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komm.Modulation

General modulation scheme. A modulation scheme of order $M = 2^m$ is defined by a constellation $\mathbf{X}$, which is a real or complex vector of length $M$, and a binary labeling $\mathbf{Q}$, which is an $M \times m$ binary matrix whose rows are all distinct. The $i$-th element of $\mathbf{X}$, for $i \in [0:M)$, is denoted by $x_i$ and is called the $i$-th constellation symbol. The $i$-th row of $\mathbf{Q}$, for $i \in [0:M)$, is called the binary representation of the $i$-th constellation symbol. For more details, see SA15, Sec. 2.5."

Parameters:

  • constellation (ArrayLike) –

    The constellation $\mathbf{X}$ of the modulation. Must be a 1D-array containing $M$ real or complex numbers.

  • labeling (ArrayLike) –

    The binary labeling $\mathbf{Q}$ of the modulation. Must be a 2D-array of shape $(M, m)$ where each row is a distinct binary $m$-tuple.

Examples:

  1. The real modulation scheme depicted in the figure below has $M = 4$ and $m = 2$.

    Example for real modulation with M = 4
    The constellation and labeling are given, respectively, by $$ \mathbf{X} = \begin{bmatrix} -0.5 \\ 0.0 \\ 0.5 \\ 2.0 \end{bmatrix} \qquad\text{and}\qquad \mathbf{Q} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}. $$

    >>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )

  2. The complex modulation scheme depicted in the figure below has $M = 4$ and $m = 2$.

    Example for complex modulation with M = 4
    The constellation and labeling are given, respectively, by $$ \mathbf{X} = \begin{bmatrix} 0 \\ -1 \\ 1 \\ \mathrm{j} \end{bmatrix} \qquad\text{and}\qquad \mathbf{Q} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{bmatrix}. $$

    >>> modulation = komm.Modulation(
...     constellation=[0, -1, 1, 1j],
...     labeling=[[0, 0], [0, 1], [1, 0], [1, 1]],
... )

constellation NDArray[T] cached property

The constellation $\mathbf{X}$ of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.constellation
array([-0.5,  0. ,  0.5,  2. ])

labeling NDArray[integer] cached property

The labeling $\mathbf{Q}$ of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.labeling
array([[1, 0],
       [1, 1],
       [0, 1],
       [0, 0]])

inverse_labeling dict[tuple[int, ...], int] cached property

The inverse labeling of the modulation. It is a dictionary that maps each binary tuple to the corresponding constellation index.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.inverse_labeling
{(1, 0): 0, (1, 1): 1, (0, 1): 2, (0, 0): 3}

order int cached property

The order $M$ of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.order
4

bits_per_symbol int cached property

The number $m$ of bits per symbol of the modulation. It is given by $$ m = \log_2 M, $$ where $M$ is the order of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.bits_per_symbol
2

energy_per_symbol float cached property

The average symbol energy $E_\mathrm{s}$ of the constellation. It assumes equiprobable symbols. It is given by $$ E_\mathrm{s} = \frac{1}{M} \sum_{i \in [0:M)} \lVert x_i \rVert^2, $$ where $\lVert x_i \rVert^2$ is the energy of constellation symbol $x_i$, and $M$ is the order of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.energy_per_symbol
1.125

>>> modulation = komm.Modulation(
...     constellation=[0, -1, 1, 1j],
...     labeling=[[0, 0], [0, 1], [1, 0], [1, 1]],
... )
>>> modulation.energy_per_symbol
0.75

energy_per_bit float cached property

The average bit energy $E_\mathrm{b}$ of the constellation. It assumes equiprobable symbols. It is given by $$ E_\mathrm{b} = \frac{E_\mathrm{s}}{m}, $$ where $E_\mathrm{s}$ is the average symbol energy, and $m$ is the number of bits per symbol of the modulation.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.energy_per_bit
0.5625

