## 4 Simulation Experiment Results and Discussion
### The Influence of the MP-WFRFT Algorithm on Signal Constellation Map Distribution  
1.1 Parameter Scanning Characteristics of the MP-WFRFT Algorithm in Two-Dimensional Constellation Map  
For the QPSK signal, the transform order of the MP-WFRFT is \(\alpha\), and the scale vector is \(V=\begin{bmatrix}MV&NV\end{bmatrix}\). The values of MP-WFRFT parameters are provided in Table 2.  
The simulation outcomes are illustrated in Figure 9. As the parameter \(\alpha\) varies, the constellation diagram undergoes transformations characterized by rotation, expansion, and compression of the original four constellation points. Firstly, the changes become increasingly evident when the values of \(\alpha\) grow larger, as depicted in Figure 9b,c. Secondly, when \(\alpha\) is modified alongside the scaling factors \(MV\) and \(NV\), the constellation points exhibit a significant transformation, evolving into nine entirely new constellation points, which is shown in Figure 9d. Thirdly, further adjustments to \(\alpha\), \(MV\), and \(NV\) induce another round of splitting, leading to the emergence of sixteen distinct constellation points, as seen in Figure 9e, the configuration of which effectively mimics a 16 QAM signal. In scenarios where the values of \(\alpha\), \(MV\), and \(NV\) are randomized, the constellation points display a mixing behavior, resembling a Gaussian distribution, as evidenced in Figure 9f,g. Notably, Figure 9h, a share identical distributions of constellation points. However, Figure 9h represents a rotated version of the constellation depicted in Figure 9a, achieved through the application of a 180\({}^{\circ}\) rotational transformation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Positive** & \(\alpha\) & \(MV\) & \(NV\) & \(Inverse\) \\ \hline (b) & 0.05 & [0 0 0 0] & [0 0 0 0] & (a) \\ (c) & 0.20 & [0 0 0 0] & [0 0 0 0] & (a) \\ (d) & 2.90 & [0 0 1.3 0] & [0 0 0 4.7] & (a) \\ (e) & 1.95 & [0 0 1 0] & [3.5 0 0 0] & (a) \\ (f) & 0.05 & [6 5 0 0] & [0 0 4 3] & (a) \\ (g) & 1.00 & [6 5 4 0] & [0 4 3 2] & (a) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & (a) \\ (h) & 2.00 & [6 5 4 3] & [6 5 4 3] & (a) \\ \hline \hline \end{tabular}
\end{table}
Table 2: MP-WFRFT parameter list of two-dimensional constellation map.  
Figure 9: _Cont._1.2 Parameter Scanning Characteristics of the MP-WFRFT Algorithm in Three-Dimensional Constellation Map  
In the case of the 4-PM signal, the MP-WFRFT applies horizontal transformation with an order \(\alpha\) and a scale vector \(V1=\begin{bmatrix}MV1&NV1\end{bmatrix}\), while vertical transformations are governed by an order \(\beta\) and a scale vector \(V2=\begin{bmatrix}MV2&NV2\end{bmatrix}\). The specific values of MP-WFRFT parameters are detailed in Table 3.  
Figure 9: Two-dimensional constellation diagram of QPSK signal after MP-WFRFT transformation.  
The simulation results are presented in Figure 10. Firstly, when the transformation orders \(\alpha\) and \(\beta\) vary, the distribution of constellation points becomes increasingly diffuse and gradually blends together, as illustrated in Figure 10b,c. Secondly, upon introducing scale vectors, the constellation points are dispersed across the Poincare sphere and trend toward a Gaussian distribution, as shown in Figure 10d,e. Thirdly, when the transformation orders \(\alpha\) and \(\beta\) are both set to one, the constellation points are reminiscent of a 2-PM constellation map, as depicted in Figure 10f. However, the coordinates of the constellation points do not align with the standard 2-PM configuration. As the complexity of the transformation orders and scale vectors increases, the distribution of constellation points increasingly resembles a Gaussian distribution, as demonstrated in Figure 10g. Furthermore, Figure 10h, a exhibit the same constellation point distribution, but Figure 10h displays the constellation points after applying a 180\({}^{\circ}\) rotational transformation to Figure 10a.