komm.RectangularPulse

Rectangular pulse. It is a pulse with waveform given by $$ p(t) = \begin{cases} 1, & 0 \leq t < w, \\ 0, & \text{otherwise}, \end{cases} $$ where $w$ is the relative width of the pulse, which must satisfy $0 < w \leq 1$.

The waveform of the rectangular pulse is depicted below for $w = 1$, and for $w = 0.5$.

Rectangular NRZ pulse. Rectangular RZ pulse.

Notes

For $w = 1$ it is also called the NRZ pulse.
For $w = 0.5$ it is also called the halfway RZ pulse.

Attributes:

width (float) –

The relative width $w$ of the pulse. Must satisfy $0 < w \leq 1$. The default value is 1.0.

`waveform()`

The waveform $p(t)$ of the pulse.

For the rectangular pulse, it is given by $$ p(t) = \rect\left(\frac{t - w/2}{w}\right). $$

Examples:

>>> pulse = komm.RectangularPulse(width=1.0)  # NRZ pulse
>>> pulse.waveform(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )
array([0., 0., 0., 0., 1., 1., 1., 1., 0.])

>>> pulse = komm.RectangularPulse(width=0.5)  # Halfway RZ pulse
>>> pulse.waveform(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )
array([0., 0., 0., 0., 1., 1., 0., 0., 0.])

`spectrum()`

The spectrum $\hat{p}(f)$ of the pulse.

For the rectangular pulse, it is given by $$ \hat{p}(f) = w \sinc(w f) \mathrm{e}^{-\mathrm{j} 2 \pi (w/2) f}. $$

Examples:

>>> pulse = komm.RectangularPulse(width=1.0)  # NRZ pulse
>>> np.abs(pulse.spectrum(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )).round(3)
array([0.   , 0.3  , 0.637, 0.9  , 1.   , 0.9  , 0.637, 0.3  , 0.   ])

>>> pulse = komm.RectangularPulse(width=0.5)  # Halfway RZ pulse
>>> np.abs(pulse.spectrum(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )).round(3)
array([0.318, 0.392, 0.45 , 0.487, 0.5  , 0.487, 0.45 , 0.392, 0.318])

`autocorrelation()`

The autocorrelation function $R(\tau)$ of the pulse.

For the rectangular pulse, it is given by $$ R(\tau) = w \tri\left(\frac{\tau}{w}\right). $$

Examples:

>>> pulse = komm.RectangularPulse(width=1.0)  # NRZ pulse
>>> pulse.autocorrelation(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )
array([0.  , 0.25, 0.5 , 0.75, 1.  , 0.75, 0.5 , 0.25, 0.  ])

>>> pulse = komm.RectangularPulse(width=0.5)  # Halfway RZ pulse
>>> pulse.autocorrelation(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... )
array([0.  , 0.  , 0.  , 0.25, 0.5 , 0.25, 0.  , 0.  , 0.  ])

`energy_density_spectrum()`

The energy density spectrum $S(f)$ of the pulse.

For the rectangular pulse, it is given by $$ S(f) = w^2 \sinc^2(w f). $$

Examples:

>>> pulse = komm.RectangularPulse(width=1.0)  # NRZ pulse
>>> pulse.energy_density_spectrum(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... ).round(3)
array([0.   , 0.09 , 0.405, 0.811, 1.   , 0.811, 0.405, 0.09 , 0.   ])

>>> pulse = komm.RectangularPulse(width=0.5)  # Halfway RZ pulse
>>> pulse.energy_density_spectrum(
...     [-1.0, -0.75, -0.5, -0.25, 0.0, 0.25, 0.5, 0.75, 1.0],
... ).round(3)
array([0.101, 0.154, 0.203, 0.237, 0.25 , 0.237, 0.203, 0.154, 0.101])