real_fourier_stf_signal

Graph.real_fourier_stf_signal(duration, initial_coefficient_lower_bound=-1, initial_coefficient_upper_bound=1, fixed_frequencies=None, optimizable_frequency_count=None, randomized_frequency_count=None)

Create a real sampleable signal constructed from Fourier components.

Use this function to create a signal defined in terms of Fourier (sine/cosine) basis signals that can be optimized by varying their coefficients and, optionally, their frequencies.

Parameters:

  • duration (float) – The total duration \(\tau\) of the signal.

  • initial_coefficient_lower_bound (float, optional) – The lower bound \(c_\mathrm{min}\) on the initial coefficient values. Defaults to -1.

  • initial_coefficient_upper_bound (float, optional) – The upper bound \(c_\mathrm{max}\) on the initial coefficient values. Defaults to 1.

  • fixed_frequencies (np.ndarray or None, optional) – A 1D array containing the fixed non-zero frequencies \(\{f_m\}\) to use for the Fourier basis. If provided, must be non-empty and specified in the inverse units of duration (for example if duration is in seconds, these values must be given in Hertz).

  • optimizable_frequency_count (int or None, optional) – The number of non-zero frequencies \(M\) to use, if the frequencies can be optimized. Defaults to 0.

  • randomized_frequency_count (int or None, optional) – The number of non-zero frequencies \(M\) to use, if the frequencies are to be randomized but fixed. Defaults to 0.

Returns:

The optimizable, real-valued, sampleable signal built from the appropriate Fourier components.

Return type:

Stf

Warning

You must provide exactly one of fixed_frequencies, optimizable_frequency_count, or randomized_frequency_count.

See also

Graph.real_fourier_pwc_signal

Corresponding operation for Pwc.

Notes

This function sets the basis signal frequencies \(\{f_m\}\) depending on the chosen mode:

  • For fixed frequencies, you provide the frequencies directly.

  • For optimizable frequencies, you provide the number of frequencies \(M\), and this function creates \(M\) unbounded optimizable variables \(\{f_m\}\), with initial values in the ranges \(\{[(m-1)/\tau, m/\tau]\}\).

  • For randomized frequencies, you provide the number of frequencies \(M\), and this function creates \(M\) randomized constants \(\{f_m\}\) in the ranges \(\{[(m-1)/\tau, m/\tau]\}\).

After this function creates the \(M\) frequencies \(\{f_m\}\), it produces the signal

\[\alpha^\prime(t) = v_0 + \sum_{m=1}^M [ v_m \cos(2\pi t f_m) + w_m \sin(2\pi t f_m) ],\]

where \(\{v_m,w_m\}\) are (unbounded) optimizable variables, with initial values bounded by \(c_\mathrm{min}\) and \(c_\mathrm{max}\). This function produces the final signal \(\alpha(t)\).

You can use the signals created by this function for chopped random basis (CRAB) optimization [1].

References