Editor: This article is the first of a three-part series. Check out Part Two here and Part Three here.

The Fourier transform is a hugely important mathematical operation that is used by scientists, engineers, financial analysts and other specialists interested in analysing patterns in data.

It was originally devised by the French mathematician Jean-Baptiste Joseph Fourier, who demonstrated that any mathematical function (e.g. y = f(x)) which repeats itself over a known window of time, space or whatever the x-axis represents – in other words, a periodic function - can be shown as the sum of many sine and cosine waves. This property allows the Fourier transform to take a function, typically describing a variable that changes over the time or spatial (meaning “space”) domain, and to display that same function as a series of waves in a totally new frequency domain.

In this article, we will take a qualitative look at how the Fourier transform works. The aim is to provide an intuitive understanding of how any function can be converted into a new domain described by frequencies, before briefly looking at its application to several important techniques in structural biology.

We will then finish the topic off by walking through the derivation of the Fourier transform for readers who would like to better understand the mathematics behind the operation.

But before we look at some qualitative examples of the Fourier transform on 2D images, it is worth reminding ourselves about the basic features of waves.

## The properties of waves

A wave is a type of mathematical function that repeats indefinitely with a given height (amplitude), wavelength (also known as period) and phase (also referred to as phase shift and can be thought of as the ‘offset’ to the function) (Check out Figure 1).

Figure 1: (Left) Sinewave; (Right) Sinewave with phase shifted by −θ.

Sine and cosine waves are both examples of periodic functions and can be used to make more complicated functions by simply adding many of them together.

Figure 2 (top left, red) shows an example of an irregularly shaped but nevertheless periodic function. This function was created simply by adding together three sine waves that have different amplitudes, periods, and phases (i.e. the yellow, green and blue lines).

Overlaying these three sine waves on top of their resulting function shows how their maxima add together to produce larger peaks whilst their minima add to give deeper troughs. When our resultant function meets the x-axis (i.e. when f(x) = 0), all three sine waves have been summated but cancel each other out.

The frequency of a wave is simply the inverse of its period, or 1/period when written mathematically. Therefore, we can represent our constituent sine waves as a plot of amplitude as a function of frequency, where each wave is shown as a line with height equal to the wave’s amplitude (Figure 2, right plots).

Notice how the frequency plot of the green sine wave lacks its phase shift (−θ). This is a very important limitation pertaining to experiments that measure only the amplitude and frequency of constituent waves, such as X-ray crystallography.

Nevertheless, the addition of these waves described by either method – amplitude as a function of x, as shown in Figure 2 (left); or amplitude as a function of frequency, Figure 2 (right) – gives us the same function when they’re added together (ignoring for the loss of phase information in the frequency plots). Infinitely repeating (or periodic) functions, such as the one in this example, can be fully described by the sum of a series of sine and cosine waves, which is also known as a Fourier series.

However, many of the functions we wish to observe in nature are not truly periodic. Even those that do theoretically repeat into infinity are not measurable within a finite interval of time – we have to stop recording at some point.

Conversely, functions that do not repeat indefinitely – known as aperiodic functions – cannot be fully described using a Fourier series because they lack a nicely defined period. In other words, they have a definite beginning and end, whilst everything in between could lack any perceptible form of regularity. Therefore, to fully describe these functions in the frequency domain, aperiodic functions require the mathematical operation known as the Fourier transform to find the sum (or more precisely, the integral) of all its constituent sine and cosine waves.

Figure 2: (Left) Visual representation of waves adding together to form periodic functions; (Right) and their corresponding Fourier transforms.

In the next section, we will take at the use of the Fourier transform from a qualitative point of view, using 2D images as our functions and decompiling them into simple sine and cosine waves.

## Fourier transforms in 2D

On a computer monitor, we typically describe images as a grid of pixels, with each pixel on that grid assigned a value between 0 and 255 to signify its brightness (assuming we are only looking at monochromatic images). But we can also describe this image as a sum of waves, where each oscillates with maxima and minima somewhere between 0 and 255, depending on the image’s features.

