## 5 Kramers-Kronig Detection Scheme
### Principles of the KK Algorithm
#### 5.1.2 Modified KK Algorithms
Consequently, the concern about the high CSPR is the additional sensitivity penalty being presented [48]. As a result, it is highly imperative to ease the CSPR constraint without making any significant sacrifice of the system performance [51]. In this regard, there have been concerted efforts on the means of lowering the required CSPR in the conventional KK system. For instance, an enhanced KK algorithm that is capable of accomplishing the conventional KK scheme's performance but with a \(\sim\)3 dB lower CSPR can be employed [49]. The scheme integrates a flexible adaptive dispersion compensation into its algorithm, and it is based on an independent, but direct intensity measurement of the received signal replicas. In this, two replicas are measured, and one of the replicas is to some extent dispersed compared with the other. However, the scheme presents additional hardware (i.e., a dispersive element, an optical splitter, and an optical demultiplexer) cost due to the requirements of two independent direct detection paths [49] and high iterative DSP [50]. It should be noted that the related cost can be insignificant in shared WDM channels [49]. The required high CSPR for the KK subsystem can also be relaxed by exploiting an enhanced SSBN mitigation algorithm that is based on the KK relation. The scheme can offer a reduction in the optimal CSPRs, and the OSNR can be improved as well. For instance, a 2-3 dB reduction in the CSPRs can be realized [51]. A modified KK algorithm that functions in comparatively low CSPR conditions has been demonstrated [52]. Unlike the conventional scheme where a strong CW is just appended to the complex SSB signal to fulfill the minimum-phase condition, in the presented reconstruction scheme, an exponential operation is applied to the complex SSB signal. Therefore, the required condition is inherently satisfied, and the original signal can be accurately reconstructed even at a low CSPR value. Nevertheless, the scheme just retrieves the information-bearing signal rather than the optical field. The scheme can also be modified to reconstruct the optical signal field by means of the KK relation. However, this is at the expense of the spectral broadening caused by the exponential operation. This results in the need for a greater system bandwidth compared with the conventional KK scheme [48]. In addition, it has been observed that most of the KK algorithm modification efforts to either prevent the use of digital upsampling or implement limited upsampling have deviated from the fundamental algorithm by employing a number of linear approximations for the nonlinear operations of the KK algorithm. Based on this, most of the modified algorithms achieve their objectives at the expense of a measurable power regarding the required CSPR for perfect reconstruction. In this context, an alternative approach to the approximation-based algorithm modification with minimal modification can be employed. Furthermore, the approach can offer suitable performance at a relatively lesser CSPR penalty [43]. Furthermore, the requirement for extensive DSP can also be addressed. This can be achieved by exploiting the scheme that is based on a SiP hardware SSBN cancellation [50]. The scheme is based on the concept that the SSBN can be removed through either software- or hardware-based approaches. The software-based SSBN cancellation approach prevents additional hardware complexity. However, they usually require an iterative process to effectively alleviate the effect of inessential elements that can hinder the desired signal [18]. On the other hand, the minimum-phase condition is not required in the hardware-based SSBN cancellation. This results in a lower achievable CSPR compared with the software-based cancellation. Besides the SSBN cancellation, a relatively lower cost and a smaller footprint can be achieved with the implementation. Compared with the enhanced KK scheme, the required CSPR can be lowered by 3 dB without any complexity increase [50].