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I'm new to Fourier Transform. Could anyone explain to me in the Short-time Fourier Transform Equation (wikipedia):

$$STFT\{x[n]\}(m,\omega) \equiv X(m,\omega) = \sum_{n = -\infty}^\infty x[n]w[n-m]e^{-j\omega n}$$

  1. What exactly is $n$? I'm confused because $n$ appears in both $x[n]$ and $\sum^\infty_{n = -\infty}$
  2. Why does it ranges from ${-\infty}$ to $\infty$? Should it not be from $0$ to some maximum $t$ since the signal duration is discrete?
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  1. $n$ is the (discrete) time variable. Its appearance on the left is not significant: the author just uses $STFT\{x[n]\}$ as notation for "STFT of the sequence $\{x[n]\}_{n=-\infty}^{\infty}$. It's just a dummy index variable there. You may want to write something like $\mathbf x$ instead to indicate this sequence.

  2. The time is understood as infinite in both directions, which of course is an abstraction, but a useful one since it makes the domain a group (which is necessary for Fourier transform). So, the time domain is $\mathbb{Z}$ here. If this were the ordinary FT, the formula would be $\sum_{n = -\infty}^\infty x[n]e^{-j\omega n}$; the window function $w$ effectively restricts the summation to some finite range of indices $n$.

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