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I am trying to understand Section 2.2.3 of the book "Fundementals of wireless communication" by Tse and Viswanath. In this section the authors explain how to go from the continuous time baseband model to the discrete time baseband model.

I am going to use equation numbers from the book. My question is the following:

If one looks at equation 2.29 it shows the sampling theorem. If I understand correctly that summation is the Whittaker–Shannon interpolation formula and it is from \$-\infty\$ to \$\infty\$.

This means in equation (2.33) the variable \$\ell\$ is from \$-\infty\$ to \$\infty\$. And this will imply the same in equation (2.35).

Does this not make the system non causal? Because if the variable \$\ell\$ is allowed to be negative, now the output sample at zero can depend on any future input samples.

May be I am missing something on why the variable \$\ell\$ is not allowed to go negative, can someone clarify?

Adding image of the pages.

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    \$\begingroup\$ Not everybody has that book. Can you post a (readable) picture? \$\endgroup\$ – a concerned citizen Nov 9 '20 at 11:33
  • \$\begingroup\$ @aconcernedcitizen I have added the image now. \$\endgroup\$ – Coniferous Nov 9 '20 at 12:13
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    \$\begingroup\$ @aconcernedcitizen the authors actually shared that book online, here. It's one of the classics! \$\endgroup\$ – Marcus Müller Nov 9 '20 at 13:50
  • \$\begingroup\$ @MarcusMüller Shame on me for not knowing, but thank you for the link! Much appreciated. \$\endgroup\$ – a concerned citizen Nov 9 '20 at 14:48
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The mathematical notation for the formula uses \$-\infty...\infty\$ as its domain and, as such, it can afford a theoretical non-causal response. Similarly, \$\text{sinc}(x)\$ is also defined with the same limits, yet in practice \$x\$ is shifted by a certain amount.

The same thing happens here: the formula may have its roots in the mathematical, ideal formulation, but in practice the shifting (that you can already see in the equations) makes the response perfectly causal, so you don't have to worry about free energy.

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  • \$\begingroup\$ I am sorry, I really do not understand your argument here. Can you explain in math? Like what do you mean by "shifting makes the response causal"?. \$\endgroup\$ – Coniferous Nov 9 '20 at 12:57
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    \$\begingroup\$ sinc(n) is non-causal, since it implies half the response lies to the left of 0. sinc(n-3) is causal and it lasts from 0 until 6. It's also bandwidth limited (i.e. truncated to those limits). Shifting means time-shifting, that n-3. \$\endgroup\$ – a concerned citizen Nov 9 '20 at 13:01

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