Fourier transform of finite aperiodic signals

Question

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

Answer 1

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.