Science, maths and computers.
If you study physics or engineering, chances are that looking at the above formula gives you painful flashbacks of either trying to prove in an exam that it’s its own inverse, or using it to program a Wiener filter (or in my case, both). But while the Fourier Transform is on the one hand an incredibly useful tool in data/image/audio analysis, it also provides some beautiful explanations for why the world is as it is.
Dubstep (wub wub wub)
Whether or not cooking with Skrillex is your thing, pretty much all music is made to sound better by the Fourier Transform. For the uninitiated, it transforms a function of time into a function of frequency. Think of a graphical equaliser - that is the Fourier Transform of a song as you listen to it in real time. It takes the waveform (which is a function of amplitude against time), and decomposes it into its constituent frequencies. It shows you how much bass, middle and treble there is at any given time, and this lets music producers have their way with your ears by altering the amplitude at different frequencies, or in other words, filtering it.
Dr. Dre is a big fan of the work of Joseph Fourier
If you’ve ever used photoshop then you might have used a Gaussian blur. What it’s mathematically doing is treating your image as a 2D grid of numbers, and convoluting it with a 2D Gaussian function. A single bright spot will spread out into a dim blur like this:
Now, if you have an image that has been blurred, knowing - or guessing - the function it’s been blurred by (doesn’t have to be digital: a camera lens is an analogue convoluting function) allows you to deconvolute the image, not too dissimilar to the zoom-enhance shtick you see in many sci-fi’s. This deconvolution can only really be performed in practice through the use of the Fourier Transform (in particular, by utilising the convolution theorem).
In fact, the Fourier Transform is used ubiquitously in data analysis, signal processing, and image/video/audio enhancement due to its ability to work magic on the crappy file you have and make it better and clearer. In fact I’m pretty sure it’s actual witchcraft.
Heisenberg’s uncertainty principle (wub wub wub?)
While the Fourier Transform is useful for many practical applications, it has a very profound impact on the nature of reality itself. This is all to do with the relationship it draws between its two variables.
Above, you can see it equates a function of frequency, ω, with a function of time, t. These variables are known as Fourier conjugates, and they’re flip sides of each other. You can’t just choose any variable you like to be on the left hand side; a transform of a function of time is always a function of frequency. However there are other pairs of variables you can use.
One such pair of Fourier conjugates are the vectors space, r, and wavenumber, k - the wavenumber is the reciprocal of wavelength, and turns out to be a more natural measure of the spatial variation of a wave.
One of the weirdest results in quantum physics is the Heisenberg Uncertainty Principle, which states that for a given particle (or more formally, a quantum system), certain pairs of variables are unknowable with arbitrary precision. For instance, a particle cannot have both an exact position and an exact momentum. The more precisely its position is defined, the more uncertain its momentum is and vice versa. It’s not a measurement problem but an inherent property of nature itself. Why?
Back to the Fourier Transform… It turns out that the transform of a sharp thin spike, is a spread out ‘blurry’ hill like the Gaussian above. Therefore, a function representing a very well defined position (or time), such as a sharp thin spike, would correspond to a delocalised, spread out function representing wavenumber (or frequency). In quantum mechanics, momentum and wavenumber are essentially the same (multiplied by a constant), and therefore Heisenberg’s Principle must necessarily hold true!
In fact, there also exists a Heisenberg Principle for time and frequency. In quantum mechanics, frequency is interchangeable with energy (again, multiplied by the same constant) and therefore the energy of a particle is uncertain over arbitrarily small time-frames. This allows particles in the quantum regime to ‘borrow’ enough energy to tunnel through a potential barrier so long as they pay it back in a small enough timeframe to be in keeping with Heisenberg.
Joseph Fourier is a big fan of the work of Dr. Dre