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In the book Modern Digital and Analog Communication Systems by B.P. Lathi and Z. Ding, on page 21, the authors wrote: "A necessary condition for the energy to be finite is that the signal amplitude $\to 0$ as $|t| \to \infty$."

The energy of signal $g(t)$ is defined as $$ E_g = \int_{-\infty}^{\infty} |g(t)|^2 dt. $$

However, there exist positive continuous functions that do not tend to $0$ but the integral is finite.

Am I missing something, or is the statement wrong?


Edit: After thinking a while about this, I think the phrase could be corrected by "A necessary condition for the energy to be finite is that the essential limit superior of the signal amplitude $\to 0$ as $|t| \to \infty$."

This includes the case for which the limit does not exist but the integral still converges. The essential supremum captures what really matters in the tail of the function.

Perhaps that is what Lathi thought when writing the sentence, and, of course, it is not sensible to use this terminology in an undergraduate book.

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  • $\begingroup$ The integral of the "function" mentioned in your link may converge over infinity. But there is no discussion about the "integral" of square value of the function to be converging to 0. Energy calculation involves summation or integration of squared value of the signal. So I believe the statement from B.P. Lathi still holds good. $\endgroup$
    – SakSath
    Dec 12, 2023 at 14:05
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    $\begingroup$ @SakSath, take the square root of the provided example. There is no problem because it is positive. $\endgroup$ Dec 12, 2023 at 15:21

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I would say that for all practical purposes Lathi is right. He would have been completely correct if he had added "if the limit exists". But for an engineering text that would be overly pedantic, at least in my opinion.

Note that the function in the answer to the question you linked to is quite artificial, and its limit as $x\to\infty$ doesn't exist. If that limit exists, it must be zero for the integral to converge, and that is what Lathi said.

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  • $\begingroup$ IMHO, when one says "A necessary condition for ..." it should be expected a minimum of rigour. I agree that "If the limit exists, then a necessary condition for the energy to be finite is that the signal amplitude $\to 0$ as $|t| \to \infty$" is correct; however, asking for the limit to exist is quite "limiting". I think we can do better than that. I'll edit the question. $\endgroup$ Dec 12, 2023 at 17:09

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