> *Definition*: a signal can be sampled at particular moments $k T_s$ in time, with $k \in \mathbb{Z}$ and $T_s \in \mathbb{R}$ the sampling period. For a signal $f: \mathbb{R} \to \mathbb{R}$ sampled with a sampling period $T_s$ may be denoted by
>
> $$
> f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}.
> $$
<br>
> *Definition*: signal transformations on a function $x: \mathbb{R} \to \mathbb{R}$ obtaining the function $y: \mathbb{R} \to \mathbb{R}$ are given by
The following signal has been derived from the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ used on a signal $f: \mathbb{R} \to \mathbb{R}$ for
> for a signal $f: \mathbb{R} \to \mathbb{R}$ continuous in $t_0$.
For sampled signals the $\delta$ signal is given by
$$
\delta[k] = \begin{cases} 1 &\text{ if } k = 0, \\ 0 &\text{ if } k \neq 0.\end{cases}
$$
## Signal sampling
We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s(t)$ that represents the sampled signal using the Dirac signal, obtaining
$$
f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}.
$$
> *Definition*: the sampling signal or impulse train $\delta_{T_s}: \mathbb{R} \to \mathbb{R}$ is defined as
> *Proposition*: the convolution product is commutative, distributive and associative.
??? note "*Proof*:"
Will be added later.
> *Theorem*: let $f: \mathbb{R} \to \mathbb{R}$ be a signal then we have for the convolution product between $f$ and the Dirac signal $\delta$ and some $t_0 \in \mathbb{R}$
Especially the auto-correlation of a continuous signal $f: \mathbb{R} \to \mathbb{R}$ given by $f(t) \star f(t)$ for all $t \in \mathbb{R}$ is useful, as it can detect periodicity without stating the proof.
For sampled/discrete signals a similar definition exists given by