# Two-Channel OTDM System for Data-Center Interconnects: A Review
## 2 Two-Channel OTDM System with Constant-Phase Pulse
### OTDM System Using a Multilayer Equalizer  
A multilayer equalizer for the two-channel OTDM system was proposed to reduce the BER floor (crucial when a higher-order modulation-formatted signal is used), as shown in Figure 6a [29]. The optical transmitter is identical to that shown in Figure 1. However, the demultiplexer is implemented with half-symbol-spaced equalizers. Thus, this system uses a multilayer equalizer consisting of a demultiplexer (as the first layer) and a MIMO  
Figure 4: Receiver sensitivities (@BER = \(3.8\times 10^{-3}\)) of 150 Gb/s OTDM-PAM-\(N\) signals depending on the pulse ER. Figure 5: Receiver sensitivities of 150 Gb/s OTDM-PAM-8 signals depending on the (**a**) power difference and (**b**) timing difference between two RZ signals. equalizer (as the second layer). The structure of the demultiplexer consists of two half-symbol-spaced FFFs, as illustrated in Figure 6b. The input-output relationship can be expressed as follows:  
\[ \begin{split} x_{\text{out,1}}[k]&=\mathbf{h}_{1}^{(k )}\mathbf{x}_{\text{in}}[2k-1]^{\text{T}}\\ x_{\text{out,2}}[k]&=\mathbf{h}_{2}^{(k)}\mathbf{x }_{\text{in}}[2k]^{\text{T}},\end{split}  \tag{3}\]  
where \(\mathbf{x}_{\text{in}}[k]\) denotes a sequence consisting of half-symbol-spaced samples (i.e., \(\mathbf{x}_{\text{in}}[k]=[x_{\text{in}}(kT_{s}+D_{1})\), \(x_{\text{in}}((k+0.5)T_{s}+D_{1})\), \(x_{\text{in}}((k+1)T_{s}+D_{1})\),..., \(x_{\text{in}}((k+L-0.5)T_{s}+D_{1})\)), \(D_{1}\): an arbitrary constant representing a delay), and the superscript 'T' represents the transpose. \(\mathbf{h}_{1}^{(k)}\) and \(\mathbf{h}_{2}^{(k)}\) are alternatively applied to the OTDM signal to reconstruct the RZ1 signal, \(\mathbf{x}_{\text{out,1}}[k]\), and RZ2 signal, \(\mathbf{x}_{\text{out,2}}[k]\), respectively [29]. \(\mathbf{h}_{1}\) and \(\mathbf{h}_{2}\) can be interpreted as matched filters that maximize the SINAD. Furthermore, errors due to power and timing mismatches can be compensated for by separating \(\mathbf{h}_{1}\) and \(\mathbf{h}_{2}\). While estimating the optimal sampling point using the FFE-based demultiplexer, the BER can be further reduced using an improved MIMO equalizer. The MIMO equalizer may consist of second-order Volterra equalizers, where the input-output relationship can be expressed as follows:  
\[ \begin{split} y[k]&=w_{0}+\sum_{k_{1}=-D_{2}}^{-D_{2 }+M_{1}-1}w_{1}(k_{1})x[k+k_{1}]+\\ &+\sum_{k_{2}=-D_{2}}^{-D_{2}+M_{1}-1\min(-D_{2}+M_{1}-1,k_{2}+M_ {2}-1)}w_{2}(k_{2},k_{3})x[k+k_{2}]x[k+k_{3}]\end{split}  \tag{4}\]  
where \(D_{2}\) denotes a constant representing a delay, \(M_{1}\) and \(M_{2}\) denote constants determining the tap length, \(w_{r}\) denotes the \(r\)-th order filter coefficient, and \(x[k]\) and \(y[k]\) represent the \(k\)-th input and output of the equalizer, respectively [29]. A parameter vector \(\mathbf{M}=[M_{1},M_{2}]\) can be introduced for convenience. Experimental Results Obtained Using a Multilayer Equalizer  
Using the data received from the setup shown in Figure 2, it was verified that the multilayer equalizer could improve BER performance. A time demultiplexer made up of two 6-tap half-symbol-spaced FFFs and a simplified symbol-spaced nonlinear MIMO equalizer made up the receiver. The 3 dB bandwidth of the receiver was set to 25 GHz. Figure 7a shows the BERs depending on the sampling position when using a symbol-spaced demultiplexer and a MIMO equalizer [29]. The received optical power was \(-1\) dBm under the back-to-back condition. When the symbol-spaced demultiplexer was used, the \(-T_{s}/4\) offset in the sampling position caused a 3.4-fold increase in BER (from \(5.8\times 10^{-4}\) to \(2.0\times 10^{-3}\)). Notably, determining the optimal sampling position was challenging when using a higher-order modulation-formatted signal and a pulse with a finite ER, as shown in Figure 7c. In addition, the nonlinear MIMO equalizer with \(\mathbf{M}=[12,12]\) could not  
Figure 6: (**a**) Schematic diagram of a two-channel OTDM system using a multilayer equalizer.