if fft is a discrete fourier transform

if the fft (fast fourier transform) is a discrete fourier transform, so Why the spectrum is not repetitive? https://en.wikipedia.org/wiki/Discrete_Fourier_transform

[Massimo Zanetti]

No one expects the spectrum is "repetitive" (I assume that by "repetitive" you mean "periodic").

Take a very simple example. The signal you want to transform is a sinusoid, then its transform is just an impulse function.

Answer 1

[Star Strider]

The fast Fourier transform operates on finite sequences with specific sampling intervals (frequencies). Unlike the analytical Fourier transform that extends from -Inf to +Inf in the frequency domain, the discrete Fourier transform extends only as far as the Nyquist frequency, the highest uniquely resolvable frequency in a sampled signal (and the frequency above which ‘aliasing’ occurs if the analog-to-digital converter does not have a hardware anti-aliasing filter on its input to prevent higher frequencies from being sampled). See signal processing textbooks by Proakis and others (or any number of books devoted to the Fourier transform and its applications) for details.

Answer 2

[Julian Oviedo]

Im studing with "Introduction To Digital Signal Processing And Filter Design (2005) - Shenoi - Wiley " as well I'm using matlab to make a better study.

************************, the discrete Fourier transform extends only as far as the Nyquist frequency, ****************************

So, that the reason why there is not periodic form on the fft.

But, If we take care that if a generate a 100 sample vector, it's not periodic so the spectum must be infinity.

[Star Strider]

The two-sided discrete Fourier transform exists only in the region of ±Nyquist frequency, not infinity. In a sampled signal, you know only what is in the signal you have. What may exist in the signal if you sampled it longer is completely unknown to you. Nothing in or derived from a sampled signal is, or ever will be, infinite.

I don’t have Introduction to Digital Signal Processing and Filter Design in my library, but by the description, it should cover this.