## 5 Computational Complexity  
In this section, the proposed algorithm is compared with the rest of the algorithms studied in this paper in terms of computational complexity. Six real multiplications and two summations are assumed to perform the 4-_th_ power operation in the V&V module [8]. Nine real multiplications and three summations are assumed in the case of the 8-_th_ power operation [14]. Two approaches are considered for the implementation of the symbol decision circuit (_DC_) module in the BPS algorithm, which is illustrated in Figure 8. In order to map a received symbol to one of the symbols in the constellation, the distance between the received symbol and all the constellation points can be performed. The received symbol is mapped to the constellation point where the calculated distance is the minimum (Figure 8b). For the rest of the paper, we consider this approach as hard decision and denote it with _DC_ = 1. For the case of Sq-mQAM constellations, the decision circuit can be implemented employing only comparators in the I and Q components of the received symbol as the  
Figure 6: Experimental setup. ECL: External Cavity Laser; PM: Phase Modulator; AWG: Arbitrary Waveform Generator; OBPF: Optical band pass filter; LO: Local Oscillator; DSO: Digital Storage Oscilloscope. Figure 7: BER _versus_ OSNR curves of a 28-Gbaud C-16 QAM back-to-back transmission employing different CPR algorithms for different laser linewidths. N\({}_{\text{errors}}\): Number of errors. decision boundaries lay on a square grid (Figure 8a) and is considered in this paper as soft decision with notation \(DC=0\). However, this approach comes at the expense of a worse process performance, as it is not resilient enough to shape distortions of the received constellation. These two approaches are considered as the best- and worst-case scenarios (in terms of computational complexity), and any other implementation of the \(DC\) module will result in a computational complexity within the range of these two cases. The hard decision approach is considered in this paper for C-mQAM constellations, as their optimal decision boundaries have a pentagonal shape in the complex plane. The hardware implementation of the hard decision approach would require two real multipliers and three real adders for each of the distance calculations, while only comparators are required in the soft decision approach. The use of sub-optimal boundaries in Figure 2 (inset \(d\)) relaxes the complexity of the \(DC\) (in terms of multiplications/additions) for C-mQAM constellations, as the decision is made based on the angle and amplitude of the symbol requiring three real multipliers and one real adder for each symbol decision. Figure 9 illustrates the flow chart of the proposed algorithm for the evaluation of its computational complexity. The red dashed line indicates the reusability of the calculations between modules. The calculations of the V&V module in the first stage are reused in the second stage, and a sign change performed on the bit representing the sign of the floating point number is performed for symbols belonging to even classes. The implementation proposed in Figure 9 forces the maximum block length of the second stage to be equal or smaller than the first stage (M\({}_{2}\leq\) M\({}_{1}\)). The computational complexity derivation of the first stage was detailed in [14], and its calculation for the second stage in terms of real-valued multiplications and summations is as follows:  
1. Each of the decision symbol circuit modules requires \(DC^{mult}\), \(DC^{sum}\), and \(DC^{comp}\) multiplications, summations, and comparisons, respectively. 2. The summation of the \(I\) and \(Q\) components of M\({}_{2}\) symbols requires 2M\({}_{2}-2\) adders. 3. The division inside the argument function requires one multiplication. A small look-up table can be used for the argument calculation. Unwrap operation requires a comparator and one summation. Multiplications and divisions of the power of two can be performed through bitwise shifting operations. 4.