komm.ASKConstellation
Amplitude-shift keying (ASK) constellation. It is a complex one-dimensional constellation in which the symbols are uniformly arranged in a ray. More precisely, the $i$-th symbol is given by $$ x_i = iA \exp(\mathrm{j} 2 \pi \phi), \quad i \in [0 : M), $$ where $M$ is the order, $A$ is the base amplitude, and $\phi$ is the phase offset of the constellation.
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
. -
phase_offset
(float
) –The phase offset $\phi$ of the constellation (in turns, not radians). The default value is
0.0
.
Examples:
-
The $4$-ASK constellation with base amplitude $A = 1$ and phase offset $\phi = 0$ is depicted below
>>> const = komm.ASKConstellation(4)
-
The $4$-ASK constellation with base amplitude $A = 2\sqrt{2}$ and phase offset $\phi = 1/8$ is depicted below.
>>> const = komm.ASKConstellation( ... order=4, ... base_amplitude=2 * np.sqrt(2), ... phase_offset=1 / 8, ... )
matrix
Array2D[complexfloating]
cached
property
The constellation matrix $\mathbf{X}$.
Examples:
>>> const = komm.ASKConstellation(4)
>>> const.matrix
array([[0.+0.j],
[1.+0.j],
[2.+0.j],
[3.+0.j]])
order
int
property
The order $M$ of the constellation.
Examples:
>>> const = komm.ASKConstellation(4)
>>> const.order
4
dimension
int
property
The dimension $N$ of the constellation.
For the ASK constellation, it is given by $N = 1$.
Examples:
>>> const = komm.ASKConstellation(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 ASK constellation is given by $$ \mathbf{m} = \frac{A}{2} (M-1) \exp(\mathrm{j}\phi). $$
Examples:
>>> const = komm.ASKConstellation(4)
>>> const.mean()
array([1.5+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 ASK constellation is given by $$ E = \frac{A^2}{6} (M - 1) (2M - 1). $$
Examples:
>>> const = komm.ASKConstellation(4)
>>> const.mean_energy()
np.float64(3.5)
minimum_distance()
Computes the minimum Euclidean distance $d_\mathrm{min}$ of the constellation. It is given by $$ d_\mathrm{min} = \min_ { i, j \in [0:M), ~ i \neq j } \lVert \mathrm{x}_i - \mathrm{x}_j \rVert. $$
For the ASK constellation, the minimum distance is given by $$ d_{\min} = A $$
Examples:
>>> const = komm.ASKConstellation(4)
>>> 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.ASKConstellation(4)
>>> const.indices_to_symbols([3, 0])
array([3.+0.j, 0.+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.ASKConstellation(4)
>>> const.closest_indices([3.1 + 0.2j, 0.1 - 0.2j])
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.ASKConstellation(4)
>>> const.closest_symbols([3.1 + 0.2j, 0.1 - 0.2j])
array([3.+0.j, 0.+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.ASKConstellation(4)
>>> const.posteriors([3.1 + 0.2j, 0.1 - 0.2j], snr=5.0).round(3)
array([0. , 0.002, 0.152, 0.846, 0.755, 0.241, 0.004, 0. ])