Why Fourier Transform is so important?
Fourier transforms is an extremely powerful mathematical tool that allows you to view your signals in a different domain, inside which several difficult problems become very simple to analyze.
At a high level, the Fourier Transform will allow us to convert our data to a different form where we can easily pick and choose the most important components and remove all of the noise.
History
Joseph Fourier discovered that given some periodic functions s(x) and an interval P, we can say
this means that an arbitrary function on a given interval can be approximated as a sum of appropriately weighted sinusoidal functions with increasingly high frequency. The main idea here is we are able to express the same function s(x) in a different system of coordinates.
Now the Fourier Transform can be viewed as an extension of the above Fourier series to non-periodic functions.
Where omega(ω) is frequency. The above function is for continuous functions, for discrete functions we use Discrete Fourier Transform.
Applications
The Fourier transform gives you answer questions about a sound that is difficult or impossible to answer with other methods. It makes hard problems easy.
A recording contains a set of three musical notes. What are the notes? If you leave the recording as a set of amplitudes over time, this is not an easy problem. If you convert the recording to a set of frequencies over time, it’s really easy.
I want to change the pitch of a recording without changing its duration. How do I do this? It’s possible, but not easy to do, by just manipulating the amplitude of an input signal. But it’s easy if you know the frequencies that comprise the signal.
Does this recording contain speech or does it contain music? Super hard to do using only amplitude-based methods. But there are good solutions that guess the right answer nearly all of the time based on the Fourier transform and its family.
Almost every question you’d like to ask about a digital sound is made easier by transforming the recording using a discrete version of the Fourier transform.
In practice, every modern digital audio processing device relies heavily on functions very similar to the Fourier transform.
In Image processing, the Fourier Transform tells you what is happening in the image in terms of the frequencies of those sinusoidal. For example, eliminating high frequencies blurs the image. Eliminating low frequencies gives you edges. And enhancing high frequencies while keeping the low frequencies sharpens the image.
Fast Fourier Transform is used extensively in image processing and computer vision. For example, convolution, a fundamental image processing operation, can be done much faster by using the Fast Fourier Transform. The Wiener filter, used for de-blur image, is defined in terms of the Fourier transform. But more importantly, even when the Fourier transform is not used directly, it provides a very useful framework for reasoning about the image processing operations.
