Fourier transforms for the practical person
In this series of posts, I plan to outline some basic ideas which I’ve learnt on the theory of Fourier transforms, and it’s practical applications in a non-rigorous manner. Once I’ve laid out the basics, I’ll then show you some interesting stuff from what I’m currently working on.
Let’s start with Fourier series. It’s actually a remarkable fact, that we can express any arbitrary periodic function, simply as the sum of the ordinary sine and cosine functions we’ve all studied at high school. If is a periodic function, then we have
The Fourier transform is an extension of this, as the period of the function approaches infinity, and the gap between successive harmonics approaches 0. So, in some sense, the fourier transform decomposes a function into it’s frequency components.
For a non-periodic function which satisfies certain conditions, there are many conventions of describing the fourier transform. Following one such convention which is widely used, the forward fourier transform is
While, the inverse fourier transform is
Notice that, if is a continuous time signal, then it’s transformed into the frequency domain by the forward transform. One of the properties of the fourier transforms is that, and are transforms of each other, and form a fourier pair, and are represented by .
This means that, if , then
if isn’t discontinuous. If it is discontinuous, then the value at that point will be the average of the value around the discontinuity. So, we can simplify our terminology and say that the fourier transform of is and vice versa.
In the next post, I’ll look at the fourier transform of some useful functions, but before that, there’s one more nice result. For an Electromagnetic wave, or a signal in a wire, the fourier transform of the voltage can be complex. However the conjugate product , is real and is proportional to the power density (or, power per unit frequency). This is know as the spectral power density.