Fourier Transforms

Up a level : Fourier Transforms
Previous page : Sine and cosine transforms
Next page : Fast Fourier transform

Say we want to combine the discrete sine and cosine transforms. One way to do that would be to do both, but let one of the results be an imaginary number, like this

$\displaystyle {{X}_{k}}=\sum\limits_{{n=0}}^{{N-1}}{{{{x}_{n}}\cdot \left( {\cos \left( {2\pi \frac{k}{N}n} \right)+i\sin \left( {2\pi \frac{k}{N}n} \right)} \right)}}$

But wait a minute… here we can recognise Euler´s formula

${e^{i\theta }} = \cos \theta + i\sin \theta$

This means that we can write our combined formula as

${{X}_{k}}=\sum\limits_{{n=0}}^{{N-1}}{{{{x}_{n}}\cdot {{e}^{{-2\pi \frac{k}{N}n}}}}}$

We have added a minus sign in the exponent, but that is just a convention. It will, in the end, just change the sign of the sine part. There are other variants of the above too, where we multiply the result by a scale factor, but for now we will use the above variant.

We call the above transformation the Discrete Fourier Transform, DFT.

It is mostly calculated in a very efficient way called the Fast Fourier Transform. We will use that in the next page.

The (continuous) Fourier Transform can be made as in the sine and cosine transforms. We have

$\displaystyle \hat{f}(k)=\int\limits_{a}^{b}{{f(t){{e}^{{-2\pi kt}}}dt}}$

For a non-periodic function, we replace the bounds with minus infinity to plus infinity. For a periodic function we can, for example, put 0 to T or –T/2 to T/2 as our lower and upper bound.

Up a level : Fourier Transforms
Previous page : Sine and cosine transforms
Next page : Fast Fourier transform

Last modified: Feb 19, 2025 @ 16:32