Fourier transform

General functions used for the discrete Fourier transform (DFT).

Added in version 1.0.4.

nexus.SamplingFrequency(orig_step)

The sampling frequency \(f_s\) for the DFT is given by the inverse of the step size in the base space.

\[f_s = \frac{1}{orig\_step}\]

Parameters:

orig_step (float) – original step size in the base space.

Returns:

sampling frequency for the DFT.

Return type:

float

nexus.Step(num_points, orig_step)

The frequency step \(\Delta f\) for the DFT is given by

\[\Delta f = \frac{f_s}{N} = \frac{1}{N * orig\_step}\]

Parameters:

• num_points (int) – number of points of the original base space.

• orig_step (float) – original step size in the base space.

Returns:

step size for the Fourier transform.

Return type:

float

nexus.Resolution(num_points, sampling_frequency)

The resolution of the DFT is the smallest visible frequency. Same as the frequency step.

\[\Delta f = \frac{f_s}{N} = \frac{1}{N * orig\_step}.\]

Parameters:

• num_points (int) – number of points of the original base space.

• sampling_frequency (float) – sampling frequency of the Fourier transform.

Returns:

Resolution of the DFT.

Return type:

float

Note

Parceval’s theorem is

\[\sum_{k=0}^{N-1} \left| x(orig\_step)_k \right|^2 = \frac{1}{N} * \sum_{n=0}^{N-1} \left|X(fourier\_step)_n \right|^2\]

nexus.NormalizationN(num_points)

Normalization factor \(\frac{1}{N}\). Used for Intensities to ensures Parseval’s theorem.

Parameters:

num_points (int) – number of points of the original base space.

Returns:

Normalization factor.

Return type:

float

nexus.NormalizationSqrtN(number_points)

Normalization factor \(\frac{1}{\sqrt{N}}\). Used for amplitudes to ensures Parseval’s theorem.

Parameters:

num_points (int) – number of points of the original base space.

Returns:

Normalization factor.

Return type:

float

nexus.NormalizationNIntegratedIntensity(number_points, orig_step)

Normalization for original-space-integrated intensity \(\frac{orig\_step}{N}\). Obeys Parseval’s theorem. Relates the Fourier components via

\[orig\_step * \sum_{k=0}^{N-1} \left| x(orig\_step)_k \right|^2 = \frac{orig\_step}{N} * \sum_{n=0}^{N-1} \left| X(fourier\_step)_n \right|^2\]

Parameters:

• num_points (int) – number of points of the original base space.

• orig_step (float) – original step size in the base space.

Returns:

Normalization factor.

Return type:

float

nexus.NormalizationNIntegratedIntensityScaled(number_points, orig_step, fourier_step)

Normalization for original-space-integrated intensity scaled by Fourier step size \(\frac{step\_size}{N * fourier\_step}\). This normalization factor is scaled by the Fourier step in order to compensate for the changing sampling rate (with changing N or time_step), for which each bin represents a changing Fourier band. Similar to the scaling of a power density spectrum.

\[\frac{orig\_step}{fourier\_step} * \sum_{k=0}^{N-1} \left| x(orig\_step)_k \right|^2 = \frac{orig\_step}{N * fourier\_step} * \sum_{n=0}^{N-1} \left|X(fourier\_step)_n \right|^2\]

Parameters:

• num_points (int) – number of points of the original base space.

• orig_step (float) – original step size in the base space.

• fourier_step (float) – step size in the Fourier space.

Returns:

Normalization factor.

Return type:

float

nexus.NormalizationIntegratedIntensityScaled(orig_step, fourier_step)

Normalization for original-space-integrated intensity scaled by Fourier step size \(\frac{orig\_step}{fourier\_step}\). Assumes that the Fourier coefficients are already normalized by \(\frac{1}{\sqrt{N}}\). Normalization does not obey Parseval’s theorem. This normalization factor is scaled by the Fourier step in order to compensate for the changing sampling rate (with changing N or time_step), for which each bin represents a changing Fourier band. Similar to the scaling of a power density spectrum.

\[\frac{orig\_step}{fourier\_step} * \sum_{k=0}^{N-1} \left| x(orig\_step)_k \right|^2 = \frac{orig\_step}{fourier\_step} * \sum_{n=0}^{N-1} \left| \frac{1}{\sqrt{N}} X(fourier\_step)_n \right|^2\]

Added in version 1.0.4.

Parameters:

• orig_step (float) – original step size in the base space.

• fourier_step (float) – step size in the Fourier space.

Returns:

Normalization factor.

Return type:

float

Note

For window functions see https://en.wikipedia.org/wiki/Window_function

nexus.Welch(num_points)

Creates a Welch window.

Added in version 1.0.4.

Parameters:

num_points (int) – number of points.

Returns:

Welch window.

Return type:

list

nexus.Sine(num_points)

Creates a Sine window.

Added in version 1.0.4.

Parameters:

num_points (int) – number of points.

Returns:

Sine window.

Return type:

list

nexus.Hann(num_points)

Creates a Hann window.

Added in version 1.0.4.

Parameters:

num_points (int) – number of points.

Returns:

Hann window.

Return type:

list

nexus.Hamming(num_points)

Creates a Hamming window.

Added in version 1.0.4.

Parameters:

num_points (int) – number of points.

Returns:

Hamming window.

Return type:

list

nexus.Kaiser(num_points, alpha)

Creates a Kaiser window.

Added in version 1.0.4.

Parameters:

• num_points (int) – number of points.

• alpha (float) – alpha value.

Returns:

Kaiser window.

Return type:

list