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I'd like to know which conditions must be satisfied in order to get a signal with a finite bandwidth.

Thank you for your time.

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  • $\begingroup$ In order to get a signal with a finite bandwidth, the bandwidth of the signal must be limited so that it is not infinite. Is that what you want to know ? $\endgroup$ – AlexTP Sep 1 '17 at 8:28
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    $\begingroup$ in the time domain $\endgroup$ – Gennaro Arguzzi Sep 1 '17 at 9:10
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    $\begingroup$ Finite support in one domain (frequency) requires infinite support in the other (time). $\endgroup$ – hotpaw2 Sep 1 '17 at 17:37
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    $\begingroup$ No. Necessary, but not sufficient. $\endgroup$ – hotpaw2 Sep 2 '17 at 5:48
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    $\begingroup$ Any Gaussian distribution/function? $\endgroup$ – hotpaw2 Sep 2 '17 at 7:04
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Let us stick to traditional functions, considering the almost everywhere equivalence class. From an actual signal perspective, I am leaving distributions aside.

According to the Paley-Wiener theorem, functions with compact frequency support (corresponding to Bernstein spaces) can be extended from the real line to the complex plane $\mathbb{C}$ as entire functions of exponential type; entire functions are holomorphic at all finite points of $\mathbb{C}$ , exponential type means that $|f(z)|\le C e^{a|z|}$. They can be called $B$-functions.

So they inherit from all their properties: no bounded support in time, infinite differentiability, Taylor expansion everywhere, controlled decay at infinity...

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