komm.SimplexConstellation
Simplex constellation. It is a real $M$-dimensional constellation of order $M$ obtained from an orthogonal constellation by subtracting from each symbol the mean of all the symbols. The $i$-th symbol is given by $$ \mathbf{x}_i = A \left( \mathbf{e}_i - \frac{1}{M} \mathbf{1} \right), \quad i \in [0 : M), $$ where $A$ is the base amplitude, and $\mathbf{e}_i$ is the $i$-th standard basis vector of $\mathbb{R}^M$. The symbols are equidistant, equicorrelated, and lie in an $(M-1)$-dimensional subspace. The simplex constellation achieves the same minimum distance as the orthogonal constellation, but with smaller energy. For more details, see PS08, Sec. 3.2–4.
Parameters:
-
order(int) –The order $M$ of the constellation.
-
base_amplitude(float) –The base amplitude $A$ of the constellation. The default value is
1.0.
Examples:
-
The $4$-ary simplex constellation with base amplitude $A = 1$ is given by
>>> const = komm.SimplexConstellation(4) >>> const.matrix array([[ 0.75, -0.25, -0.25, -0.25], [-0.25, 0.75, -0.25, -0.25], [-0.25, -0.25, 0.75, -0.25], [-0.25, -0.25, -0.25, 0.75]]) -
The $2$-ary simplex constellation with base amplitude $A = 1$ is given by
>>> const = komm.SimplexConstellation(2) >>> const.matrix array([[ 0.5, -0.5], [-0.5, 0.5]])
matrix Array2D[floating]
cached
property
The constellation matrix $\mathbf{X}$.
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.matrix
array([[ 0.75, -0.25, -0.25, -0.25],
[-0.25, 0.75, -0.25, -0.25],
[-0.25, -0.25, 0.75, -0.25],
[-0.25, -0.25, -0.25, 0.75]])
order int
property
The order $M$ of the constellation.
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.order
4
dimension int
property
The dimension $N$ of the constellation.
For the simplex constellation, it is given by $N = M$. Note, however, that the symbols lie in an $(M-1)$-dimensional subspace.
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.dimension
4
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 simplex constellation is given by $$ \mathbf{m} = \mathbf{0}. $$
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.mean()
array([0., 0., 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 simplex constellation is given by $$ E = A^2 \left( 1 - \frac{1}{M} \right). $$
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.mean_energy()
np.float64(0.75)
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 simplex constellation, the minimum distance is given by $$ d_{\min} = A \sqrt{2}. $$
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.minimum_distance()
np.float64(1.4142135623730951)
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.SimplexConstellation(4)
>>> const.indices_to_symbols([1, 3])
array([-0.25, 0.75, -0.25, -0.25, -0.25, -0.25, -0.25, 0.75])
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.SimplexConstellation(4)
>>> const.closest_indices([0.6, -0.1, -0.2, -0.3])
array([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[floating]) –The symbols closest to the received points. Has the same shape as
received.
Examples:
>>> const = komm.SimplexConstellation(4)
>>> const.closest_symbols([0.6, -0.1, -0.2, -0.3])
array([ 0.75, -0.25, -0.25, -0.25])
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.SimplexConstellation(4)
>>> const.posteriors([0.6, -0.1, -0.2, -0.3], noise_power=0.5).round(3)
array([0.62 , 0.153, 0.125, 0.102])