Skip to content

komm.QAModulation

Quadrature-amplitude modulation (QAM). It is a complex modulation scheme in which the constellation is given as a Cartesian product of two PAM constellations, namely, the in-phase constellation, and the quadrature constellation. More precisely, the $i$-th constellation symbol is given by $$ \begin{aligned} x_i = \left[ A_\mathrm{I} \left( 2i_\mathrm{I} - M_\mathrm{I} + 1 \right) + \mathrm{j} A_\mathrm{Q} \left( 2i_\mathrm{Q} - M_\mathrm{Q} + 1 \right) \right] \exp(\mathrm{j}\phi), \quad & i \in [0 : M), \\ & i_\mathrm{I} = i \bmod M_\mathrm{I}, \\ & i_\mathrm{Q} = \lfloor i / M_\mathrm{I} \rfloor, \end{aligned} $$ where $M_\mathrm{I}$ and $M_\mathrm{Q}$ are the orders (powers of $2$), and $A_\mathrm{I}$ and $A_\mathrm{Q}$ are the base amplitudes of the in-phase and quadrature constellations, respectively. Also, $\phi$ is the phase offset. The order of the resulting complex-valued constellation is $M = M_\mathrm{I} M_\mathrm{Q}$, a power of $2$. The QAM constellation is depicted below for $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 4)$ with ($A_\mathrm{I}, A_\mathrm{Q}) = (A, A)$; and for $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 2)$ with $(A_\mathrm{I}, A_\mathrm{Q}) = (A, 2A)$; in both cases, $\phi = 0$.

16-QAM constellation. 8-QAM constellation.

Parameters:

  • orders (tuple[int, int] | int) –

    A tuple $(M_\mathrm{I}, M_\mathrm{Q})$ with the orders of the in-phase and quadrature constellations, respectively; both $M_\mathrm{I}$ and $M_\mathrm{Q}$ must be powers of $2$. If specified as a single integer $M$, then it is assumed that $M_\mathrm{I} = M_\mathrm{Q} = \sqrt{M}$; in this case, $M$ must be an square power of $2$.

  • base_amplitudes (tuple[float, float] | float) –

    A tuple $(A_\mathrm{I}, A_\mathrm{Q})$ with the base amplitudes of the in-phase and quadrature constellations, respectively. If specified as a single float $A$, then it is assumed that $A_\mathrm{I} = A_\mathrm{Q} = A$. The default value is $1.0$.

  • phase_offset (float) –

    The phase offset $\phi$ of the constellation. The default value is 0.0.

  • labeling (Literal['natural_2d', 'reflected_2d'] | 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 either 'natural_2d' or 'reflected_2d'. The default value is 'reflected_2d', corresponding to the Gray labeling.

Examples:

The square $16$-QAM modulation with $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 4)$ and $(A_\mathrm{I}, A_\mathrm{Q}) = (1, 1)$, and Gray labeling is depicted below.

16-QAM modulation with Gray labeling.

>>> qam = komm.QAModulation(16)
>>> qam.constellation
array([-3.-3.j, -1.-3.j,  1.-3.j,  3.-3.j,
       -3.-1.j, -1.-1.j,  1.-1.j,  3.-1.j,
       -3.+1.j, -1.+1.j,  1.+1.j,  3.+1.j,
       -3.+3.j, -1.+3.j,  1.+3.j,  3.+3.j])
>>> qam.labeling
array([[0, 0, 0, 0], [1, 0, 0, 0], [1, 1, 0, 0], [0, 1, 0, 0],
       [0, 0, 1, 0], [1, 0, 1, 0], [1, 1, 1, 0], [0, 1, 1, 0],
       [0, 0, 1, 1], [1, 0, 1, 1], [1, 1, 1, 1], [0, 1, 1, 1],
       [0, 0, 0, 1], [1, 0, 0, 1], [1, 1, 0, 1], [0, 1, 0, 1]])
>>> qam.modulate([0, 0, 1, 1, 0, 0, 1, 0])
array([-3.+1.j, -3.-1.j])

The rectangular $8$-QAM modulation with $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 2)$ and $(A_\mathrm{I}, A_\mathrm{Q}) = (1, 2)$, and Gray labeling is depicted below.

8-QAM modulation with Gray labeling.

>>> qam = komm.QAModulation(orders=(4, 2), base_amplitudes=(1.0, 2.0))
>>> qam.constellation
array([-3.-2.j, -1.-2.j,  1.-2.j,  3.-2.j,
       -3.+2.j, -1.+2.j,  1.+2.j,  3.+2.j])
>>> qam.labeling
array([[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0],
       [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]])
>>> qam.modulate([0, 0, 1, 1, 0, 0, 1, 0, 1])
array([-3.+2.j, -1.-2.j, -1.+2.j])