In this article, we will follow on from our derivation of the Fourier series – a way to describe any periodic (repeating) function as a sum of cosine and sine waves.
Here, we will re-image the Fourier series in order to apply it not only to functions that repeat themselves over a defined period, but also to functions that appear seemingly irregular.
This part of the article series will be most important for experimental applications that use the operation because many of the functions we wish to analyse either fail to adopt a definite period or experimental limitations mean it simply isn’t possible to record all the data we need to describe a function’s full period.
From Fourier series to transform
Let us briefly revise our definitions for the function f(t) as the infinite sum of exponential waves and our derivation of their corresponding amplitudes (Fourier coefficients):
We have so far been looking at periodic functions that repeat themselves every T periods. However, this is typically not the case in the natural world, where the functions for which we wish to find constituent waves are non-repeating, or in other words, aperiodic.
This statement can be represented mathematically by thinking about what an aperiodic function truly is: if a periodic function repeats every T periods, then an aperiodic function will have a period that tends to infinity. i.e. a non-repeating function has an infinitely large period.
We can, therefore, state that the period for an aperiodic function, T→∞. With this in mind, let’s rearrange our equation for cn, which we derived in Part 2:
We have already stated that ω0=2π/T in our previous discussion of the Fourier series and now equipped with our notion that T→∞ for aperiodic functions, we see that ω0 becomes vanishingly small as the period of our function is increased.
n·ω0 effectively becomes a continuous variable under these conditions, allowing it to take any real value between +∞ and -∞ (consider n=±∞).
As a result, we will define one new variable and one new function that will allow us to describe any aperiodic function. These are:
By substituting these two equations into one equation, we obtain our formula for the Fourier transform for any aperiodic function:
This equation tells us the properties (amplitude, frequency and phase) of each wave required to construct the aperiodic function f(t), which is a form of analysis. But what about going in the reverse direction? In other words, can we take a bunch of waves (with known amplitude, frequency and phase) and combining them together to make a previously unknown function - a form of synthesis?
Well, yes! This is called the inverse Fourier transform and can be derived starting from our initial description of f(t) as a sum of waves:
Now, let’s multiply the right-hand side (RHS) by both T and 1/T:
Recall that F(ω)=T·cn and ω=n·ω0 as T→∞. Furthermore, we know that 1/T=ω0/2π and so as the period (T) gets infinitely large, our value of ω0 becomes infinitely small. We can, therefore, infer from fundamental calculus that under these conditions:
In conclusion, we can substitute these new variables into the equation to find:
which can be written in integral form as:
Therefore, we denote this equation as the inverse Fourier transform.
Fourier transforms of even, odd, real and imaginary functions
Based on what we initially know about a given function g(t), we can start to simplify the Fourier transform of g(t) and eventually be able to plot the transform. Before we begin to simplify G(ω) (the Fourier transform of g(t)) let’s describe g(t) as a sum of its real and imaginary parts:
Now, let’s find a general expression for the Fourier transform of g(t) by using our complex notation from above:
For the sake of simplicity, let’s consider g(t) to be either even or odd and exist in either the real or imaginary planes. This gives us four possible permutations for g(t): even and real, odd and real, even and imaginary, or odd and imaginary.
Let’s remind ourselves that even functions require only cosine terms to be fully described and odd functions require only sine terms. Therefore, when g(t) is even and real:
its Fourier transform is similarly real and even.
When g(t) is odd and real:
its Fourier transform is odd and imaginary.
When g(t) is even and imaginary:
its Fourier transform is even and imaginary.
And finally, when g(t) is odd and imaginary:
its Fourier transform is odd and real.
Next, we will finish our mathematical explanation of the Fourier transform by looking at functions that do not fit nicely into one of these four categories described above. In other words, we will look at the Fourier transform of complex functions.
Fourier transforms of complex functions
Consider the complex wave g(x). We can represent g(x) as a sum of its real and imaginary parts:
where we will denote the real and imaginary parts of g(x) as:
Knowing that the real and imaginary parts of a complex wave correspond to the cosine and sine parts of Euler’s formula respectively, we can generalise g(x) to:
where A is the amplitude, ω0=2π/T (T is the period) and θ denotes the phase shift.
In order to see how the complex function g(x) relates to its Fourier transform, we will look at a simplified form of the radio frequency signal detected from a structural nuclear magnetic resonance (NMR) experiment.
Decaying wave: worked example
Take our complex, decaying wave q(x) to be:
where b=1, A=1, ω0=5π/6 and θ=0:
Figure 1 (left) shows the plots of the real and imaginary parts of q(x) for x≥0.
In practice, we can only detect the real part of the wave and the imaginary component is essentially an abstract concept in the context of experimental NMR. Nevertheless, let’s now continue and find the Fourier transform of q(x), before plotting its real and imaginary parts.
Plugging our known values for q(x) into our Fourier transform equation gives us:
for x≥0. Let’s complete the integral to get:
We must now rearrange the equation to get Q(ω) as its sum of real and imaginary parts:
where the real and imaginary parts can be separated,
Figure 1: Real and imaginary parts of the complex function q(x) and its Fourier transform, Q(ω).
Have a go at deriving the real and imaginary parts of the Fourier transform of q(x) yourself. But this time, use b=b, A=A, ω0= ω0 and θ=θ to find a general equation for each part of the transform.
Throughout this three-part series of articles, we have looked at the Fourier transform from a conceptual point of view, aiming to give you an intuitive understanding of the interconversion between space and frequency domains via this mathematical operation. We then derived the Fourier series, an elegant way to describe a wave-like function that repeats into infinity using only cosine and sine terms. To end this topic, the derivation of the Fourier transform was presented to show how any function can be represented using a continuous wave variable in the frequency domain.
This topic can be incredibly confusing and difficult to conceptualise when learning it for the first time. It might take you several re-reads to fully grasp the mathematics behind the operation. In reality, programs have been written to allow you to perform the Fourier transform on any data you might want to analyse and so the detailed steps outlined here will not be necessary outside of a university course.
Although not strictly relevant to structural biology, 3Blue1Brown has two excellent YouTube videos that provide a beautifully elegant way of explaining the mathematics behind the Fourier series and transform. I highly recommend giving them a watch.
Author: Joseph I. J. Ellaway, BSc Biochemistry with a Year in Research