komm.BiorthogonalConstellation

Biorthogonal constellation. It is a real $(M/2)$-dimensional constellation of order $M$ (which must be even) obtained from the $M/2$ symbols of an orthogonal constellation together with their negatives. The $i$-th symbol is given by $$ \mathbf{x}_i = \begin{cases} A \mathbf{e}_i, & i \in [0 : M/2), \\ -A \mathbf{e}_{i - M/2}, & i \in [M/2 : M), \end{cases} $$ where $A$ is the amplitude, and $\mathbf{e}_i$ is the $i$-th standard basis vector of $\mathbb{R}^{M/2}$. For more details, see PS08, Sec. 3.2–4.

Parameters:

• order (int) –

  The order $M$ of the constellation. Must be an even integer.

• amplitude (float) –

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

Examples:

1. The $4$-ary biorthogonal constellation with amplitude $A = 1$ is given by

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

2. The $2$-ary biorthogonal constellation with amplitude $A = 3$ is given by

   >>> const = komm.BiorthogonalConstellation(2, amplitude=3.0)
   >>> const.matrix
   array([[ 3.],
          [-3.]])
   

matrix Array2D[floating] cached property

The constellation matrix $\mathbf{X}$.

Examples:

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

order int property

The order $M$ of the constellation.

Examples:

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

dimension int property

The dimension $N$ of the constellation.

For the biorthogonal constellation, it is given by $N = M/2$.

Examples:

>>> const = komm.BiorthogonalConstellation(4)
>>> const.dimension
2

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[floating]) –

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

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

Examples:

>>> const = komm.BiorthogonalConstellation(4)
>>> const.mean()
array([0., 0.])

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 biorthogonal constellation is given by $$ E = A^2. $$

Examples:

>>> const = komm.BiorthogonalConstellation(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 $$ d_\mathrm{min} = \min_ { i, j \in [0:M), ~ i \neq j } \lVert \mathbf{x}_i - \mathbf{x}_j \rVert. $$

For the biorthogonal constellation, the minimum distance is given by $$ d_{\min} = \begin{cases} 2A, & M = 2, \\ A \sqrt{2}, & M \geq 4. \end{cases} $$

Examples:

>>> const = komm.BiorthogonalConstellation(4)
>>> const.minimum_distance()
np.float64(1.4142135623730951)

>>> const = komm.BiorthogonalConstellation(2)
>>> const.minimum_distance()
np.float64(2.0)

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[floating]) –

  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.BiorthogonalConstellation(4)
>>> const.indices_to_symbols([0, 3])
array([ 1.,  0.,  0., -1.])

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.BiorthogonalConstellation(4)
>>> const.closest_indices([0.7, 0.2, -0.1, -0.9])
array([0, 3])

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[floating]) –

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

Examples:

>>> const = komm.BiorthogonalConstellation(4)
>>> const.closest_symbols([0.7, 0.2, -0.1, -0.9])
array([ 1.,  0.,  0., -1.])

posteriors()

Returns the posterior probabilities of each constellation symbol given received points, the noise power, and prior probabilities.

The posteriors are computed under the Gaussian channel model $Y = X + Z$, assuming that each received point is the transmitted symbol corrupted by additive Gaussian noise of power $\sigma_Z^2$. For real-valued constellations the noise is real Gaussian with variance $\sigma_Z^2$; for complex-valued constellations it is circularly symmetric complex Gaussian, with the noise power equally divided between the real and imaginary parts, i.e., $\mathrm{E}[\mathrm{Re}\{Z_n\}^2] = \mathrm{E}[\mathrm{Im}\{Z_n\}^2] = \sigma_Z^2/2$.

Parameters:

• received (ArrayLike) –

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

• noise_power (float) –

  The noise power (variance) $\sigma_Z^2$.

• 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.BiorthogonalConstellation(4)
>>> const.posteriors([0.7, 0.2], noise_power=0.5).round(3)
array([0.627, 0.231, 0.038, 0.104])