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