Time representation and frequency representation The time representation is usually the first (and the most natural) description of a signal we consider, since almost all physical signals are obtained by receivers recording variations with time. The frequency representation, obtained by the Fourier transform $$X(\nu) = \int_{-\infty}^{+\infty} x(t)\ e^{-j2\pi \nu t}\ dt,$$ is also a very powerful way to describe a signal, mainly because the relevance of the concept of frequency is shared by many domains (physics, astronomy, economics, biology ...) in which periodic events occur. But if we look more carefully at the spectrum $X(\nu)$, it can be viewed as the coefficient function obtained by expanding the signal $x(t)$ into the family of infinite waves, $\exp\{j2\pi \nu t\}$, which are completely unlocalized in time. Thus, the spectrum essentially tells us which frequencies are contained in the signal, as well as their corresponding amplitudes and phases, but does not tell us at which times these frequencies occur.
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