## 3 KK-Based FSO Communications  
In a KK-based FSO receiver, because a direct detection receiver is used, the optical signal phase is lost. However, it is possible to retrieve the signal phase from the KK relation. Let \(m(t)\) be a complex message signal whose spectrum lies between \(-B/2\) and \(B/2\) (see Figure 2b), where \(B/2\) is the electrical carrier frequency. The carrier is modulated by \(m(t)\) and a DC bias \(K\) is added (see Figure 2a). Let  
\[y(t)=K+m(t)e^{-j\pi Bt} \tag{1}\]The absolute of \(y(t)\), \(Z(t)\) is the input of a dual drive Mach-Zehnder modulator. Let the complex field output of the optical transmitter be [6]  
\[s(t)=\sqrt{P_{0}}Z(t)e^{-j(2\pi f_{c}t+\theta(t))}, \tag{2}\]  
where \(f_{c}\) is the optical carrier frequency, \(\theta(t)\) is the laser phase noise at the transmitter and \(P_{0}\) is the carrier power. The DC bias, \(K\) leads to a pilot beam as shown in Figure 2d. It can be shown that \(y(t)\) is a minimum phase signal if and only if the winding number of the time trajectory in the complex plane is zero [12; 13; 14]. The condition \(|K|>\max\left[\,|\,m(t)\,|\,\right]\) is sufficient for guaranteeing the minimum phase properties [14]. Let  
\[y(t)=Z(t)e^{j\varphi(t)}, \tag{3}\]  
where \(Z(t)\) and \(\varphi(t)\) are the amplitude and phase of \(y(t)\). When \(y(t)\) is a minimum phase signal, \(Z(t)\) and \(\varphi(t)\) are related by the Hilbert transform,  
\[\varphi(t)=\frac{1}{\pi}p.v.\int_{-\infty}^{\infty}dt^{\prime}\frac{\log[Z(t^ {\prime})]}{t-t^{\prime}}, \tag{4}\]  
where \(p.v.\) stands for principal value.  
Figure 3 shows the FSO system based on KK relation. At the receiver, a direct detection receiver is employed. The photocurrent is given by  
\[I(t)=Rr(t)=RH|s(t)|_{2}, \tag{5}\]  
where \(R\) is the responsivity of the photodiode, \(r(t)\) is the received power, and \(H\) is the channel gain. Using Equation (2) in Equation (5), we see that the photocurrent does not depend on the transmitter laser phase noise, \(\theta(t)\). In Equation (5), noise is neglected. Using the Equations (2) and (3) in Equation (5), we find  
\[Z(t)=\sqrt{\frac{I(t)}{RHP_{0}}}. \tag{6}\]  
Figure 2: (**a**) Schematic of the KK digital transmitter, (**b**) message spectrum, (**c**) spectrum of \(y(t)\), and (**d**) spectrum of optical transmitter output, i.e., after optical modulation.  
Using Equation (6) in Equation (4), we obtain  
\[\varphi(t)=\frac{1}{2\pi}p.v.\int_{-\infty}^{\infty}dt^{\prime}\frac{\log[I(t)/( RHP_{0})]}{t-t^{\prime}}, \tag{7}\]  
Thus, we can retrieve the phase of \(y(t)\). From Equations (1), (3) and (6), we have  
\[m(t)=\left[\sqrt{\frac{I(t)}{RHP_{0}}}e^{i\varphi(t)}-K\right]e^{j\pi Bt}. \tag{8}\]  
The convolution in Equation (7) can be conveniently implemented in frequency domain as  
\[\widetilde{\varphi}(\omega)=\frac{i}{2}sign(\omega)F\left\{\log\left[\frac{I( t)}{RHP_{0}}\right]\right\}, \tag{9}\]  
\[\varphi(t)=F^{-1}\{\widetilde{\varphi}(\omega)\}, \tag{10}\]  
where \(F\) denotes the Fourier transform and \(sign(\cdot)\) is the sign function,  
\[sign(\omega)=\left\{\begin{array}{ccc}-1&for&\omega>0\\ 0&for&\omega=0\\ 1&for&\omega<0\end{array}\right. \tag{11}\]  
As shown in Figure 3, the receiver photocurrent passes through an analog-to-digital converter (ADC) and then through the receiver digital signal processing (DSP) unit (Rx-DSP). The phase retriever (Equations (9) and (10)) is implemented in the Rx-DSP. The complex message signal \(m(t)\) is estimated using Equation (8). The logarithm appearing in Equation (7) introduces spectral broadening and hence, digital up-sampling of the received photocurrent is required. The block diagram of KK transceiver can be more generalized for wavelength division multiplexing (WDM) by adding a multiplexer and a demultiplexer at the transmitter and receiver, respectively.  
Figure 3: Schematic of the KK transceiver for FSO communications.