## 5 Kramers-Kronig Detection Scheme
### Principles of the KK Algorithm
#### 5.1.2 Modified KK Algorithms  
As stated earlier, some technical challenges require significant attention for the effective implementation of the KK algorithm. Table 6 presents some modification efforts to address the issues. They are grouped by the matter that they tried to address. For instance, some schemes tried to address the required upsampling, while some focused on the demanded high CSPR. However, it has been observed that for perfect reconstruction, most of the efforts usually achieve their aims at the expense of additional penalties that are related to factors such as CSPR, complexity, latency, bandwidth, and cost. As previously mentioned, the spectral broadening is a result of nonlinear operations, and a viable solution has to find a way of getting around their execution. This can be achieved by exploiting a DSP algorithm that isdigital upsampling-free [18]. For instance, suitable mathematical approximations can be employed to get around the associated nonlinear operations of the conventional KK algorithm. In this context, the square root of \(I(t)\) can be approximated to a second-order binomial expansion. Furthermore, by exploiting the minimum-phase condition, the phase of the signal can also be approximated. The function, \(\ln\sqrt{I(t)}\), can also be approximated based on the second-order Taylor expansion [45; 46]. Unlike the conventional KK algorithm that operates at 6 Sa/symbol, the algorithm that is based on mathematical approximations can operate at 2 Sa/symbol [46]. Moreover, the implementation is capable of exhibiting similar performance to that of the conventional KK scheme [46]. Besides the logarithm function that can be approximated, the exponential function can also be eliminated through the Cartesian-form representation of the complex signal [44]. It is noteworthy that some approaches may still comprise some nonlinear operations such as square and square root functions in the modification process. Nevertheless, a comparative study has established that the associated nonlinear operations of the upsampling-free KK algorithm present a negligible contribution to the spectral broadening. However, the upsampling-free algorithm limitation is that the performance difference between its implementation and the conventional KK algorithm can be significant when the required SNR is absolutely high [44]. Owing to the related multiple iterations of some approaches, considerable complexity and latency can be incurred by their application [18]. Additionally, to satisfy the minimum-phase condition, the conventional KK system demands a strong CW tone whose amplitude is comparatively greater than that of the information-bearing signal. When the minimum-phase condition is met, the field reconstruction can be accurately achieved through the KK relation at the expense of a high CSPR [48; 49]. However, the requirement for a strong carrier imposes a significant limitation  
\begin{table}
\begin{tabular}{p{42.7 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Reference** & **Related Effect** & **Adopted Technique** & **Advantages** & & & **Tradeoff** \\ \hline \multicolumn{2}{p{42.7 pt}}{**Digital upsampling**} & & & & & & \\ \hline
[18] & & & DSP algorithm without digital upsampling & & & & \\ \(\bullet\) & high complexity spectral broadening & & & & & \\ \(\bullet\) & high power consumption & & & & & \\ \([\)45; 46\(]\) & & & & & & & \\ \([\)44\(]\) & & & & & & & \\ \([\)43\(]\) & & & & & & & \\ \hline \multicolumn{2}{p{42.7 pt}}{**High CSPR**} & & & & & & \\ \hline
[49] & & & flexible adaptive dispersion compensation & & & & \\ \([\)50\(]\) & \(\bullet\) & limited power efficiency & & & & & \\ \(\bullet\) & nonlinear effect increase & & & & & \\ \(\bullet\) & highers reach receiver sensitivity & & & & & \\ \([\)51\(]\) & & & & & & \\ \([\)52\(]\) & & &  
on the transmitter and can also increase the nonlinear fiber propagation effects in the direct detection systems [49].