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komm.PAMConstellation

Pulse-amplitude modulation (PAM) constellation. It is a real one-dimensional constellation in which the symbols are uniformly arranged in the real line and centered about the origin. For more details, see SA15, Sec. 2.5.1.

Parameters:

  • order (int) –

    The order $M$ of the constellation.

  • delta (float) –

    The distance $\Delta$ between adjacent symbols. The default value is 2.0.

Examples:

  1. The $4$-PAM constellation with $\Delta = 2$ is depicted below.

    4-PAM constellation. 

    >>> const = komm.PAMConstellation(4)

  2. The $7$-PAM constellation with $\Delta = 5$ is depicted below.

    7-PAM constellation. 

    >>> const = komm.PAMConstellation(7, delta=5)

matrix Array2D[floating] cached property

The constellation matrix $\mathbf{X}$.

Examples:

>>> const = komm.PAMConstellation(4)
>>> const.matrix
array([[-3.],
       [-1.],
       [ 1.],
       [ 3.]])

order int property

The order $M$ of the constellation.

Examples:

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

dimension int property

The dimension $N$ of the constellation.

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

Examples:

>>> const = komm.PAMConstellation(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[floating]) –

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

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

Examples:

>>> const = komm.PAMConstellation(4)
>>> const.mean()
array([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 PAM constellation is given by $$ E = \frac{\Delta^2}{12}(M^2 - 1). $$

Examples:

>>> const = komm.PAMConstellation(4)
>>> const.mean_energy()
np.float64(5.0)

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 PAM constellation, the minimum distance is given by $$ d_{\min} = \Delta. $$

Examples:

>>> const = komm.PAMConstellation(4)
>>> 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.PAMConstellation(4)
>>> const.indices_to_symbols([3, 0])
array([ 3., -3.])
>>> const.indices_to_symbols([[3, 0], [1, 2]])
array([[ 3., -3.],
       [-1.,  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.PAMConstellation(4)
>>> const.closest_indices([-0.8, 2.4])
array([1, 3])
>>> const.closest_indices([[-0.8, 2.4], [0.0, 10.0]])
array([[1, 3],
       [2, 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.PAMConstellation(4)
>>> const.closest_symbols([-0.8, 2.4])
array([-1.,  3.])
>>> const.closest_symbols([[-0.8, 2.4], [0.0, 10.0]])
array([[-1.,  3.],
       [ 1.,  3.]])

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.PAMConstellation(4)
>>> const.posteriors([-0.8, 2.4], snr=2.0).round(3)
array([0.103, 0.7  , 0.195, 0.002, 0.   , 0.007, 0.343, 0.65 ])
>>> const.posteriors([[-0.8, 2.4], [0.0, 10.0]], snr=2.0).round(3)
array([[0.103, 0.7  , 0.195, 0.002, 0.   , 0.007, 0.343, 0.65 ],
       [0.02 , 0.48 , 0.48 , 0.02 , 0.   , 0.   , 0.   , 1.   ]])