komm.QAMConstellation

Quadrature amplitude modulation (QAM) constellation. It is a complex one-dimensional constellation obtained by a Cartesian product of two PAM constellations, namely, the in-phase constellation, and the quadrature constellation. For more details, see SA15, Sec. 2.5.1.

Parameters:

orders (tuple[int, int] | int) –

A tuple $(M_\mathrm{I}, M_\mathrm{Q})$ with the orders of the in-phase and quadrature constellations, respectively. If specified as a single integer $M$, then it is assumed that $M_\mathrm{I} = M_\mathrm{Q} = \sqrt{M}$; in this case, $M$ must be a perfect square.
deltas (tuple[float, float] | float) –

A tuple $(\Delta_\mathrm{I}, \Delta_\mathrm{Q})$ with the distances of the in-phase and quadrature constellations, respectively. If specified as a single float $\Delta$, then it is assumed that $\Delta_\mathrm{I} = \Delta_\mathrm{Q} = \Delta$. The default value is 2.0.
phase_offset (float) –

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

Examples:

The square $16$-QAM constellation with $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 4)$ and $(\Delta_\mathrm{I}, \Delta_\mathrm{Q}) = (2, 2)$ is depicted below.
```
>>> const = komm.QAMConstellation(16)
```
The rectangular $8$-QAM constellation with $(M_\mathrm{I}, M_\mathrm{Q}) = (4, 2)$ and $(\Delta_\mathrm{I}, \Delta_\mathrm{Q}) = (2, 4)$ is depicted below.
```
>>> const = komm.QAMConstellation(orders=(4, 2), deltas=(2.0, 4.0))
```

`matrixArray2D[complexfloating]` `cached` `property`

The constellation matrix $\mathbf{X}$.

Examples:

>>> const = komm.QAMConstellation(16)
>>> const.matrix
array([[-3.-3.j],
       [-3.-1.j],
       [-3.+1.j],
       [-3.+3.j],
       [-1.-3.j],
       [-1.-1.j],
       [-1.+1.j],
       [-1.+3.j],
       [ 1.-3.j],
       [ 1.-1.j],
       [ 1.+1.j],
       [ 1.+3.j],
       [ 3.-3.j],
       [ 3.-1.j],
       [ 3.+1.j],
       [ 3.+3.j]])

`orderint` `property`

The order $M$ of the constellation.

For the QAM constellation, it is given by $$ M = M_\mathrm{I} M_\mathrm{Q}. $$

Examples:

>>> const = komm.QAMConstellation(16)
>>> const.order
16

`dimensionint` `property`

The dimension $N$ of the constellation.

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

Examples:

>>> const = komm.QAMConstellation(16)
>>> 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 QAM constellation is given by $$ \mathbf{m} = 0. $$

Examples:

>>> const = komm.QAMConstellation(16)
>>> 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 QAM constellation is given by $$ E = \frac{\Delta_\mathrm{I}^2}{12}(M_\mathrm{I}^2 - 1) + \frac{\Delta_\mathrm{Q}^2}{12}(M_\mathrm{Q}^2 - 1). $$ For the special case of a square QAM constellation, this simplifies to $$ E = \frac{\Delta^2}{6}(M - 1). $$

Examples:

>>> const = komm.QAMConstellation(16)
>>> const.mean_energy()
np.float64(10.0)

`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 QAM constellation, the minimum distance is given by $$ d_{\min} = \min(\Delta_\mathrm{I}, \Delta_\mathrm{Q}). $$

Examples:

>>> const = komm.QAMConstellation(16)
>>> 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[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.QAMConstellation(16)
>>> const.indices_to_symbols([3, 0])
array([-3.+3.j, -3.-3.j])
>>> const.indices_to_symbols([[3, 0], [1, 2]])
array([[-3.+3.j, -3.-3.j],
       [-3.-1.j, -3.+1.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.QAMConstellation(16)
>>> const.closest_indices([-3.1 + 2.9j, -3 - 3.5j])
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.QAMConstellation(16)
>>> const.closest_symbols([-3.1 + 2.9j, -3 - 3.5j])
array([-3.+3.j, -3.-3.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.QAMConstellation(16)
>>> const.posteriors([-3.1 + 2.9j], snr=2.0).round(3)
array([0.   , 0.021, 0.219, 0.449, 0.   , 0.009, 0.091, 0.186, 0.   ,
       0.001, 0.008, 0.016, 0.   , 0.   , 0.   , 0.   ])

komm.QAMConstellation

matrix Array2D[complexfloating] cached property

order int property

dimension int property

mean()

mean_energy()

minimum_distance()

indices_to_symbols()

closest_indices()

closest_symbols()

posteriors()

`matrixArray2D[complexfloating]` `cached` `property`

`orderint` `property`

`dimensionint` `property`

`mean()`

`mean_energy()`

`minimum_distance()`

`indices_to_symbols()`

`closest_indices()`

`closest_symbols()`

`posteriors()`