Firstly, let’s say our image is a plot of pixel intensity along two orthogonal (i.e. perpendicular) axis, namely x and y. Each (x,y) position will have a corresponding value of pixel intensity at that spot on the image. Just like we saw in Figure 2 (right) for our 1-dimensional function, we can describe our image (which is also a function) as a set of waves that travel along the x- and y-axes.

We will plot the waves that extend across the x-axis in their frequency domain along a new axis called h, and those that process along the y-axis will be plotted in the frequency domain along the axis k. Each point along the h- and k-axes corresponds to a wave with frequency equal to its coordinates at that point. For example, a point in the (h,k) space at position (3,5) is a single wave that oscillates across the x-axis with a frequency of 3 and along the y-axis with a frequency of 5. The pixel intensity at a given (h,k) position in the frequency domain (again, given a value between 0 and 255) is the amplitude of the wave at that position and subsequently relates to the extent the wave’s peaks and troughs contribute to the image (Figure 3).

Figure 3: Diagram of waves, as shown along the h- and k-axes. (Top) A wave that varies in pixel intensity along the y-axis only (i.e. no change in x) is shown as two dots in the frequency domain (h,k). Each dot is at h=0 to show the wave did not oscillate across the x-axis. The wave is described by the dots at positions k = +1 and k = -1 as we see a wave with a frequency of 1 processing across the y-axis in both directions (it is impossible to know if this wave is going from positive y to negative y or vice versa, so we plot both!). (Middle) This wave oscillates across the x-axis with a frequency of 1 but does not change across the y-axis. Therefore, the frequency domain of this wave has two dots to describe this single wave (again, processing in both positive and negative x directions) with k = 0 and h values of +1 and -1. (Lower) Finally, a wave that moves across the x-y plane diagonally is shown in the frequency domain to the right. This time, the wave is oscillating in both x- and y-axes with a frequency of 1. As a result, the wave is shown as two dots with h= ±1 and k=±1.

An important thing to note is that waves with low frequencies will be responsible for describing the gross structures within an image, whilst higher frequency waves will contribute to fine details.

Take Figure 4 as a guide; it shows how an image of my brooding cat can be reconstructed using waves with progressively higher frequency waves along the h- and k-axes. At the start, low-frequency waves generate the body of the cat, before higher frequency waves reveal features such as the ears, nose, and eyes. Very high-frequency waves show details such as whiskers and even individual body hairs.

Figure 4: Reconstruction of an image (left) by adding higher and higher frequency waves together (right). Eventually, the image will be fully reconstructed once an infinite series of waves have been summated. This process is the inverse Fourier transform operation - where instead of finding the waves that make up a function, we’re adding them together to make the function.

Figure 5 (right) shows the Fourier transform of this moody furball with the incredibly high-frequency waves included. This effectively represents the image with no loss of information.

In practice, we will have forfeited some of the image’s details because they were encoded by even higher frequency waves that were not shown in our 2D representation of the frequency domain (Figure 5, right). To fully reconstruct the image, we would need to include all of the waves on the h- and k-axes, of which there are infinitely many...

Figure 5: (Left) 2D image represented as a mathematical function. Here, the pixel intensity is a function of its position along two spatial axes: x and y. (Right) The Fourier transform of the image to the left. Low-frequency waves are found in the centre of the plot.

### Pseudo-repeating functions

So far, we have shown how an image (or any other 2D function for that matter) can be deconstructed into a continuum of waves along two orthogonal frequency axes: h and k).

However, what if our original image comprised of elements that were repeated many times to form a pattern or lattice? This kind of function is referred to as pseudo-repeating and the most obvious example from chemistry is the lattice found in a crystal.

To form an understanding of a 2D crystal as a pseudo-repeating function, let us take a single dot that exists along the x- and y-axes (Figure 6, top). This dot is analogous to one molecule or ion that is yet to form a crystal. The Fourier transform of this single dot is shown on its right and demonstrates how a continuous series of waves along the h- and k-axes are needed just to reconstruct this dot.

Adding another dot to our image changes the pattern of waves needed to reconstruct this image, even though our original dot is unchanged – there are simply two of them.

