the Transmission of Colour Image Over Double Generalized Gamma FSO Channel

1 Abstract —In this paper performance analysis of colour image Free Space Optics (FSO) transmission over Double Generalized Gamma (DGG) turbulence communication channel is carried out. At the reception side, we have used an average bit error rate (ABER) for reconstructed image performance measure, as the function of FSO link transmission parameters, such as propagation distance, Rytov variance and turbulence shaping and severity parameters (γ 1 , γ 2 , m 1 , m 2 ). Obtained results cover a large number of colour image FSO transmission scenarios, for Gamma-Gamma, Double-Weibull and K turbulence models channels considered as special cases.

fluctuations of received FSO signal [3]- [5]. The scintillation index is the measure of the turbulence strength, where lower values lead to less intensity variations and, vice versa, higher index values result in the higher values of turbulences. Refraction index Cn 2 [6], is a variable that depends on the geographic location, altitude and time of the day. The refraction deviation index results in scintillation, the deviation and spread of the beam. The characteristic values of refraction index in terrestrial communication are in the range 10 -17 -10 -13 m -2/3 [4], [5], while in moderate turbulences it has a value of about 10 -15 m -2/3 [7]. In order to obtain accurate modelling of FSO propagation, various mathematical and numerical models, have been provided in the literature. Weibull distribution model is mostly used for modelling of scintillation of signal with different intensities of turbulences and is often applied in systems with large aperture on the receiving side [8]. Log-Normal distribution model [9] is used for modelling scintillation related to the regimes of weak atmospheric turbulence [3], [10]. Rician distribution model is used for description of scintillation that occurs in terrestrial communication channels in sparsely populated areas and suburbs [11]. Gamma-Gamma distribution is a very simplified model of scintillation that can be applied for other regimes of turbulences [12]. However, recently Double Generalized Gamma (DGG) turbulence model [13], [14] has been considered since it generalizes many existing turbulence channel model and provides an excellent fit to the plane and spherical waves simulation data. This model can be reduced to other previously mentioned models (except for the Rician turbulence model) of scintillation by setting the corresponding values to the DGG model parameters. In [15] FSO image transmission over Rician turbulence channel was considered, so this analysis can be considered as an extension and supplementation of results provided in [15].
In this paper we have analysed performances of image (colour image-fire) transmission over DGG channel as the function of FSO transmission parameters within its theoretic boundaries. In the Section II basic assumptions of Double Generalized Gamma FSO transmission have been presented. Obtained simulation results and performance analysis are presented in the Section III. Concluding remarks are given in the Section IV.

II. SYSTEM MODEL
At the aperture plane of the receiver, the received FSO signal is modeled as where r is the position vector on the receive aperture plane, fc is the optical carrier frequency, and us(t)exp(jθ(t)) denotes the complex envelope of the modulation signal. Here, χ(r) is the turbulence-induced amplitude fluctuations and ϕ(r) is the phase variations of the channel. At the output photocurrent can be modelled as where nT(t) denotes total noise at the receiver expressed as [16], [17] ( ) , where Th n denotes thermal noise and sh n denotes shot noise.
Thermal noise can be modeled as the stationary Gaussian random process with the zero-mean value expressed as [16], where B k denotes Boltzmann constant and n F denotes amplifier noise figure, T and L R are temperature and load resistance respectively, f  is the effective noise bandwidth.
The effective bandwidth is dependent on the bit rate, b R , as Shot noise can be modelled as the stationary zero-mean Gaussian random process expressed as [16], [17] 2 2 2 , where q represents an electron charge, g and R represent gain and responsivity, respectively, Pt denotes transmitted optical power, m denotes modulation index, I represent accounted normalized irradiance, and A F denotes the excess noise factor expressed as [17] (1 )(2 1 ), where A k is the ionization factor.
The total noise is obtained as a sum of thermal and shot noise variances and expressed as Information carrying part of signal can be modeled as where fIF = fc -fLO is the equivalent signal frequency, fLO is local oscillator frequency, θIF is the equivalent signal phase and α represents the effective FSO fading fluctuation of the channel. As shown in [14], in order to model simultaneous effects of turbulence-induced amplitude fluctuations and phase aberrations, the Probability Density Function (PDF) of FSO fading amplitude will be modelled with Double Generalized Gamma distribution, as 2 1 [18], Γ(x) denotes special Gamma function [18], p and q are positive integer numbers that satisfy p/q = γ1/γ2, and , Ω 1, Ω 2, m1, m2, are parameters of Generalized Gamma distributions, which model statistically independent random processes arising respectively from large scale and small scale turbulent eddies. Parameters m1, m2 are shaping parameters defining the turbulence-induced fading, while parameters γ1, γ2 are defining the severity levels of statistically independent irradiances forming the DGG. These parameters, and their corresponding constant values along the FSO can be identified using the moments of small and large scale irradiance fluctuations and are directly tied to the atmospheric parameters (the ratio of Fresnel zone, Rytov variance, σ 2 Rytov, proportional to the scintillation index, and function of wavenumber, k = 2π/λ, the refractive index structure constant, and the propagation distance) as shown in (9a), (9b) and (10) from [13]. Parameters Ω 1 and Ω 2 depend on and directly are tied to the atmospheric conditions. Assuming a plane wave when inner scale effects are considered, the variances for the large-scale and the small-scale scintillations are given by [13] in the forms of the ratio of Fresnel zone to finite inner scale and Rytov variance (σ 2 where k = 2π/λ is the wave-number, λ is the wavelenght, Lpropagation distance and Cn 2 refraction index. Finally, received instantaneous SNR of the system after demodulation is given as and Is=|us(t)| 2 is the average intensity of the optical field, and Px is the output signal power. Algorithm for simulating FSO transmission of colour image (image of fire) is accomplished in steps explained in [19].

