# How are Fourier Transform and Fourier Series used in practicality?

First post here.

So, I've always been curious as to how mathematics can be applied in the real world, solving difficult issues. And right now, I'm trying to understand how Fourier Transform (FT) and Fourier Series (FS) are being used. The problem is, information is rather scarce in this area of math.

FT is quite easy to understand as to how it's used. Take for example noise cancellation; using FT to convert time-domain to frequency-domain to get the frequencies we want to block and create an "opposite" sound wave. In this example, is FS used to create the opposite sound wave or is inverse FT used? Or maybe something else happens here?

Help is much appreciated!

Fourier transform: something to note is that sinusoids are eigenfunctions of convolution operators. Convolution can be used to represent blur in imaging systems (e.g. xray detectors). So representing signals in the frequency domain is particularly convenient for quantifying detector performance: blur is represented by merely attenuating certain spatial frequencies.

Fourier series: can be used to represent musical notes of different timbres. Square waves, saw tooth, etc. This is used in additive synthesis of synthesizer instruments for example.

P.S. you don't need any transforms to noise cancel - you simply invert the amplitude of the sound wave so it's negative :P

• Hey! Excellent answer! Do you think you could expand on the usage of Fourier Series? I feel like I still don’t have the full picture… Do you know of any article that could explain it all? Commented Apr 7, 2022 at 10:01
• Sure. Well, personally, I understand things like so: There is only one Fourier decomposition, the Fourier transform (FT). The FT takes you from a signal which varies in either time (or space), and gives you an alternate representation in terms of sinusoidal oscillations per unit time (or space). Fourier series (FS) may be derived as a special case of the FT, in which your signal only contains a finite number of frequencies, i.e. it is a finite sum of sinusoids, i.e. it is periodic. The FT is more general, and also can be used to decompose aperiodic functions. Has this helped?
– SSD
Commented Apr 7, 2022 at 13:37
• Not exactly. I was wondering more about the part where you said how FS can be used to represent musical notes. Do you know exactly how they work or maybe a source that explains that? Commented Apr 9, 2022 at 0:06