## 5 Kramers-Kronig Detection Scheme
### Principles of the KK Algorithm
#### 5.1.1 Conventional KK Algorithm  
A typical block diagram of the conventional KK scheme is depicted in Figure 8. Complex optical data \(s(t)\exp(\mathrm{i}\omega_{c}t)\) are superimposed on a copolarized reference optical carrier \(c\exp(\mathrm{i}\omega_{c}t)\) with a real-valued amplitude \(c\) at the transmitter. In this regard, an optical signal, \(E(t)\), that impinges on the direct detection receiver can be defined as [21]:  
\[E(t)=\left(c+s(t)\mathrm{e}^{\mathrm{i}\omega_{c}t}\right)\mathrm{e}^{\mathrm{ i}\omega_{c}t}, \tag{20}\]  
where \(\omega_{Z}=\omega_{s}-\omega_{c}\).  
In addition, the KK concept is based on the application of the Hilbert transform to an analytic time signal with known amplitude information only. Moreover, if the spectrum of the signal, \(a(t)=s(t)\exp(\mathrm{i}\omega_{Z}t)\), does not extend to negative frequencies, then it is analytic. Furthermore, in accordance with the KK concept, the carrier has to be sufficiently large to fulfill the minimum-phase condition (i.e., \(|s(t)|<c\)) with a reasonably high degree of confidence. In reality, this indicates that the CSPR must be larger than the signal PAPR to ensure a perfect DSP-based phase retrieval. Furthermore, it should be noted that the optimal CSPR required for a related PAPR depends on parameters such as the OSNR, the employed modulation format, and the received signal pulse shape [21].  
Figure 8: KK DSP algorithm chain for full-optical field reconstruction with: (1) received photocurrent two-sided power spectrum, (2) downconverted optical spectrum at a carrier frequency, and (3) reconstructed two-sided data spectrum. \(|S|^{2}\): signal–signal beating; SC\({}^{*}\): signal–carrier beating; \(|C|^{2}\): carrier–carrier beating.  
In addition, for a receiver with responsivity \(R\) with \(R=1\), after the square law detection by a single-ended PD, a current, \(I(t)\), that is proportion to the received electrical signal power can be realized as [44; 45; 46]:  
\[I(t)=|E(t)|^{2}=|c+s(t)\mathrm{e}^{\mathrm{i}\omega_{Z}t}|^{2} \tag{21}\]  
It should be noted that there are some contributions from the signal-signal beating \(|s(t)|^{2}\), the signal-carrier beating \(2\Re\{c\,s(t)\exp(\mathrm{i}\omega_{Z}t)\}\), as well as carrier-carrier mixing, \(c^{2}\), to the power spectrum of \(I(t)\), as illustrated in Figure 8[21]. Furthermore, an ADC is then used for sampling and quantizing the photocurrent \(I(t)\). Subsequently, the function \(\ln\sqrt{I(t)}\) is estimated from the measured data, where \(\ln\left(\bullet\right)\) denotes the natural logarithm operation. It is noteworthy that an unlimited number of higher harmonics will be generated due to the associated nonlinearities of the operation. Therefore, before executing the nonlinear operations, the output data of the ADC has to be sufficiently upsampled to inhibit aliasing errors [21; 44; 45; 46; 119].  
The conventional KK algorithm retrieves the phase information, \(\varphi(t)\), from the intensity information via the Hilbert transform as:  
\[\varphi(t)=\mathcal{H}[\ln\sqrt{I(t)}]. \tag{22}\]  
Consequently, the electrical field, \(E(t)\), of the downconverted optical signal, can be recovered as:  
\[E(t)=\sqrt{I(t)}\exp[\mathrm{i}\varphi(t)]=c+s(t)\ \exp(\mathrm{i}\omega_{Z}t). \tag{23}\]  
The next process is the removal of the reference carrier. Furthermore, the subsequent signal can then be shifted from the intermediate frequency, \(\omega_{Z}\), to the baseband. Thus, the obtained complex data signal \(s(t)\) can be defined as [21]:  
\[s(t)=\bigg{[}\sqrt{i(t)}\exp[\mathrm{i}\varphi(t)]-c\bigg{]}e^{-\mathrm{i} \omega_{Z}t}. \tag{24}\]  
Moreover, digital downsampling can also be implemented and further DSP such as electrical equalization, carrier recovery, and demodulation can then follow [21; 44; 45; 46]. In the following subsection, we discuss a number of modifications to the conventional KK algorithm.