Fourier Transform

The fourier command transforms EXAFS data from k-space to R-space (radial distance space) via Fourier transform, enabling analysis of the local coordination environment around the absorbing atom.

The transform is applied to k-weighted chi data with optional apodization:

\[F(R) = \int k^n \cdot \chi(k) \cdot w(k) \cdot e^{2ikR} dk\]

where \(n\) is the k-weight, \(w(k)\) is an apodization window, and the integral is over the specified k-range.

Basic Usage

fourier <k-range> [options]

where:

• <k-range> is a mandatory two-value range in k-space (required)
• [options] control k-weighting, window function, and r-axis spacing

Command Options

Option	Description
<k-range>	k-space range for the transform (required). Both limits must be specified; use .. for open bounds within the available data. Accepts k units. See range specification for details.
--kweight <n> -k <n>	k-weighting exponent (default: 0.0). Common values: 1 (linear), 2 (quadratic), 3 (cubic).
--apodizer <func> -a <func>	Window function for apodization (default: hanning). See Apodization Functions for choices.
--parameter <p> -p <p>	Shape parameter for window functions that support it (default: 3.0).
--width <distance> -w <distance>	Ramp width for the apodization window edges (default: half the k-range span). Specified in Å.
--maxR <distance>	Maximum R value for the output r-axis (default: auto-computed from k-range). Specified in Å.
--dr <distance> --spacing <distance>	Fixed r-axis spacing (default: Nyquist-optimal). Specified in Å.

Apodization Functions

Apodization windows reduce spectral ringing from the sharp k-range cutoffs. EstraPy supports:

Function	Description	Parameter use
rectangular	No apodization (sharp cutoff)	-
hanning	Cosine-based window (default)	-
hamming	Variant of Hanning with reduced sidelobe	-
welch	Parabolic window	-
sine	Pure sine window	-
blackman	Very low sidelobe window	-
gaussian	Gaussian shape with tunable width	See definition below.
exponential	Exponential decay with tunable rate	See definition below.

Window names are matched fuzzily (minimum 3 characters required).

Window definitions

Rectangular

No apodization is applied (sharp cutoff):

\[w(r) = \begin{cases} 1 & r \leq 1 \\ 0 & r > 1 \end{cases}\]

Hanning

Cosine-based smooth window:

\[w(r) = \frac{1}{2} + \frac{1}{2}\cos(\pi r)\]

Hamming

Variant of Hanning with reduced first sidelobe:

\[w(r) = 0.54 - 0.46\cos(2\pi r)\]

Welch

Parabolic window:

\[w(r) = 1 - r^2\]

Sine

Pure sine taper:

\[w(r) = \cos\left(\frac{\pi r}{2}\right)\]

Blackman

Very low sidelobe window with excellent stopband attenuation:

\[w(r) = 0.42 - 0.5\cos(2\pi r) + 0.08\cos(4\pi r)\]

Gaussian

Gaussian shape with tunable width controlled by parameter \(p\) (default: 3.0):

\[w(r) = \exp(-r^2 p)\]

Higher \(p\) values create narrower windows with sharper cutoffs.

Exponential

Exponential decay with tunable rate controlled by parameter \(p\) (default: 3.0):

\[w(r) = \exp(-rp)\]

Higher \(p\) values create faster decay rates.

Parameter definition

For gaussian and exponential windows, the --parameter option controls the shape:

• Gaussian: Controls the width; higher values (\(p \geq 5\)) create sharper transitions
• Exponential: Controls the decay rate; higher values create steeper roll-off

The normalized distance \(r\) ranges from 0 (at the flat region edge) to 1 (at the window outer edge).

R-axis Specification

The r-axis (output radial distance grid) can be specified via:

Case	r-axis behavior
No options	Nyquist-optimal: spacing and extent determined from k-range via \(\Delta R = \pi / (2 \Delta k_{\max})\)
--maxR <Rmax> only	Nyquist spacing computed, axis spans \([0, R_{\max}]\)
--dr <spacing> only	Fixed spacing, extent computed from k-range
Both --maxR and --dr	Fixed spacing and extent: axis spans \([0, R_{\max}]\) with steps of --dr

Results

After executing fourier, a new Fourier domain is created for each page containing:

• r - Radial distance axis (in Ångströms)
• f - Fourier transform magnitude

The magnitude is typically plotted to identify coordination shells.

Examples

# Basic Fourier transform with Hanning window
fourier 3k 15k

# Quadratic k-weighted transform (emphasizes higher-k)
fourier 2k 14k --kweight 2

# Custom window with narrow ramp (sharp edges)
fourier 3k 14k --apodizer kaiser --parameter 5 --width 0.5A

# Fine r-space sampling up to 6 Ångströms
fourier 2k 16k --maxR 6A --dr 0.01A

# Linear k-weighted with Blackman window
fourier 3k 13k --kweight 1 --apodizer blackman

Tips and Best Practices

1. k-range selection:
   • Start where the data quality is good (typically 2–3 k)
   • End where noise dominates (often 12–15 k, depending on material and temperature)
   • Wider ranges improve r-resolution per Nyquist
2. k-weight choice:
   • k=1: Emphasizes low-k structure
   • k=2: Default choice, balanced emphasis
   • k=3: Emphasizes high-k (farther-distance) features
3. r-axis:
   • Nyquist default provides good efficiency
   • Reduce --maxR to focus on near-neighbor shells
   • Increase --dr for smoother curves at the cost of larger output (does not generate new information beyond Nyquist)

See also:

• Background Removal
• General syntax