Disc-Shaped Golden Angle Modulation

The core design of GAM with \(N\) constellation points defines the complex amplitude of its $n$-th constellation point with radius $r$_$n$ as follows: $$ x_{n} = r_{n} \cdot e^{i2\pi\phi n}, \quad n \in \{1, 2, \ldots, N\} $$

The value of ϕ is denoted as the golden ratio, which equals \(\frac{1 + √5}{2} \approx 1.618\) and is responsible for the spiral winding and uniform packing of the symbols in the constellation diagram. The two distintive designs of GAM are presented in the upcoming sections.

The Disc-GAM aims to distribute its constellation points in the shape of a disc, in order to utilize the entirity of the two-dimensional plane. It's distintive distribution is achieved by defining their radius $r$_$n$ as: $$ r_{n} = c_{disc} \cdot \sqrt n , \quad n \in \{1, 2, \ldots, N\} $$ $$ c_{\text{disc}} \triangleq \sqrt{\frac{2\overline{P}}{N+1}} $$