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Periodicity of the Fourier transform

If X[n] is, as above, a signal that repeats every $N$ samples, the Fourier transform of X[n] also repeats itself every $N$ units of frequency, that is,

{\cal FT}\left \{ X[n] \right \} (k+N) =
{\cal FT}\left \{ X[n] \right \} (k)

for all real values of $k$. This follows immediately from the definition of the Fourier transform, since the factor

V = \cos(-k\omega) + i\sin(-k\omega)

is unchanged when we add $N$ (or any multiple of $N$) to $k$.

Miller Puckette 2006-12-30