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Table 1 Coordinate expression in Euler’s formula

From: Secure spatial modulation based on two-dimensional generalized weighted fractional Fourier transform encryption

Constellation

\(x_1\)

\(x_2\)

\(x_3\)

\(x_4\)

\(x_1\)

\(x_{11}=e^{\theta _1i}(1+i)\)

\(x_{12}=e^{\theta _1i}i+e^{\theta _2i}\)

\(x_{13}=e^{\theta _2i}(1+i)\)

\(x_{14}=e^{\theta _1i}+e^{\theta _2i}i\)

\(x_2\)

\(x_{21}=e^{\theta _1i}i-e^{\theta _2i}\)

\(x_{22}=e^{\theta _1i}(-1+i)\)

\(x_{23}=-e^{\theta _1i}+e^{\theta _2i}i\)

\(x_{24}=e^{\theta _2i}(-1+i)\)

\(x_3\)

\(x_{31}=e^{\theta _2i}(-1-i)\)

\(x_{32}=-e^{\theta _1i}-e^{\theta _2i}i\)

\(x_{33}=e^{\theta _1i}(-1-i)\)

\(x_{34}=-e^{\theta _1i}-e^{\theta _2i}i\)

\(x_4\)

\(x_{41}=e^{\theta _1i}-e^{\theta _2i}i\)

\(x_{42}=e^{\theta _2i}(1-i)\)

\(x_{43}=-e^{\theta _1i}i+e^{\theta _2i}\)

\(x_{44}=e^{\theta _2i}(1-i)\)