next
up
previous
contents
index
Next: Fourier transform as additive
Up: Fourier analysis of periodic
Previous: Fourier analysis of periodic
Contents
Index
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,
\begin{displaymath} {\cal FT}\left \{ X[n] \right \} (k+N) = {\cal FT}\left \{ X[n] \right \} (k) \end{displaymath}
for all real values of $k$. This follows immediately from the definition
of the Fourier transform, since the factor
\begin{displaymath} V = \cos(-k\omega) + i\sin(-k\omega) \end{displaymath}
is unchanged when we add $N$ (or any multiple of $N$) to $k$.
Miller Puckette
2006年12月30日