0

I know the fourier transform of most of the signals,but how about the fourier transform of aperiodic finite signals?

kikki
  • 1
  • Finite is a keyword here. You take the full length of the signal as a single period and assume it repeats indefinitely. That will allow you to see the signal's frequency spectrum. – kbau Aug 26 '15 at 11:07
  • Can you give me some other clues? This isnt a homework questio by the way – kikki Aug 26 '15 at 11:11
  • I would need to know better how well you understand the Fourier transform in general. Do you see how it represents the spectrum of the signal? Do you see how any function can be represented as a Fourier series if it isn't periodic? – kbau Aug 26 '15 at 11:14

1 Answers1

2

Fourier series is not the same as Fourier transform. Fourier transform never requires periodicity: continuous distribution of frequencies ensures that the signal never repeats. In fact, Fourier transform is mostly used for nonperiodic signals. They do have to be $L^2$ integrable, which includes all finite signals. If you slacken the rules of convergence, you can handle distributions such as the delta function and non-integrable functions such as sign(x).

Fourier series only uses multiples of a base frequency, so it always produces periodic signals. If you take a finite signal and compute the fourier series, you will not get the same as what Fourier transform would give you. But it's close. Fourier transform of a periodically extended finite signal would be a set of spikes (delta functions) at the same frequencies that the Fourier series gives you. Taking only one period (and the rest is silence), is the same as multiplying with a square window. Multiplication in real space is the same as convolution in fourier space, so Fourier transform of a finite signal can be computed as a sum sinc ($\sin x /x$) functions, centered at the Fourier series frequencies, with amplitudes defined by the series coefficients.

orion
  • 15,781