Advanced Math: Fourier Transforms


One of the most useful mathematical techniques ever invented is called the Fourier Transform. To motivate this, imagine that you are listening to a concert, and that you record the intensity of the sound u(t) of the orchestra as a function of the time t. The orchestra is composed of instruments that all make sounds of a frequency \omega with each such sound having an intensity U(\omega). A sound of frequency \omega has the mathematical expression

e^{i \omega t}

where i  is the square-root of -1. The total sound that we hear is the combination of all of these sounds and it can be expressed as an integral in the form

u(t) = \frac{1}{2\pi}\int^\infty_{-\infty}U(\omega)e^{i \omega t}\ d\omega


This integral which links the two functions U(\omega) and u(t), is called the Inverse Fourier Transform after the French mathematician Fourier. Remarkably, it is easy to find the inverse to this process. This is called the Fourier Transform and it is given by

U(\omega) = \int^\infty_{-\infty} u(t) e^{-i \omega t}\ dt

The Fourier Transform has countless applications ranging from telecommunications to crystallography, from speech recognition to astronomy, from Radar to mobile phones, and from meteorology to archaeology. It even has important applications in cryptography. The whole signal processing industry (indeed much of the digital revolution) owes its existence to the Fourier Transform. The reason is that it links the intensity of a function to the waves that make it up. As so much of what we do involves sound or light waves, the applications are universal. It is so important that calculating the Fourier Transform was one of the first tasks given to the early computers in use in the 1960s.

However, these implementations had a lot of difficulties in calculating the Fourier Transform and were so slow that they were absorbing a huge amount of computing time. Roughly speaking if you wanted to find the values of the Fourier Transform at N points then you had to do N^2 calculations. Unfortunately, for accuracy you have to take large values of N which means that N^2 is very large, too large to make it possible to calculate easily. However, in 1965 there was a remarkable breakthrough when a technique called the Fast Fourier Transform or FFT was invented to evaluate the Fourier Transform. This was much faster, taking a time proportional to N \mbox{ log}{(N)} which is a lot smaller than N^2.  With the FFT available to calculate the Fourier Transform quickly, its applications became almost unlimited.