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I'm studying Digital Communication and I couldn't understand the relationship between temporal orthogonality and sampling theorem.

Definition of the Temporal Orthogonality is as follows:

$$\mathrm{ACF}= \int s_i(t)\cdot s_l(t-kT) \ \mathrm{d}t= E_{il}\cdot \delta [k]$$

In the text, it is stated that:

[...] This means that with respect to the represented symbol sequence lossless discrete-time signal processing is possible using only one sample per symbol interval, even if the sampling theorem is not fulfilled.

We are at the output of the transmitter.

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Say you have two pulses, $p(t)$ and $q(t)$, such that the following three statments are true:

\begin{align} \int_{-\infty}^\infty p(t) p(t) dt &= 1,\\ \int_{-\infty}^\infty q(t) q(t) dt &= 1,\\ \int_{-\infty}^\infty p(t) q(t) dt &= 0. \end{align}

Then these pulses are said to be orthonormal. Now, let us say you want to transmit two information-bearing symbols $a_1$ and $a_2$. (The symbols carry information when the receiver does not know their values in advance). You can acomplish this by transmitting the signal $s(t) = a_1 p(t) + a_2 q(t)$.

How does the receiver recover the symbols? By performing these operations:

\begin{align} a_1 &= \int_{-\infty}^\infty s(t) p(t) dt, \\ a_2 &= \int_{-\infty}^\infty s(t) q(t) dt. \\ \end{align}

I think it's clear that this scheme can be extended to any number of symbols; all you need to do is find a set of orthonormal pulses of the required size. The most common approach is to find a "prototype pulse" $p(t)$ with the property that, when time-shifted $lT_p$ seconds with $l$ integer, it is orthonormal to all other pulses time-shifted $mT_p$ seconds, for any integer $l \neq m$. So, you can transmit any number of symbols with the signal

\begin{align} s(t) &= a_1 p(t - T_p) + a_2 p(t - 2T_p) + a_3 p(t - 3T_p) + \cdots \\ &= \sum_k a_k p(t-kT_p). \end{align}

The symbols $a_k$ are recovered in the same way as before, but this operation can be implemented with a filter "matched" to the prototype pulse $p(t)$ (see numerous questions related to matched filters on this site). When the matched filter's output is sampled at time $t=kT_p$, you obtain back the symbol $a_k$.

Note that the filter's output is not necessarily sampled at a rate that meets Nyquist's sampling theorem (I have said nothing about the bandwidth of $p(t)$ or $s(t)$), but that is OK because you're not interested in reconstructing $s(t)$ from its samples, all you want is the symbols $a_k$.

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