## 5 Kramers-Kronig Detection Scheme
### Principles of the KK Algorithm  
An SCOH KK scheme is a direct detection Co-Rx in which the cost-effectiveness of the direct detection system and the high-performance benefits of COH transmission are jointly exploited. As previously mentioned, for a minimum-phase optical SSB signal, the phase information can be retrieved from its intensity by the KK relation. This offers a high potential for the direct detection optical SSB signal that is really robust to the linear transmission impairments and SSBN. Nonetheless, one of the related technical challenges of the conventional KK scheme implementation is the requirement for the DSP block to function at a sampling rate that is faster than the one stipulated by Nyquist. This challenge can be associated with certain nonlinear operations such as the exponential and logarithm functions that are required in the KK algorithm [119]. The associated nonlinear operations can considerably broaden the spectrum of the signal. Consequently, to address the induced spectral broadening, it is imperative to implement a digital upsampling prior to the DSP [18; 43; 44; 120]. Furthermore, detailed information on why digital upsampling is  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Technique & \multicolumn{2}{c}{**Advantages**} & \multicolumn{2}{c}{**Disadvantages**} & **Reference** \\ \hline Single-Stage Linearization Filter & \(\bullet\) & no need for multiple symbol decision-based SSBN reconstruction procedures & \(\bullet\) & offers partial compensation presents limited compensation performance & [22; 29; 31] \\  & \(\bullet\) & low DSP complexity & \(\bullet\) & reduced performance difference between the Tx-EDC and Rx-EDC \\ \hline Two-Stage Linearization Filter & \(\bullet\) & improved compensation gain & \(\bullet\) & possesses residual beating terms & [29] \\ \hline Iterative Linearization Filter & \(\bullet\) & enhanced compensation gain & \(\bullet\) & high DSP complexity & [29; 50; 57; 58; 118] \\  & \(\bullet\) & offers identical performance for the Rx-EDC and Tx-EDC & \(\bullet\) & incurs significant latency & \\ \hline SSBN Estimation and Cancellation & \(\bullet\) & presents no additional unwanted beating product & \(\bullet\) & depends on the symbol decision-making accuracy & [29; 59] \\  & \(\bullet\) & no need for multiple symbol decision-based SSBN reconstruction procedures & \(\bullet\) & exhibits performance degradation at lower OSNRs & \\  & \(\bullet\) & offers improved compensation gain & & OSNRs & \\ \hline Kramers-Kronig & \(\bullet\) & offers the best performance & \(\bullet\) & demands a high digital upsampling rate & [29; 45; 46; 50; 55] \\  & & & requires a high CSPR value & \\  & & & exhibits a bandwidth broadening effect & \\ \hline DSP: digital signal processing: CSPR: carrier-to-signal-power ratio; OSNR: optical signal-to-noise ratio; SSBN: signal–signal beat noise; EDC: electronic dispersion compensation; Tx-EDC: transmitter-based EDC; Rx-EDC: receiver-based EDC. & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: A summary of SSBN mitigation techniques.  
essential was given in [43]. Based on this, a larger memory and faster processing speed are demanded in the KK algorithm's DSP chip. Experimental demonstrations have also revealed that to realize an optimum receiver sensitivity, typical values of the required samples per symbol are in the range of 4-6 Sa/symbol, [29; 30; 45; 46; 61; 109]. Therefore, the KK scheme's real-time implementation may be hindered by the increase in the number of samples at DSP. Moreover, DSP can be implemented in parallel to lower the specified processing speed of the chip. However, the power consumption and complexity will be considerably increased [44; 46]. Based on this, it is highly important to modify the KK DSP algorithm. In the following subsections, the conventional KK algorithm, as well as the modified alternatives are discussed.