\(\renewcommand\AA{\unicode{x212B}}\)

RealFFT v1

Enable screenshots using DOCS_SCREENSHOTS in CMake

Summary

Performs real Fast Fourier Transform

See Also

ExtractFFTSpectrum, FFT, FFTDerivative, MaxEnt, SassenaFFT, FFTSmooth

Properties

Name	Direction	Type	Default	Description
InputWorkspace	Input	MatrixWorkspace	Mandatory	The name of the input workspace.
OutputWorkspace	Output	MatrixWorkspace	Mandatory	The name of the output workspace. It contains three spectra: the real, the imaginary parts of the transform and their modulus.
WorkspaceIndex	Input	number	0	The index of the spectrum in the input workspace to transform.
Transform	Input	string	Forward	The direction of the transform: “Forward” or “Backward”. Allowed values: [‘Forward’, ‘Backward’]
IgnoreXBins	Input	boolean	False	Ignores the requirement that X bins be linear and of the same size. FFT result will not be valid for the X axis, and should be ignored.

Description

This is an algorithm for Fourier transform of real data. It uses the GSL routines gsl_fft_real_transform and gsl_fft_halfcomplex_inverse. The result of a forward transform is a three-spectra workspace with the real and imaginary parts of the transform in position 0 and 1 correspondingly. The third spectrum has the modulus of the transform (\(\sqrt{Re^2+Im^2}\)). Only positive frequencies are given and as a result the output spectra are twice as short as the input one.

An input workspace for backward transform must have the form of the output workspace of the forward algorithm, i.e. its first two spectra must have the real part and the imaginary parts. The output workspace contains a single spectrum with the real inverse transform.

The forward transform doesn’t use the absolute values of the X axis but rather assumes that the data starts at X = 0 and ends at X = XMax - XMin, where XMin and XMax are the lower and the upper limits on the X axis. As a result the output is a transform of the input spectrum shifted along the X axis so that the first bin is centered at 0.

The output of the backward transform always starts at X = 0.

Usage

import numpy as np

# Create a workspace with a Gaussian peak in the centre.
ws = CreateSampleWorkspace(Function='User Defined',UserDefinedFunction='name=Gaussian,Height=1,PeakCentre=0,Sigma=1',XMin=-10,XMax=10,BinWidth=0.1)
# Forward transform its 11-th spectrum
transform = RealFFT( ws, 10, "Forward")
# Backward transform must look like the original spectrum only shifted along the X axis
ws_back = RealFFT(transform, Transform="Backward")

# Check the outputs

# Check the sizes
print('Number of bins in the input workspace    {}'.format(ws.blocksize()))
print('Number of bins in the forward transform  {}'.format(transform.blocksize()))
print('Number of bins in the backward transform {}'.format(ws_back.blocksize()))

# Check the X axes
print('Input starts at {:.1f} , ends at {:.1f} , the width is {:.1f}'.format(
      ws.readX(10)[0], ws.readX(10)[-1], ws.readX(10)[-1] - ws.readX(10)[0]))
print('Forward starts at  {:.1f} , ends at {:.2f} , the width is {:.2f}'.format(
      transform.readX(0)[0], transform.readX(0)[-1], transform.readX(0)[-1] - transform.readX(0)[0]))
print('Backward starts at {:.1f} , ends at {:.1f} , the width is {:.1f}'.format(
      ws_back.readX(0)[0], ws_back.readX(0)[-1], ws_back.readX(0)[-1] - ws_back.readX(0)[0]))

# Check that the backward transform restores the original data.
# The input spetrum values
y10 = ws.readY(10)
# The spectrum returned from the backward RealFFT
y_back = ws_back.readY(0)
# Check that they are almost equal.
# Using numpy array calculations show that all elements of arrays y_back and y10 are very close
print(np.all(np.abs(y_back - y10) < 1e-15))
# but not equal
print(np.all(y_back == y10))

Output

Number of bins in the input workspace    200
Number of bins in the forward transform  101
Number of bins in the backward transform 200
Input starts at -10.0 , ends at 10.0 , the width is 20.0
Forward starts at  0.0 , ends at 5.05 , the width is 5.05
Backward starts at 0.0 , ends at 20.0 , the width is 20.0
True
False

Categories: AlgorithmIndex | Arithmetic\FFT

Source

C++ header: RealFFT.h

C++ source: RealFFT.cpp