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komm.APSKConstellation

Amplitude- and phase-shift keying (APSK) constellation. It is a complex one-dimensional constellation obtained by the union of $K$ component PSK constellations, called rings.

Parameters:

  • orders (tuple[int, ...]) –

    A $K$-tuple with the orders $M_k$ of each ring, for $k \in [0 : K)$.

  • 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.

Examples:

  1. The $8$-APSK constellation with $(M_0, M_1) = (4, 4)$, $(A_0, A_1) = (1, 2)$, and $(\phi_0, \phi_1) = (0, 0)$ is depicted below.

    (4,4)-APSK constellation. 

    >>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))

  2. The $16$-APSK constellation with $(M_0, M_1) = (8, 8)$, $(A_0, A_1) = (1, 2)$, and $(\phi_0, \phi_1) = (0, 1/16)$ is depicted below.

    (8,8)-APSK constellation. 

    >>> const = komm.APSKConstellation(
...     orders=(8, 8),
...     amplitudes=(1.0, 2.0),
...     phase_offsets=(0.0, 1 / 16)
... )

  3. The $16$-APSK constellation with $(M_0, M_1) = (4, 12)$, $(A_0, A_1) = (\sqrt{2}, 3)$, and $(\phi_0, \phi_1) = (1/8, 0)$ is depicted below.

    (4,12)-APSK constellation. 

    >>> const = komm.APSKConstellation(
...     orders=(4, 12),
...     amplitudes=(np.sqrt(2), 3.0),
...     phase_offsets=(1 / 8, 0.0)
... )

matrix Array2D[complexfloating] cached property

The constellation matrix $\mathbf{X}$.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.matrix
array([[ 1.+0.j],
       [ 0.+1.j],
       [-1.+0.j],
       [ 0.-1.j],
       [ 2.+0.j],
       [ 0.+2.j],
       [-2.+0.j],
       [ 0.-2.j]])

order int property

The order $M$ of the constellation.

For the APSK constellation, it is given by $$ M = \sum_{k \in [0:K)} M_k. $$

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.order
8

dimension int property

The dimension $N$ of the constellation.

For the APSK constellation, it is given by $N = 1$.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.dimension
1

mean()

Computes the mean $\mathbf{m}$ of the constellation given prior probabilities $p_i$ of the constellation symbols. It is given by $$ \mathbf{m} = \sum_{i \in [0:M)} p_i \mathbf{x}_i. $$

Parameters:

  • priors (ArrayLike | None) –

    The prior probabilities of the constellation symbols. Must be a 1D-array whose size is equal to the order $M$ of the constellation. If not given, uniform priors are assumed.

Returns:

  • mean (Array1D[complexfloating]) –

    The mean $\mathbf{m}$ of the constellation.

For uniform priors, the mean of the APSK constellation is given by $$ \mathbf{m} = 0. $$

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.mean()
array([0.+0.j])

mean_energy()

Computes the mean energy $E$ of the constellation given prior probabilities $p_i$ of the constellation symbols. It is given by $$ E = \sum_{i \in [0:M)} p_i \lVert \mathbf{x}_i \rVert^2. $$

Parameters:

  • priors (ArrayLike | None) –

    The prior probabilities of the constellation symbols. Must be a 1D-array whose size is equal to the order $M$ of the constellation. If not given, uniform priors are assumed.

Returns:

  • mean_energy (floating) –

    The mean energy $E$ of the constellation.

For uniform priors, the mean energy of the APSK constellation is given by $$ E = \frac{1}{M} \sum_{k \in [0:K)} A^2 M_k. $$

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.mean_energy()
np.float64(2.5)

minimum_distance()

Computes the minimum Euclidean distance $d_\mathrm{min}$ of the constellation. It is given by $$ d_\mathrm{min} = \min_ { i, j \in [0:M), ~ i \neq j } \lVert \mathrm{x}_i - \mathrm{x}_j \rVert. $$

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.minimum_distance()
np.float64(1.0)

indices_to_symbols()

Returns the constellation symbols corresponding to the given indices.

Parameters:

  • indices (ArrayLike) –

    The indices to be converted to symbols. Must be an array of integers in $[0:M)$.

Returns:

  • symbols (NDArray[complexfloating]) –

    The symbols corresponding to the given indices. Has the same shape as indices, but with the last dimension expanded by a factor of $N$.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.indices_to_symbols([3, 0])
array([0.-1.j, 1.+0.j])

closest_indices()

Returns the indices of the constellation symbols closest to the given received points.

Parameters:

  • received (ArrayLike) –

    The received points. Must be an array whose last dimension is a multiple of $N$.

Returns:

  • indices (NDArray[integer]) –

    The indices of the symbols closest to the received points. Has the same shape as received, but with the last dimension contracted by a factor of $N$.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.closest_indices([0.1 - 1.1j, 1.2 + 0.1j])
array([3, 0])

closest_symbols()

Returns the constellation symbols closest to the given received points.

Parameters:

  • received (ArrayLike) –

    The received points. Must be an array whose last dimension is a multiple of $N$.

Returns:

  • symbols (NDArray[complexfloating]) –

    The symbols closest to the received points. Has the same shape as received.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.closest_symbols([0.1 - 1.1j, 1.2 + 0.1j])
array([0.-1.j, 1.+0.j])

posteriors()

Returns the posterior probabilities of each constellation symbol given received points, the signal-to-noise ratio (SNR), and prior probabilities.

Parameters:

  • received (ArrayLike) –

    The received points. Must be an array whose last dimension is a multiple of $N$.

  • snr (float) –

    The signal-to-noise ratio (SNR) of the channel (linear, not decibel).

  • priors (ArrayLike | None) –

    The prior probabilities of the symbols. Must be a 1D-array whose size is equal to $M$. If not given, uniform priors are assumed.

Returns:

  • posteriors (NDArray[floating]) –

    The posterior probabilities of each symbol given the received points. Has the same shape as received, but with the last dimension changed by a factor of $M / N$.

Examples:

>>> const = komm.APSKConstellation(orders=(4, 4), amplitudes=(1.0, 2.0))
>>> const.posteriors([0.1 - 1.1j], snr=2.0).round(3)
array([0.104, 0.015, 0.076, 0.516, 0.011, 0.   , 0.006, 0.272])