result for the shared source configuration at a data rate of 1.25 Gbps; as can be seen, the performance is limited by the high beat noise from the centralized frequencies. The BER versus the number of concurrent users when a distributed, randomly assigned central frequency source is used is shown in Figure 9. The figure shows the system is less susceptible to beat noise since the likelihood of the lasers beating has decreased. However, the turbulence strength has a clear effect on the system performance.

Figure 10 shows the performance of a distributed, finely regulated central frequency source with a 3 GHz accuracy SAC OCDMA system. The beat noise is significantly mitigated as a result of each laser’s center frequency being controlled; thus, the performance is mainly limited by turbulence strength. As can be seen, when only beat noise is considered, all of the 31 users can be active and also achieve error-free transmission (BER less than 10⁻⁹). Whereas, for strong and moderate turbulence, no error-free transmission is possible, with moderate strength giving a relatively better performance than strong turbulence. For weak turbulence, an error-free transmission can be achieved for only 10 concurrent users.

Figure 11 shows BER versus the active number of users for the same source configuration as in Figure 10 but at a data rate of 10 Gbps and central frequency position precision of 2GHz. Similarly, the beat noise is mitigated due to the higher precision of writing the frequency of the laser despite the higher data rate. It is also clear from the figure that weak turbulence can support a modest number of users, while moderate and strong turbulence leads to totally unintelligible communication.

4. Conclusions

In conclusion, in this paper, we investigated the performance of the FSO SAC OCDMA system employing coherent light sources. We showed that when both beat noise and channel turbulence are taken into account, a gamma–gamma pdf cannot adequately characterize the received signal and must be replaced by an empirical pdf. We also demonstrated that when there are more than 16 overlapping lasers, a gamma pdf may be utilized as a reliable approximation for the pdf. Additionally, the results showed that while beat noise can be reduced even at high data rates by precisely controlling the central frequencies, performance is still primarily constrained by the strength of the turbulence.