komm.PSKModulation
Phase-shift keying (PSK) modulation. It is a complex modulation scheme in which the constellation symbols are uniformly arranged in a circle. More precisely, the the $i$-th constellation symbol is given by $$ x_i = A \exp \left( \mathrm{j} \frac{2 \pi i}{M} \right) \exp(\mathrm{j} \phi), \quad i \in [0 : M), $$ where $M$ is the order (a power of $2$), $A$ is the amplitude, and $\phi$ is the phase offset of the modulation.
Parameters:
-
order(int) –The order $M$ of the modulation. It must be a power of $2$.
-
amplitude(float) –The amplitude $A$ of the constellation. The default value is
1.0. -
phase_offset(float) –The phase offset $\phi$ of the constellation. The default value is
0.0. -
labeling(Literal['natural', 'reflected'] | ArrayLike) –The binary labeling of the modulation. Can be specified either as a 2D-array of integers (see base class for details), or as a string. In the latter case, the string must be either
'natural'or'reflected'. The default value is'reflected', corresponding to the Gray labeling.
Examples:
-
The $4$-PSK modulation with base amplitude $A = 1$, phase offset $\phi = 0$, and Gray labeling is depicted below.
>>> psk = komm.PSKModulation(4) >>> psk.constellation.round(3) array([ 1.+0.j, 0.+1.j, -1.+0.j, -0.-1.j]) >>> psk.labeling array([[0, 0], [0, 1], [1, 1], [1, 0]]) -
The $8$-PSK modulation with base amplitude $A = 0.5$, phase offset $\phi = \pi/8$, and natural labeling is depicted below.
>>> psk = komm.PSKModulation( ... order=8, ... amplitude=0.5, ... phase_offset=np.pi/8, ... labeling='natural', ... ) >>> psk.constellation.round(3) array([ 0.462+0.191j, 0.191+0.462j, -0.191+0.462j, -0.462+0.191j, -0.462-0.191j, -0.191-0.462j, 0.191-0.462j, 0.462-0.191j]) >>> psk.labeling array([[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]])
constellation NDArray[complexfloating]
cached
property
The constellation $\mathbf{X}$ of the modulation.
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.constellation.round(3)
array([ 1.+0.j, 0.+1.j, -1.+0.j, -0.-1.j])
labeling NDArray[integer]
cached
property
The labeling $\mathbf{Q}$ of the modulation.
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.labeling
array([[0, 0],
[0, 1],
[1, 1],
[1, 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:
>>> psk = komm.PSKModulation(4)
>>> psk.inverse_labeling
{(0, 0): 0, (0, 1): 1, (1, 1): 2, (1, 0): 3}
order int
cached
property
The order $M$ of the modulation.
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.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:
>>> psk = komm.PSKModulation(4)
>>> psk.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.
For the PSK, it is given by $$ E_\mathrm{s} = A^2. $$
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.energy_per_symbol
1.0
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:
>>> psk = komm.PSKModulation(4)
>>> psk.energy_per_bit
0.5
symbol_mean complex
cached
property
The mean $\mu_\mathrm{s}$ of the constellation. It assumes equiprobable symbols. It is given by $$ \mu_\mathrm{s} = \frac{1}{M} \sum_{i \in [0:M)} x_i. $$
For the PSK, it is given by $$ \mu_\mathrm{s} = 0. $$
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.symbol_mean
0j
minimum_distance float
cached
property
The minimum Euclidean distance $d_\mathrm{min}$ of the constellation. It is given by $$ d_\mathrm{min} = \min_ { i, j \in [0:M), ~ i \neq j } \lVert x_i - x_j \rVert. $$
For the PSK, it is given by $$ d_\mathrm{min} = 2A\sin\left(\frac{\pi}{M}\right). $$
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.minimum_distance
1.414213562373095
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[complexfloating]) –The output sequence(s). Has the same shape as the input, with the last dimension divided by $m$.
Examples:
>>> psk = komm.PSKModulation(4)
>>> psk.modulate([0, 0, 1, 1, 0, 0, 0, 1]).round(3)
array([ 1.+0.j, -1.+0.j, 1.+0.j, 0.+1.j])
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 $m$.
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 $m$.