About discrete time model for wireless channel

Question

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?

Answer 1

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.