>>> modulation = komm.Modulation(
...     constellation=[0, -1, 1, 1j],
...     labeling=[[0, 0], [0, 1], [1, 0], [1, 1]],
... )
>>> modulation.energy_per_bit
0.375

symbol_mean float | complex cached property

The mean $\mu_\mathrm{s}$ of the constellation. It assumes equiprobable symbols. It is given by $$ \mu_\mathrm{s} = \frac{1}{M} \sum_{i \in [0:M)} x_i. $$

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.symbol_mean
0.5

>>> modulation = komm.Modulation(
...     constellation=[0, -1, 1, 1j],
...     labeling=[[0, 0], [0, 1], [1, 0], [1, 1]],
... )
>>> modulation.symbol_mean
0.25j

minimum_distance float cached property

The minimum Euclidean distance $d_\mathrm{min}$ of the constellation. It is given by $$ d_\mathrm{min} = \min_ { i, j \in [0:M), ~ i \neq j } \lVert x_i - x_j \rVert. $$

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.minimum_distance
0.5

>>> modulation = komm.Modulation(
...     constellation=[0, -1, 1, 1j],
...     labeling=[[0, 0], [0, 1], [1, 0], [1, 1]],
... )
>>> modulation.minimum_distance
1.0

modulate()

Modulates one or more sequences of bits to their corresponding constellation symbols.

Parameters:

  • input (ArrayLike) –

    The input sequence(s). Can be either a single sequence whose length is a multiple of $m$, or a multidimensional array where the last dimension is a multiple of $m$.

Returns:

  • output (NDArray[T]) –

    The output sequence(s). Has the same shape as the input, with the last dimension divided by $m$.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.modulate([0, 0, 1, 1, 0, 0, 1, 0])
array([ 2. ,  0. ,  2. , -0.5])
>>> modulation.modulate([[0, 0, 1, 1], [0, 0, 1, 0]])
array([[ 2. ,  0. ],
       [ 2. , -0.5]])

demodulate_hard()

Demodulates one or more sequences of received points to their corresponding sequences of hard bits ($\mathtt{0}$ or $\mathtt{1}$) using hard-decision decoding.

Parameters:

  • input (ArrayLike) –

    The input sequence(s). Can be either a single sequence, or a multidimensional array.

Returns:

  • output (NDArray[integer]) –

    The output sequence(s). Has the same shape as the input, with the last dimension multiplied by $m$.

Note

This method implements the general minimum Euclidean distance hard demodulator, assuming uniformly distributed symbols.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.demodulate_hard([2.17, -0.06, 1.94, -0.61])
array([0, 0, 1, 1, 0, 0, 1, 0])
>>> modulation.demodulate_hard([[2.17, -0.06], [1.94, -0.61]])
array([[0, 0, 1, 1],
       [0, 0, 1, 0]])

demodulate_soft()

Demodulates one or more sequences of received points to their corresponding sequences of soft bits (L-values) using soft-decision decoding. The soft bits are the log-likelihood ratios of the bits, where positive values correspond to bit $\mathtt{0}$ and negative values correspond to bit $\mathtt{1}$.

Parameters:

  • input (ArrayLike) –

    The received sequence(s). Can be either a single sequence, or a multidimensional array.

  • snr (float) –

    The signal-to-noise ratio (SNR) of the channel. It should be a positive real number. The default value is 1.0.

Returns:

  • output (NDArray[floating]) –

    The output sequence(s). Has the same shape as the input, with the last dimension multiplied by $m$.

Note

This method implements the general soft demodulator, assuming uniformly distributed symbols.

Examples:

>>> modulation = komm.Modulation(
...     constellation=[-0.5, 0.0, 0.5, 2.0],
...     labeling=[[1, 0], [1, 1], [0, 1], [0, 0]],
... )
>>> modulation.demodulate_soft([2.17, -0.06, 1.94, -0.61], snr=100.0).round(1)
array([ 416. ,  245.3,  -27.6,  -16.9,  334.2,  184. , -108.4,   32. ])
>>> modulation.demodulate_soft([[2.17, -0.06], [1.94, -0.61]], snr=100.0).round(1)
array([[ 416. ,  245.3,  -27.6,  -16.9],
       [ 334.2,  184. , -108.4,   32. ]])