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Is it possible for a signal to have finite length in both the time domain and the frequency domain? Or does the finite length of one necessarily imply that the other has infinite length?

(By "finite length" I mean that the signal is zero everywhere except over a finite range.)

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The discrete Fourier transform is a mapping from $\mathbb C^N$ to $\mathbb C^N$. What signal on $N$ samples would be of infinite length? –  Rahul Jun 1 '12 at 12:28
    
Quite right. I guess I didn't have my brain in gear... –  MathematicalOrchid Jun 1 '12 at 12:34

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up vote 3 down vote accepted

The Paley-Wiener theorem implies that a function and its Fourier transform cannot both have compact support. Terence Tao has a blog post that proves this fact without the use of complex analysis.

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I'm not sure whether this is what you are aiming at, but there is a well known theorem which says that if $f\neq 0$ is a function with compact support on the real line, then the Fourier transform of $f$ cannot be zero on an interval and vice versa. See e.g. Theorem 2.6 in Stéphane Mallats 'a wavelet tour of signal processing'

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