Following on from the post that I made a couple of weeks ago introducing Fourier Transforms – this post will go into a little bit more detail and I will be looking at a simple example of a particular transform to illustrate what is happening. I will try to keep things as simple as possible – there are certain regularity conditions that would need to be applied in the general case, but here I will be choosing functions that satisfy these conditions anyway.

For a function $f(x)$ its Fourier Transform is defined to be $$\hat{f}(k)=\int_{-\infty}^{\infty}\!{f(x)e^{-2 \pi ikx}} \;\mathrm{d}x$$

The function $f(x)$ is in the time-domain with $x$ representing time. The transform $\hat{f}(k)$ is in the frequency-domain with $k$ representing the frequency.

I decided to make a video to visually present Fourier Transforms. They are something that I have always personally had difficulty visualising and making this video has helped me to understand them even better and I hope that it will help others likewise. Fourier Transforms are generally complex-valued functions and in some cases can be very difficult to find as an explicit formula. However, in the video below, I have chosen the function $\mathbf{cos}(2\pi(2x))\mathbf{e}^{-x^2}$ to work with which has a real-valued Fourier Transform that can be explicitly stated; I will typing-up the derivation of the Fourier Transform over the coming weeks.

I would love to hear any comments about how the video could be improved and any feedback on the video.

I have been taking quite an interest in Fourier Transforms lately. Fourier Analysis was one of the courses that I did during my final year at Warwick – I didn’t know what to expect at the time since I had heard about Fourier Series and Fourier Transforms but had never really worked with them or studied any of the theory. The course, like many of the other maths courses at Warwick, was very fast-paced and extremely demanding. I spent most of my time trying to keep up with all of the definitions, theorems and proofs not just from this course but from all of the other courses that I was doing at the time. I more or less just had to accept that there existed these things called Fourier Transforms but I never really got round to settling down to try and visualise them to really understand what they were (after all, there are only 24 hours in a day).

Well, a few years later and I have managed, to some degree, to do just that – better late than never; but I realise now how exciting and interesting Fourier Transforms are (no, really).

Fourier Transforms are a way of breaking a function down. The function can (and often does) represent a sound wave and the Fourier Transform enables you to analyse the frequencies that are involved – the transform tells you which frequencies are present and how much a particular frequency contributes to the overall sound wave being analysed.

From the transform it is then possible to see which frequencies are dominant in the sound wave and what gives it its characteristic feel. For example, a sound wave produced by a guitar will be very different from the sound wave produced by a saxophone – even though they may be playing the same note, neither one is playing a completely pure note; each will have various other frequencies present and in different amounts which all interact with each other to produce the sound that you recognise as either a guitar or a saxophone. This explains why it is possible to distinguish the sound of one instrument from the sound of a different instrument just by listening to them – as your ear receives the various frequencies all interacting with each other your brain is then able to recognise the characteristic pattern of the wave and determine whether you are listening to a violin or a piano.

Fourier transforms take the original function (which is in the time-domain) and converts it to a function in the frequency-domain.

Here is an example

Rectangle Function

Rectangle Function

Sinc Function

Sinc Function

The picture on the left is the rectangle wave (very important in digital signal processing) and the picture on the right is its Fourier Transform, the sinc function defined by $$f(x)=\dfrac{\mathrm{sin}(\pi x)}{\pi x}$$

The sinc function tells us that the dominant frequencies in the rectangle wave are between 0 and 1 and as the frequency gets higher then its contribution generally decreases. This is just the beginning – Fourier theory finds its way into many different and unexpected areas of science from pure mathematics, to signal processing, astronomy and music technology.

I have been playing around with Fourier Transforms on SAGE Math over the last couple of days so I will be posting a bit more about them over the coming weeks and going into a bit more detail with them.