III. SIMULATION AND PERFORMANCE ANALYSIS
To simulate FSO transmission of the fire images over the Double Generalized Gamma turbulence fading channel the following experiment is conducted: Step 1: The original colour images ( Fig. 1(a) and Fig. 1(b)) are imported from the base and the source coding is done.
Step 2: The obtained binary signal was transmitted by BPSK modulation.
Step 3: The BPSK-modulated signal is transmitted over the DGG fading channel with added noise and with different values of the Rytov variance.
Step 4: At the receiver side, the signal is decoded and images were reconstructed.
Step 5: Obtained images are analysed.
(a) (b) Fig. 1. Basic Images: a) im1; b) where xij -pixel of original image, yij -pixel of transmitted image, n-number of bits, M  N-the size of the image, and  denotes EXOR operator over each of n pair of bits from xij and yij. For the purpose of the experiment we have used the image set shown in the Fig. 1 [20]. The values of the Rytov variance (obtained for different values of propagation distance and refraction index) σ 2 Rytov are varied in the interval {0.3, 0.83, 1.0, 1.2, 2.29, 3.96} and applied for each transmitted image.
In the Fig. 2-Fig. 4 performance of reconstructed Image im1 is shown, as the function of refraction index and propagation length change for the given sets of parameters γ1, γ2, m1, m2 defined to match most common scenarios of plane wave turbulence as explained in [13]. From the figures it can be seen how performance deteriorates as the length of FSO propagation link increases. Also it is visible how change of refraction structure parameter Cn 2 affects the BER for given turbulence channel performance. The increase of the parameter Cn 2 value, causes further increase of the BER for all observed values of SNR.  In the Fig. 5 BER performance of reconstructed Image 1 are given, in the function of turbulence shaping and severity parameters γ1, γ2, m1, m2. From the figure it can be seen that decrease of the parameter γ2 leads to the increase of the BER for all the observed values of SNR (curve denoted with -☆-in comparison with curve denoted with -■-). It can be seen that, the increase of the parameter m2 leads to decrease of the BER for all the observed values of SNR (curve denoted with -▲-in comparison with curve denoted with -■-), while the increase of the parameter m1 leads to the increase of the BER (curve denoted with -•-in comparison with curve denoted with -■-).  Our goal was to determine theoretical BER values for some last mile FSO transmission realizations in such observed scenarios. In this way we have determined just lower boundary values of the quality of transmitted image, since by introducing the usage of some of well-known error correction code techniques (as explained i.e. in [21]), additional quality improvement can be achieved.

IV. CONCLUSIONS
In this paper, we have investigated the BER performance of colour image FSO transmission over DGG turbulence channels. We have presented the obtained BER performance as the function of various FSO link parameters. It is shown that the increase of the parameter Cn 2 value causes an increase of the BER. Also it can be concluded that the decrease of parameter γ2 value leads to the increase of the BER for all the observed values of SNR.
The analysis has been carried out for very general turbulence scenario, which can be reduced to many other ones. By applying the analysis presented in this work, FSO link designers can determine boundary values for achieving required BER values in the wide span of colour image last-mile transmission scenarios. Presented analysis could also serve as good starting point for FSO transmission analysis of the video signal over turbulence channels in corresponding last mile realizations.