## 2 Principle of the Proposed Scheme  
### Geometric Shaping Based on Iterative Polar Modulation  
Geometric shaping could achieve shaping gain by optimizing the shape of the high-dimensional signal constellation. Most of the GS algorithms are based on optimizing a specific objective function, such as maximizing mutual information, minimizing BER, or minimizing mean square error (MMSE) to determine the location of constellation points satisfying some conditions.  
The IPM modulation format considers the MMSE based on iterative polar quantization procedure [17]. The constellation coordinates determined by a numerical algorithm like Arimoto-Blahut [22] are optimum in the transmission system with significant influence of thermal noise and amplified spontaneous emission noise. The nonuniform iterative polar quantization considers implementing MMSE in two dimensions, including scalar nonuniform amplitude (\(r\)) and scaler uniform phase quantization (\(\varphi\)). The calculation formula of MSE in polar quantization consists of two parts, and could be computed as Equation (1).  
\[MSE=D_{granular}+D_{overload} \tag{1}\]  
where \(D_{granular}\) represents the granulation noise and \(D_{overload}\) represents the overload noise. The granulation noise and overload noise are given by Equations (2) and (3).  
\[D_{granular}=\sum\limits_{i=1}^{L_{r}}\sum\limits_{j=1}^{L_{i}}\int\limits_{ \theta_{j,i}^{j+1}}^{\theta_{j+1,i}}\int\limits_{r_{i}}^{r_{i+1}}\left[r^{2}+ m_{i}^{2}-2 rm_{i}cos\big{(}\varphi-\varphi_{j,i}\big{)}\right]\cdot p_{r}(r, \varphi)drd\varphi \tag{2}\]  
\[D_{overload}=\int\limits_{r_{L_{r}+1}}^{\infty}(r-m_{L_{r}})^{2}p_{r}(r)dr \tag{3}\]  
where \(L_{i}\) is the number of constellation points on \(i\)-th circle; \(L_{r}\) is the number of circles; \(\theta_{j,i}\) is the phase of \(j\)-th points on \(i\)-th circle; \(p_{r}(r)\) represents the source of probability density function (PDF). \(m_{i}\) is the optimum radius of \(i\)-th circle. The optimum number of constellation points of \(i\)-th circle is determined by using Lagrange multiplier method to implement MMSE. Accordingly, we have  
\[L_{i}=\sqrt[3]{m_{i}^{2}\int_{r_{i}}^{r_{i+1}}p_{r}(r)dr}/\left[\sum\
olimits _{i=2}^{L_{r}}\frac{1}{M}\sqrt[3]{m_{i}^{2}\int_{r_{i}}^{r_{i+1}}p_{r}(r)dr} \right];\;i=1,2,\cdots L_{r} \tag{4}\]  
As for PDF function of the optimum source, we consider the case of Gaussian distribution, that is  
\[p_{r}(r)=\frac{1}{2\pi\sigma^{2}}\cdot\exp\left[-\frac{r^{2}}{\sigma^{2}}\right] \tag{5}\]We assumed the partial derivatives to \(m_{i}\) and \(r\), respectively, of Equations (2) and (3) are equal to zero, then the optimum radius is given by  
\[m_{i}=\left[2\sin(\Delta\theta_{i}/2)\int_{r_{i}}^{r_{i+1}}rp(r)dr\right]/\left[ \Delta\theta_{i}\int_{r_{i}}^{r_{i+1}}p(r)dr\right];\;\Delta\theta_{i}=2\pi/L_ {i};\;i=1,2,\cdots L_{r} \tag{6}\]  
Repeat calculating Equations (4) and (5) until these equations satisfied the following limits of integration determined by Equation (7).  
\[r_{i}=\left[\pi\left(m_{i}^{2}-m_{i-1}^{2}\;\right)\right]/\left[m_{i}L_{i} \sin(\Delta\theta_{i}/2)-m_{i-1}L_{i-1}\sin(\Delta\theta_{i-1}/2)\right];\;i=1,2,\cdots L_{r} \tag{7}\]  
We could obtain the optimum IPM constellation coordinates using the above Equations (2)-(7). In this paper, three kinds of constellations are generated based on nonuniform iterative polar quantization; the orders of these constellations \(M\) are equal to 16, 64, 256, respectively. The relevant constellation is shown in Figure 1, which is consistent with [17; 18; 19; 20]. The IPM constellations show that it is a kind of circle modulation method. Still, the performance is better than circle QAM (CQAM) because of its MMSE and larger channel capacity, especially in the IPM-256. We find out that there is a point at the origin of the coordinate axis in constellations of IPM-16 and IPM-256, respectively, as shown in Figure 1a,c, respectively, which makes it dramatically reduce the average transmission power, which could be called centered-IPM (CIPM).