# DSP-Based 40 GB/s Lane Rate Next-Generation Access Networks +
## 3 Simulation Results
### DSP Complexity  
Similar to other short- and medium-reach applications [20; 21], it is important to assess the various DSP algorithms from the complexity point of view for access networks. Here, the complexity is measured as the required number of real-valued multiplications and summations per symbol. Considering the first term of the right-hand side (RHS) of Equation (1), the required multiplication (summation) count per symbol is \(M+N\left(M+N-1\right)\) for FFE/DFE with linear taps. For the second-order FFE terms, as indicated in the second term of RHS of Equation (1), two types of multiplications are performed: the first type is \(y[k-i]y[k-j]\), representing the product between two symbols, which can be expressed by a \(M\times M\) lower triangular matrix. \(M\) such multiplications are required for each symbol, since the \((M-1)\times(M-1)\) lower triangular matrix terms from the previous symbol period are reserved, and only \(M\) new product terms are updated. The second type is \(a_{i,j}y[k-i]y[k-j]\) which is the multiplication with tap coefficient that requires \(\frac{M^{2}+M}{2}\) multiplications. In total, there are \(\frac{M^{2}+3M}{2}\) multiplications per symbol for nonlinear FFE. Similarly, for the second-order nonlinear DFE terms as shown by the last term of the RHS of Equation (1), the required multiplication count is \(\frac{N^{2}+N-2}{2}\), with the difference that the symbols' square terms are not considered. The total number of sums for the second-order FFE&DFE is \(\frac{M^{2}+M+N^{2}-N}{2}\). For Viterbi MLSE, the key parameters include the constellation alphabet size \(K\), the memory length \(L\), and the trellis length \(R\) which also determines the backtrack length. The trellis has \(K^{L}\) states and analyzes \(K^{L+1}\) branches per each symbol. The multiplication takes place when calculating branch metrics by using typical Euclidian distance [22; 23]:  
\[\left(y^{\prime}[k]-\sum_{l=0}^{L}c_{l}\cdot x[k-l]\right)^{2} \tag{2}\]  
Figure 5: Link power budget for each modulation format under various DSP configurations for transmission distances of (**a**) 10 km, (**b**) 20 km, (**c**) 30 km, and (**d**) 40 km. The optical power sensitivity shown in Figure 4 is considered to calculate the link power budget. where \(c\) is the channel impulse response coefficient, and \(x\) is transmitted symbol sequence. Simplifying MLSE using the absolute value of the distance instead of the square of the distance in branch metric can save complexity without performance degradation [22], and thus it is used in this work. The required number of multiplications is \((L+1)\cdot K^{L+1}\), where \(L+1\) multiplications are for the convolution operation of \(\sum_{l=0}^{l}c_{l}\cdot x[k-l]\). The number of sums is also \((L+1)\cdot K^{L+1}\). After \(R\) symbols, it backtracks the trellis to determine the MLSE solution, which involves \(K^{L+1}\) if-statements, which can be equivalently viewed as a sum operation. Table 1 summarizes the complexity of various DSP configurations for all of the modulation schemes. Note that the complexity of FFE/DFE is the same for all three schemes, since the tap count is the same, but the MLSE complexity depends on the modulation constellation size \(K\) and memory length \(L\). It clearly indicates that the majority of complexity is from the FFE/DFE whose second-order nonlinear equalizations account for approximately 50% of overall multiplication or sums. Linear FFE/DFE has the least complexity. MLSE with a small memory length shows moderate complexity regarding both the required multiplication and sum operations. The complexity of the second-order FFE/DFE can be significantly reduced by neglecting the terms whose corresponding tap coefficients are much less weighted [17; 24]. Recent experimental demonstrations show that a 75% complexity reduction is possible [24].