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ATHENA and IFEFFIT determine the background function using a definition based in Fourier theory. In short, the background is that part of the μ(E) data containing low frequency Fourier components, while the data consists of the remaining, higher-frequency Fourier components. The AUTOBK algorithm, then, is a Fourier-based technique. It varies the parameters of a piece-wise spline such that, when that spline is subtracted from the μ(E) data, what is left has its low frequency Fourier components optimized. As we will see shortly, the default behavior of ATHENA is to simply minimize the low frequency components.
Before discussing how ATHENA defines low and high frequency in the context of background removal, we can look at a simple illustration of why background removal is necessary and why it can be difficult. To understand the background removal algorithm, we need to understand a couple of things about the Fourier transform
ATHENA has a teaching tool which uses simple sine waves to demonstrate some of the most important features of the Fourier transform operation. In the Help menu, you will find an entry labeled "Explain Fourier transforms". When you select that, the main window will be replaced with the teaching tool shown here
The Fourier transform teaching tool.
The point of a Fourier transform is to distinguish the frequency components of a signal. The left side of The next figure shows a simple example of a mixed frequency signal along with its components. The right side shows that the two frequency signal Fourier transforms to a spectrum with two peaks, one for each component frequency.
(Left) When you click the button labeled “Plot waves in k”, this is what gets plotted. This shows two waves of phase 2 and 3 (i.e. the sine waves repeat after 2 or 3 on the scale of the x-axis). The third trace shows the sum of those two sine waves. In the summed data, the two signals are intermixed in a way that confuses the fact that there are two sine ways involved. (Right) When you Fourier transform the sum of the two sine waves, this gets plotted. That there are two peaks in the Fourier transform indicates that there were two sine waves involved in the sum. That the peaks are at 2 and 3 on the x-axis, indicate the frequencies of the sine waves in the sum. Try changing the values of the sine waves or adding a third sine wave of a different frequency and replotting the sum and its Fourier transform. The Fourier transform, then, is a tool that isolate different frequencies in a signal.
So, can we use a Fourier transform to distinguish the sine wave from the step-like function? The left side of The next figure shows an artificial XAS signal, consisting of a step-like function (actually an error function) multiplied by 1 plus a sine function, much like the equation for mu(E). The right side shows the Fourier transform of this XAS-like function.
(Left) This figure is of (poorly) simulated μ(E) spectrum. It was made by clicking on the button that says “Plot edge step”. In fact, this is a slightly broadened error function multiplied by one plus a sine wave of frequency 2. As such, it is an idealization of real XAS data. (Right) Clicking on the button labeled “Plot FT of edge step” results in this plot.
As you can see, the Fourier transform of the entire spectrum is dominated by the signal near 0 on the x-axis. The signal from the sine wave is a little bump barely recognizable above the ripple from the step-like function. Apparently we need to do something a bit more clever than simply Fourier transforming the μ(E) data if we want to examine the oscillatory chi function without excessive interference from the step-like part of the μ(E) spectrum.
The solution is to somehow determine the shape of the step-like part of the μ(E) spectrum and subtract it from the data. Ideally, this will remove all the stuff near 0 on the x-axis, leaving peaks associated with the oscillatory part of the signal.
The AUTOBK algorithm is based on the principles discussed in this section. Using a cutoff frequency, it varies a piece-wise spline and subtracts that spline from the data. When the Fourier components below that cutoff frequency are minimized, the spline is subtracted one last time from the data. The difference between the data and the spline are normalized, converted from energy to photoelectron wavenumber and called χ(k) .
In the next section, we will discuss how to choose that cutoff frequency in ATHENA and how to choose it sensibly.
The Fourier transform teaching tool, has some other features not discussed above. It lets you explore the effect of the extent of the data range on the Fourier transform, play with different windowing function, see how close together in frequency sine waves can be and still be distinguished in a Fourier transform, and explore the different parts of the complex Fourier transform. These are all important topics that will be touched upon in later sections.
