## 5 Computational Complexity
In order to retrieve the \(I\) and \(Q\) components a small size look up table is required. Two comparators and an adder are needed to keep the calculated value in the range of [0,2\(\pi\)). 5. \(4M_{2}\) multiplications and \(2M_{2}\) summations are required for phase correction of the input symbols. Figure 8: Two different implementation approaches of the symbol decision circuit corresponding to the best- and worst-case scenarios in terms of computational complexity. (**a**) Decision based on I and Q levels comparison. (**b**) Decision based on distance calculation to all the points in the constellation. The computational complexity calculations of the proposed algorithm and the BPS algorithm are illustrated in Table 1. Block-based concept, unwrapping function and the final symbol decision module are assumed in all the calculations, as explained for the proposed two-stage n-PSK partitioning case. Table 2 shows the computational complexity reduction factors of the proposed algorithm, utilizing sub-optimal decision boundaries (see Figure 2; inset \(d\)) relative to the BPS algorithm. In the case of C-16 QAM compared to Sq-16 QAM, the computational complexity reduction factor is in the range of 3.8-24.5 in the required number of real multipliers, and this factor is between 3.3 and 30.1 for the number of real adders depending on the implementation of the \(DC\) module. In the case of C-64 QAM compared to Sq-64 QAM, the computational complexity reduction factor is in the range of 2.6-59.1 in the number of real multipliers, and in the range of 1.9-63.3 for the number of real adders depending on the implementation of the \(DC\) module. Computational complexity reduction factors of 24.5 \(\mid\) 30.1 and 29.1 \(\mid\) 32.2 for the proposed two-stage n-PSK partitioning are achieved compared to BPS applied to C-16 QAM and C-64 QAM constellations, respectively. The computational complexity reduction factors are summarized, for the case of optimal boundaries, in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline _Algorithm_ & _Reduction Factors [DC = 1, M\({}_{1}\) = M\({}_{2}\) = 19] (Multipliers 1 [ Address)]_ & _Specifications_ \\ \hline _BPS Sq-16 QAM_ & 3.8 \(\mid\) 3.3 & M = 19, \(\beta=32\), \(DC=0\) \\  & 24.5 \(\mid\) 30.1 & M = 19, \(\beta=32\), \(DC=1\) \\  & 2.6 \(\mid\) 1.9 & M = 19, \(\beta=64\), \(DC=0\) \\  & 59.1 \(\mid\) 63.3 & M = 19, \(\beta=64\), \(DC=1\) \\  & 24.5 \(\mid\) 30.1 & M = 19, \(\beta=32\), \(DC=1\) \\  & 29.1 \(\mid\) 32.2 & M = 19, \(\beta=32\), \(DC=1\) \\ \hline \hline \end{tabular} Note: \(DC\) = 1 implies hard decision (worst-case complexity). \(DC\) = 0 implies soft decision.  
\end{table} 
Table 2: Computational complexity reduction factors relative to two-stage n-PSK partitioning algorithm employing sub-optimal decision boundaries in the symbol decision circuit module. \begin{table}
\begin{tabular}{c c c} \hline \hline _Ral Multipliers_ & _Real Address_ & _Compursurs_ \\ \hline _BPS Sq-16 QAM_ & 64\(\beta\) + \(M(\beta+1)\cdot D^{\text{CM}}_{\text{Sq}}+4M\) & 64\(\beta\) + \(M(\beta+1)\cdot D^{\text{CM}}_{\text{Sq}}-\beta\) + 24 \(\beta\) + \(M(\beta+1)\cdot D^{\text{CM}}_{\text{Sq}}+2\) \\  & 64\(\beta\) + \(M(\beta+1)\cdot D^{\text{CM}}_{\text{Sq}}+4M\) & 64\(\beta\) + \(M(\beta+1)\cdot D^{\text{