4

I'm familiar with the slogan "the Fourier transform decomposes signals into frequencies", but I've still somehow managed to get by without really understanding what the Fourier transform is or why it is useful.

My understanding is that it is a sort of change of basis on the space of functions. Is this accurate? What exactly is the basis we are changing to, and which basis are we changing from?

I'm also familiar with the characteristic function from probability theory, or the Fourier transform of the pdf of a random variable, and the fact that it determines the distribution of a random variable. Other than making a couple proofs a bit nicer, what exactly is the significance of the characteristic function?

Powerspawn
  • 564
  • 1
    By Parseval's theorem the Fourier transform is an unitary linear operator $L^2 \to L^2$. It is diagonal in the Hermite functions (orthogonal) basis – reuns Dec 26 '16 at 05:57
  • 1
    I think it was originally intended to get a sense of the harmony in signals by extracting their frequencies, and then the theory was extended/expanded to become in the form of what is known as general Fourier transform theory. $$\quad$$ P.S. You may also find this interesting: http://math.stackexchange.com/questions/1987998 – polfosol Dec 26 '16 at 07:07
  • @user1952009: Your comment struck me as funny. – marty cohen Dec 26 '16 at 07:09
  • 1
    @martycohen ?? ${}{}$ – reuns Dec 26 '16 at 07:15
  • 1
    @polfosol The Fourier transform is the unitary linear operator $L^2 \to L^2$ diagonalizing the derivative (more generally the convolution) operators – reuns Dec 26 '16 at 07:16
  • 1
    If you can transform a signal into its frequencies you can filter out unwanted frequencies and the inverse transform will give you a clean signal. This is used a lot in signal processing to filter out high and/or low frequency noise. – John Douma Jul 31 '21 at 20:58

1 Answers1

0

Consider a function $h$. This function can be made by adding functions of the form $\exp(i \omega)$ together. You might ask, “How much of each $\omega$ do I need?” I could tell you this answer by giving you a complex number for each $\omega$; that is, I could give you a new function of $\omega$. That new function is the Fourier transform of $h$ and the different $\omega$s are called frequencies.

NicNic8
  • 6,951