FourierExplorer
Joachim Köppen DF3GJ ... Kiel, March 2025

Some brief explanations

• This script explores the Fourier transform, showing for some examples the relation between a time-varying signal and its frequency spectrum, and likewise between the brightness distribution of an image and its spatial frequency spectrum, which is the visibility function obtained by interferometric observations of a (radio) source.
• The spectrum is passed through a user-defined band-pass filter, and the signal/image reconstructed from the filtered spectrum is shown. In this manner, one may inspect which part of the spectrum/visibility influences which part of the data, and where the information for which aspect of the data feature is stored in the spectrum.
• Various simple time-series signals and brightness distributions can be chosen.
  One may keep constant the maximum frequency fmax, umax or the time/space window of the data window, by clicking on the corresponding textfield, which will get a green background.
• The image sources can be treated as linear 1-D objects or as 2-D objects, hence a circular disc with a given radial brightness profile.
• nbin is the number of bins in the FFT (FastFourierTransformation) used here.
• fmax, umax is the maximum temporal/spatial frequency. Entering data and clicking on this textfield will keep this value constant, marked by a green background.
• window is the width of the time/space window (or the field of view) of the original data. Entering data and clicking on this textfield will keep this value constant, marked by a green background.
• Depending on the sample data, there will be one or several parameters describing the data
• The spectrum displayed in the middle plot is always normalized to a peak value of 1.0 (or 0 dB).
• bandpass filter enter the lower and upper frequency limit of the filter. The spectrum passed-through is shown in red, while the remaining parts in grey.
• Phase is shown with a magenta curve, between -0.5 and +0.5, meaning -π/2 and +π/2
• Xrange buttons will set the full x-range corresponding to the nbin setting.
  Yrange buttons will set the default y-ranges.
• A magenta color of the spectrum display's Xrange button indicates that the spectral range shown in the plot differs from the fmax/umax setting. Clicking the button will show the full range.
• mouse position displays the coordinates of the mouse position in each plot.


• Also shown is the observed visibility function of the radio source 3C33.1 observed on 1425 MHz (from Fig.1 of E.B.Fomalont (1968, Ap.J.Suppl. 15, 203)), either by a simple polygonal interpolation between the observed visibilities or by the fit curve from the article. Note that the left hand plot is intentionally left empty.