# Temporal Orthogonality

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.

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 $a_k$.