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komm.PSKConstellation

Phase-shift keying (PSK) constellation. It is a complex one-dimensional constellation in which the symbols are uniformly arranged in a circle. More precisely, the $i$-th symbol is given by $$ x_i = A \exp \left( \mathrm{j} \frac{2 \pi i}{M} \right) \exp(\mathrm{j} 2 \pi \phi), \quad i \in [0 : M), $$ where $M$ is the order, $A$ is the amplitude, and $\phi$ is the phase offset of the constellation.

Parameters:

  • order (int) –

    The order $M$ of the constellation.

  • amplitude (float) –

    The amplitude $A$ of the constellation. The default value is 1.0.

  • phase_offset (float) –

    The phase offset $\phi$ of the constellation (in turns, not radians). The default value is 0.0.

Examples:

  1. The $4$-PSK constellation with amplitude $A = 1$ and phase offset $\phi = 0$ is depicted below.

    4-PSK constellation. 

    >>> const = komm.PSKConstellation(4)

  2. The $8$-PSK constellation with amplitude $A = 0.5$ and phase offset $\phi = 1/16$ is depicted below.

    8-PSK constellation. 

    >>> const = komm.PSKConstellation(8, amplitude=0.5, phase_offset=1 / 16)

matrix Array2D[complexfloating] cached property

The constellation matrix $\mathbf{X}$.

Examples:

>>> const = komm.PSKConstellation(4)
>>> const.matrix
array([[ 1.+0.j],
       [ 0.+1.j],
       [-1.+0.j],
       [ 0.-1.j]])

order int property

The order $M$ of the constellation.

Examples:

>>> const = komm.PSKConstellation(4)
>>> const.order
4

dimension int property

The dimension $N$ of the constellation.

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

Examples:

>>> const = komm.PSKConstellation(4)
>>> 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 PSK constellation is given by $$ \mathbf{m} = 0. $$

Examples:

>>> const = komm.PSKConstellation(4)
>>> 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 PSK constellation is given by $$ E = A^2. $$

Examples:

>>> const = komm.PSKConstellation(4)
>>> const.mean_energy()
np.float64(1.0)

minimum_distance()

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

For the PSK constellation, the minimum distance is given by $$ d_\mathrm{min} = 2A\sin\left(\frac{\pi}{M}\right). $$

Examples:

>>> const = komm.PSKConstellation(4)
>>> const.minimum_distance()
np.float64(1.414213562373095)

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.PSKConstellation(4)
>>> 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.PSKConstellation(4)
>>> 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.PSKConstellation(4)
>>> 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.PSKConstellation(4)
>>> const.posteriors([0.1 - 1.1j, 1.2 + 0.1j], snr=2.0).round(3)
array([0.018, 0.   , 0.008, 0.974, 0.982, 0.012, 0.   , 0.005])