# High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDEE Equalizers
## 3 Proposed Partially-Joint Single-Ended DQPSK Equalization
A low complexity alternative of proposed three port partially joint single ended VDFE, is a degenerated form of PVDFE, the so called skimmed VDFE (SVDFE) [10], where extreme pruning is applied to the feed-forward Volterra kernels, keeping only the coefficients that correspond to the diagonal of the full sized counterpart. VDFE employing skimmed Volterra kernels in place of the full-sized counterparts, have been proposed as a viable solution in high-speed, low-cost implementation for the equalization of OOK keying signaling, as well as for the DQPSK modulation, operating at bit rates as high as 40 Gb/s [2; 10; 12; 20]. Assuming the same example of Equation (4), let the signals processed given by the triplet (\(l_{c}\), \(Q_{c}\), \(Q_{d}\)), i.e., the destructive port of the I channel, be ignored. Now the three-port partially-joint single-ended SVDFE equalizers VDFE[\(M_{f}\),\(M_{b}\)]-\(I_{d}\) is described as: \((l=1,2)\). \[ \begin{array}{l}u_{l}(n)=\sum\limits_{i=1}^{2}\sum\limits_{k=1}^{3}\sum \limits_{m=0}^{M_{f}-1}f_{i,m}^{k,l}y_{i,k}(n-m)+\sum\limits_{i=1}^{2}\sum \limits_{k=1}^{3}\sum\limits_{m_{1}=1}^{M_{f}-1}f_{i,m_{1},m_{1}}^{k,l}y_{i,k} ^{2}(n-m_{1})+\\ \sum\limits_{i=1}^{2}\sum\limits_{m_{1}=1}^{M_{b}}\sum\limits_{m_{2}=m_{1}}^{M _{b}}y_{i,m_{1},m_{2}}^{l}\hat{l}_{i}(n-m_{1})\hat{l}_{i}(n-m_{2})\end{array}  \tag{7}\]  
Figure 4: (**a**) Schematic illustration of utilized signals and (**b**) signal flow graph for the proposed partially-joint single-ended DQPSK VDFE[\(M_{f}\),\(M_{b}\)]-\(I_{d}\). The cases of all three-port partially-joint single-ended skimmed VDFE equalizer, (SVDFE[\(M_{f}\),\(M_{b}\)]-\(I_{x}\) or SVDFE[\(M_{f}\),\(M_{b}\)]-\(Q_{x}\)) are treated similarly. In Figure 5 the differences in structure of the feed-forward part between VDFE (Figure 5a) and SVDFE (Figure 5b) are depicted. While, in this particular block diagram the pruning procedure is presented only for one of the received signals it should be noted that the same technique implies for every different input signal. The computational complexity for the estimation of \(u_{1}(n)\) and \(u_{2}(n)\) is now reduced to:  
\[C_{SVDFE(M_{f},M_{b})}=24M_{f}+2M_{b}(M_{b}+1) \tag{8}\]  
Compared to the complexity requirements of the fully-deployed counterpart (Equation (2)), a significant reduction is achieved, as in the latter case the computational requirement depends linearly on \(M_{f}\), noting that the contribution of the last term in the summation is rather marginal as \(\hat{I}_{1}(n)\) and \(\hat{I}_{2}(n)\) represent binary digits. Apart from the complexity reduction, the proposed partially-joint constructive/destructive equalization scheme offers better numerical behavior compared to the full counterpart. The use of all four signals \(I_{c}(t)\), \(I_{d}(t)\), \(Q_{c}(t)\), and \(Q_{d}(t)\), available at the DQPSK receiver, improves performance by maximizing the signal diversity. MLSE equalizers certainly benefit from this approach, however, in the case of symbol by symbol equalizers, such as DFE and VDFE, some extra attention is required. Following the low-pass equivalent description of a DQPSK link [24], we notice that in the case of ideal identical noise free photodetectors, the four electrical signals available at the receiver, \(I_{c}(t)\), \(I_{d}(t)\), \(Q_{c}(t)\), and \(Q_{d}(t)\) are linearly dependent, as it can be easily shown that \(I_{c}(t)+I_{d}(t)-Q_{c}(t)-Q_{d}(t)=0\). In a realistic situation, the presence of noise at the receiver results in (marginally) linear independence, as \(I_{c}(t)+I_{d}(t)-Q_{c}(t)-Q_{d}(t)=n(t)\), with \(n(t)\) denoting the contribution of the noise signals from all four photodiodes. Noting that the coefficients of the VDFE equalizer (Equation (2)) are estimated by the solution of a linear system of equations either implicitly or explicitly, the condition number of the associated matrix which is formulated using the available signals (Equations (1)), is of crucial importance concerning the numerical accuracy of the estimated output [22].