komm.APSKModulation

Amplitude- and phase-shift keying (APSK) modulation. It is a complex modulation scheme in which the constellation is the union (concatenation) of component PSK constellations, called rings. More precisely, consider $K$ rings $\mathbf{X}_k$, for $k \in [0 : K)$, where the $k$-th ring has order $M_k$, amplitude $A_k$, and phase offset $\phi_k$. The $i$-th constellation symbol of the $k$-th ring is given by $$ x_{k,i} = A_k \exp \left( \mathrm{j} \frac{2 \pi i}{M_k} \right) \exp(\mathrm{j} \phi_k), \quad k \in [0 : K), \quad i \in [0 : M_k). $$ The resulting APSK constellation is therefore given by $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_0 \\ \vdots \\ \mathbf{X}_{K-1} \end{bmatrix}, $$ which has order $M = M_0 + M_1 + \cdots + M_{K-1}$. The order $M_k$ of each ring need not be a power of $2$; however, the order $M$ of the constructed APSK modulation must be. The APSK constellation is depicted below for $(M_0, M_1) = (8, 8)$ with $(A_0, A_1) = (A, 2A)$ and $(\phi_0, \phi_1) = (0, \pi/8)$; and for $(M_0, M_1) = (4, 12)$ with $(A_0, A_1) = (\sqrt{2}A, 3A)$ and $(\phi_0, \phi_1) = (\pi/4, 0)$.

(8,8)-APSK constellation. (4,12)-APSK constellation.

Parameters:

orders (tuple[int, ...]) –

A $K$-tuple with the orders $M_k$ of each ring, for $k \in [0 : K)$. The sum $M_0 + M_1 + \cdots + M_{K-1}$ must be a power of $2$.
amplitudes (tuple[float, ...]) –

A $K$-tuple with the amplitudes $A_k$ of each ring, for $k \in [0 : K)$.
phase_offsets (float | tuple[float, ...]) –

A $K$-tuple with the phase offsets $\phi_k$ of each ring, for $k \in [0 : K)$. If specified as a single float $\phi$, then it is assumed that $\phi_k = \phi$ for all $k \in [0 : K)$. The default value is 0.0.
labeling (Literal['natural'] | ArrayLike) –

The binary labeling of the modulation. Can be specified either as a 2D-array of integers (see base class for details), or as a string. In the latter case, the string must be equal to 'natural'. The default value is 'natural'.

Examples:

>>> apsk = komm.APSKModulation(orders=(8, 8), amplitudes=(1.0, 2.0), phase_offsets=(0.0, np.pi/8))
>>> apsk.constellation.round(4)
array([ 1.    +0.j    ,  0.7071+0.7071j,  0.    +1.j    , -0.7071+0.7071j,
       -1.    +0.j    , -0.7071-0.7071j, -0.    -1.j    ,  0.7071-0.7071j,
        1.8478+0.7654j,  0.7654+1.8478j, -0.7654+1.8478j, -1.8478+0.7654j,
       -1.8478-0.7654j, -0.7654-1.8478j,  0.7654-1.8478j,  1.8478-0.7654j])

>>> apsk = komm.APSKModulation(orders=(4, 12), amplitudes=(np.sqrt(2), 3.0), phase_offsets=(np.pi/4, 0.0))
>>> apsk.constellation.round(4)
array([ 1.    +1.j    , -1.    +1.j    , -1.    -1.j    ,  1.    -1.j    ,
        3.    +0.j    ,  2.5981+1.5j   ,  1.5   +2.5981j,  0.    +3.j    ,
       -1.5   +2.5981j, -2.5981+1.5j   , -3.    +0.j    , -2.5981-1.5j   ,
       -1.5   -2.5981j, -0.    -3.j    ,  1.5   -2.5981j,  2.5981-1.5j   ])