tanh_ramp

The Boulder Opal Toolkits are currently in beta phase of development. Breaking changes may be introduced.

tanh_ramp(duration, end_value, start_value=None, ramp_duration=None, center_time=None)

Create a Signal object representing a hyperbolic tangent ramp.

Parameters:

duration (float) – The duration of the signal, \(T\).
end_value (float or complex) – The asymptotic value of the ramp towards \(t \to +\infty\), \(a_+\).
start_value (float or complex, optional) – The asymptotic value of the ramp towards \(t \to -\infty\), \(a_-\). Defaults to minus end_value.
ramp_duration (float, optional) – The characteristic time for the hyperbolic tangent ramp, \(t_\mathrm{ramp}\). Defaults to \(T/6\).
center_time (float, optional) – The time at which the ramp has its greatest slope, \(t_0\). Defaults to \(T/2\).

Returns:

The hyperbolic tangent ramp.

Return type:

Signal

See also

signals.linear_ramp(): Create a Signal object representing a linear ramp.
signals.tanh_ramp_pwc(): Graph operation to create a Pwc representing a hyperbolic tangent ramp.
signals.tanh_ramp_stf(): Graph operation to create a Stf representing a hyperbolic tangent ramp.

Notes

The hyperbolic tangent ramp is defined as

\[\mathop{\mathrm{Tanh}}(t) = \frac{a_+ + a_-}{2} + \frac{a_+ - a_-}{2} \tanh\left( \frac{t - t_0}{t_\mathrm{ramp}} \right) ,\]

where the function’s asymptotic values \(a_\pm\) are defined by:

\[a_\pm := \lim_{t\to\pm\infty} \mathop{\mathrm{Tanh}}(t) ,\]

and \(t_0\) is related to \(t_\mathrm{ramp}\) by:

\[\left.\frac{{\rm d}\mathop{\mathrm{Tanh}}(t)}{{\rm d}t}\right|_{t=t_0} = \frac{ (a_+ - a_-)}{2 t_\mathrm{ramp}} .\]

Note that if \(t_0\) is close to the edges of the ramp, for example \(t_0 \lesssim 2 t_\mathrm{ramp}\), then the first and last values of the outputted array will differ from the expected asymptotic values.

With the default values of start_value (\(a_-\)), ramp_duration (\(t_\mathrm{ramp}\)), and center_time (\(t_0\)), the ramp expression simplifies to

\[\mathop{\mathrm{Tanh}}(t) = A \tanh\left( \frac{t - T/2}{T/6} \right),\]

where \(A = a_+\) is the end value (the start value is then \(-A\)). This defines a symmetric ramp (around \((T/2, 0)\)) between \(-0.995 A\) (at \(t=0\)) and \(0.995 A\) (at \(t=T\)).

Examples

Define a tanh ramp.

>>> signal = qctrl.signals.tanh_ramp(
...     duration=4, end_value=2, start_value=1, ramp_duration=0.4, center_time=2.
... )
>>> signal.export_with_time_step(time_step=0.4)
array([1.00012339, 1.00091105, 1.00669285, 1.04742587, 1.26894142,
   1.73105858, 1.95257413, 1.99330715, 1.99908895, 1.99987661])