Repeating this process causes our plot of (h, k) to appear fragmented as the number of dots increases, effectively creating patches of waves in the frequency domain that are needed to reconstruct the image, rather than the continuum required for a single dot.

Figure 6: (Left) 2D image as a function of x and y; (Right) The Fourier transform of each image.

## The applications of the Fourier transform

Several experimental techniques rely heavily on the mathematics behind the Fourier transform in order to extract meaningful information about the intricate structures of biological molecules, such as proteins. Have a look at Figure 6 to give you a better idea.

### Nuclear magnetic resonance

One such technique is biomolecular nuclear magnetic resonance (NMR). A sample of a pure biological molecule is placed inside a powerful electromagnet (aligning the spins of the molecules’ nuclei) and exposed to pulses of radio-frequency radiation. These pulses flip the spin of selected nuclei before they are allowed to return to their alignment with the electromagnet. During this process, the nuclei emit a sinusoidal-like wave of radio-frequency radiation that decays over time (Figure 7a, left).

The Fourier transform of this signal is determined and further interpretation generates the NMR spectrum of chemical shifts. These chemical shifts are then analysed to find the structure of relatively small proteins.

(On a side note, if you are interested to learn more about NMR, do check out our introductory article here).

### X-ray crystallography

Crystals of a single protein are regular, repeating lattices of molecules, analogous to our dot example in Figure 6.

Shining a bright beam of X-rays directly at a protein crystal causes the electrons in every molecule to oscillate and re-emit X-rays that interfere with each other. These X-rays are observed as a diffraction pattern on a detector. This diffraction pattern is equivalent to the Fourier transform of the crystal. And by rotating our sample, we can obtain an adequate set of Fourier terms needed to reconstruct our original crystal as a lattice of protein molecules in 3D (Fig. 7b).

However, we only record the amplitude and frequency of the waves that make up our crystal, so the phase information will be lost. It is good to know that there are several different ways to approximate each waves phase or even measure them experimentally, but the details are outside the scope of this article to explain. For an introduction to the subject, I encourage you to take a look at Professor Stephen Curry’s address to The Royal Institute.

### Cryo-electron microscopy

Arguably the most influential technique in the field of structural biology at the moment, electron cryo-microscopy (cryoEM) involves freezing a thin layer of water that suspends many individual biological molecules.

This layer is then exposed to high-energy electrons that strongly interact with these molecules and scatter to create the 2D Fourier transform of the molecule (much like our cat example). Electromagnets act as lenses to magnify, focus and perform the inverse Fourier transform (i.e. reconstructing the image from all of its constituent waves) of the molecule, with a detector capturing a low-resolution image of all the molecules that are suspended across the layer of frozen water (Fig. 7c).

Subsequent steps make use of algorithms to average the images of 2D molecules before using them to generate a 3D reconstruction of the final molecule. Overall, this process is computationally challenging and also implements the Fourier transform in order to complete the reconstruction of the molecule (Fig. 7c).

The details of this process are well outside the remit of this article and the subject has been covered well by Professor Grant Jensen’s fantastic YouTube series.

Figure 7: Applications of the Fourier transform in structural biology. For example, (a) NMR workflow from signal detection to finding a solution to the structure of the SARS-unique region of the bat coronavirus HKU9 (Hammond, Tan, and Johnson, 2017); (b) Outline of general X-ray crystallography experiment using the structure of the SARS-COV-2 spike receptor-binding domain (red) in a complex with a neutralising antibody (Wu , et al., 2020); (c) Outline of cryoEM experiment using the structure of SARS-COV-2 RNA polymerase as an example (Hillen, et al., 2020).

In conclusion, the Fourier transform is a way of taking any function and decompiling it into a series of waves, each with a different amplitude, frequency and phase. Integrating over a series of these waves will give us the original function.

In the next article, we will take a look at how the Fourier transform is described mathematically to give you an understanding of how experimental data can be used to find the structures of molecules.

## Author

Joseph I. J. Ellaway

BSc Biochemistry with a Year in Research

Imperial College London