# Improved Performance in the Detection of ACO-OFDM Modulated Signals Using Deep Learning Modules
## 2 ACO-OFDM System Model
### ACO-OFDM Signal Transmission over FSO Turbulent Channel  
A schematic of the ACO-OFDM encoder-decoder system is shown in Figure 1.  
At the transmitter, the input data bits are converted to complex symbols by M-QAM and then converted to N parallel symbols, where N is the number of the subcarriers of the OOFDM system. Figure 1: Asymmetrically clipped optical orthogonal frequency division multiplexing (ACO-OFDM) system. After this, Hermitian symmetry is realized for the N parallel symbols, as in Equations (6) and (7). In ACO-OFDM, only the odd subcarriers are used, and the even subcarriers are zeroed in order to get real values of the signals in the output of the N point Inverse fast Fourier transform (N-IFFT) [6]:  
\[X_{0}=X^{*}_{\frac{\text{N}}{2}}=0, \tag{6}\]  
\[X_{\text{N-i}}=X^{*}_{\text{i}}\ \ \text{i}=1,2,\ldots,\ \frac{\text{N}}{2}-1. \tag{7}\]  
After achieving the Hermitian symmetry, we obtain  
\[0,X_{1},0,\ X_{3},\ldots,X_{\frac{\text{N}}{2}-1},0,\ X^{*}_{\frac{\text{N}}{ 2}-1},0,\ldots,\ X^{*}_{1}. \tag{8}\]  
Due to the use of Hermitian symmetry with only the odd subcarriers, after the IFFT (in the time domain of the signals), we obtain the anti-symmetry property, as obtained in (9), and the imaginary parts of the time domain signals are zero:  
\[x_{\text{i}}=-x_{\text{i}+\frac{\text{N}}{2}}. \tag{9}\]  
After the IFFT, the N parallel time domain signals are converted to one serial stream to be transmitted in time domain. Then, a cyclic prefix (CP) is added to the transmitted signals in order to prevent ISI. The negative values are clipped. This does not lead to any loss of information due to the anti-symmetry property in (9). Then, the transmitted data passes through the FSO atmospheric turbulent channel with additive white Gaussian noise (AWGN) and arrives at the receiver. At the receiver, a photo detector detects the received power of the signal, and a reverse process back to the transmitter is implemented. The BER performance of ACO-OFDM modulated signals with any M-QAM constellation which propagates through a turbulent channel can be calculated as follows [26; 27; 28]:  
\[P_{\text{error}}\cong\frac{4\left(\sqrt{\text{M}}-1\right)}{\sqrt{\text{M}} \log_{2}^{\sqrt{\text{M}}}}\text{erfc}\left(\sqrt{\frac{3\log_{2}^{\text{M}} \gamma(1)}{2(\text{M}-1)}}\right), \tag{10}\]  
where M is the number of the constellation, \(\gamma(\text{I})\) is the electrical signal to noise ratio (SNR) per bit, and \(\text{erfc}(. )\) is the complementary error function. Then, the average BER of the ACO-OFDM transmitted data over the noisy channel can be obtained by averaging (10) over the scintillation distribution:  
\[\text{BER}=\int_{0}^{\infty}f_{\text{I}}(\text{I}).\frac{4\left(\sqrt{\text{ M}}-1\right)}{\sqrt{\text{M}}\log_{2}^{\sqrt{\text{M}}}}\text{erfc}\left(\sqrt{ \frac{3\log_{2}^{\text{M}}\gamma(1)}{2(\text{M}-1)}}\right)\text{d}I, \tag{11}\]  
where \(f_{\text{I}}(\text{I})\) is the PDF of the received signal at the receiver, and I is the intensity of the transmitted signal. When the channel is characterized by weak or strong turbulence, the PDF of the received signal is lognormal and Gamma-Gamma distributed, as shown in (3) and (4), respectively. In the two cases, \(f_{\text{I}}(\text{I})\) depends on the scintillation index parameter \(\sigma_{\text{R}}^{2}\) and affects the BER performance of the ACO-OFDM system. When the value of \(\sigma_{\text{R}}^{2}\) increases, the BER also increases. Therefore, in order to recover the data with a minimum BER resulting from the turbulence, it is very important to estimate accurately the CSI of the turbulent channel in order to help mitigate the turbulence. When the channel is with strong turbulence or the turbulence over the channel is unknown or variable, it is difficult to estimate the CSI accurately and to implement ACO-OFDM over FSO turbulence channels. Therefore, there is a great need to enable the use of this modulation scheme over FSO communication channels without knowing the parameters of the channel.