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What is the graphic meaning of the transform's area?

$$\int_{-\infty}^{+\infty}{X(f)df}$$

Where $X(f)$ is the continuous Fourier transform of the signal $x(t)$.

Thank you very much.

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There is an implicit $e^{\jmath 2 \pi f t }$ with $t=0$ in your integral which means it equals $x(0)$.

If $X(f)$ were a probability density instead of a Fourier Transform, it would have an interpretation in the context of the axioms of probability. When you say $graphic$ meaning , the context is somewhat vague. In calculus, the general meaning of an integral without using terms like measure, is the area under the curve, allowing for negative areas for negative values of the function.

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    $\begingroup$ i think what Stanley might be trying to say is that, with the Fourier Transform and inverse defined as $$X(f) = \int_{-\infty}^{+\infty} x(t) e^{-j2\pi ft} dt \\ x(t) = \int_{-\infty}^{+\infty} X(f) e^{+j2\pi ft} df $$ in the inverse, if one sets $t=0$ you get simply $$\begin{align} x(0) &= \int_{-\infty}^{+\infty} X(f) e^{+j2\pi f0} df \\ &= \int_{-\infty}^{+\infty} X(f) \, 1 \, df \end{align}$$ $\endgroup$ – robert bristow-johnson Jun 24 '17 at 4:13
  • $\begingroup$ There are some different conventions for the normalizing factor $1/ 2 \pi$, one is to include it in the inverse transform $\endgroup$ – user28715 Jun 24 '17 at 4:42
  • $\begingroup$ Hi @Stanley Pawlukiewicz, my goal is to understant what kind of area is it on the complex plane. Is an area on complex plane? Is a volume on the complex plane+time axis? $\endgroup$ – Gennaro Arguzzi Jun 24 '17 at 4:54
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    $\begingroup$ notational convention is important, in my opinion, to limit confusion. i wouldn't wanna call "$f$" angular frequency. $\endgroup$ – robert bristow-johnson Jun 24 '17 at 6:26
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    $\begingroup$ How old are you in Martian years. $\endgroup$ – user28715 Jun 24 '17 at 6:39

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