Geometric-Bell Golden Angle Modulation

GB-GAM exhibits a higher density of constellation points near the center of the complex two-dimensional plane, increasing the Euclidean distance between them as they get further away from the center. This allows for a close approximation of the Gaussian distribution of a random variable. The complex amplitude of its $n$-th constellation point with radius $r$_$n$ is defined as follows: $$ x_{n} = r_{n} \cdot e^{i2\pi\phi n}, \quad n \in \{0, 1, \ldots, N-1\} $$

Specifically, the radius is defined as: $$ r_{n} = c_{gb} \sqrt{\ln\left(\frac{N}{N-n}\right)}, \quad n \in \{0, 1, \ldots, N-1\} $$ $$ c_{gb} \triangleq \sqrt{\frac{N\overline{P}}{N \cdot \ln(N)-\ln(N!)}} $$

When using simulations to estimate the values of the Bit-Error Rate (BER), no labeling methods were applied to GAM. As a result, the BER values of Disc-GAM and GB-GAM may result higher than other considered modulation schemes such as QAM and PSK when gray labeling is applied.