# Signals 

## Definitions

> *Definition*: a signal is a function of space and time.
> 
> * Output can be analog or quantised.
> * Input can be continuous or discrete.

<br>

> *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
>
> | Signal transformation | Time | Amplitude |
> | :-: | :-: | :-: |
> | Reversal | $y(t) = x(-t)$ | $y(t) = -x(t)$ |
> | Scaling | $y(t) = x(at)$ | $y(t) = ax(t)$ |
> | Shifting | $y(t) = x(t - b)$ | $y(t) = x(t) + b$ |
>
> for all $t \in \mathbb{R}$. 

For sampled signals similar definitions hold. 

### Symmetry

> *Definition*: consider a signal $f: \mathbb{R} \to \mathbb{R}$ which is defined in an interval which is symmetric around $t = 0$, we define.
>
> * $f$ is *even* if $f(t) = f(-t)$, $\forall t \in \mathbb{R}$.
> * $f$ is *odd* if $f(t) = -f(-t)$, $\forall t \in \mathbb{R}$. 

For sampled signals similar definitions hold.

> *Theorem*: every signal can be decomposed into symmetric parts.

??? note "*Proof*:"

    Will be added later.

### Periodicity

> *Definition*: a signal $f: \mathbb{R} \to \mathbb{R}$ is defined to be periodic in $T$ if and only if 
>
> $$
>   f(t + T) = f(t), \qquad \forall t \in \mathbb{R}. 
> $$

For sampled signals similar definitions hold.

> *Theorem*: a summation of two periodic signals with periods $T_1, T_2 \in \mathbb{R}$ respectively is periodic if and only if 
>
> $$
>   \frac{T_1}{T_2} \in \mathbb{Q}.
> $$

??? note "*Proof*:"

    Will be added later.

### Signals

> *Definition*: the Heaviside step signal $u: \mathbb{R} \to \mathbb{R}$ is defined by 
>
> $$
>   u(t) = \begin{cases} 1 &\text{ if } t > 0,\\ 0 &\text{ if } t < 0,\end{cases}
> $$
>
> for all $t \in \mathbb{R}$. 

For a sampled function the Heaviside step signal is given by

$$
    u[k] = \begin{cases} 1 \text{ if } k \geq 0, \\ 0 \text{ if } k < 0, \end{cases}
$$

for all $k \in \mathbb{Z}$. 

> *Definition*: the rectangular signal $\text{rect}: \mathbb{R} \to \mathbb{R}$ is defined by
>
> $$
>   \text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \\ 0 &\text{ if } |t| > \frac{1}{2},\end{cases}
> $$
>
> for all $t \in \mathbb{R}$. 

The rect signal can be normalised obtaining the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ defined by

$$
    D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases}
$$

for all $t \in \mathbb{R}$. 

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

$$
    \lim_{\varepsilon \;\downarrow\; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon \;\downarrow\; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0),
$$

using the mean [value theorem for integrals](../../../mathematics/calculus/integration.md#the-mean-value-theorem-for-integrals).

> *Definition*: the Dirac signal $\delta$ is a generalized signal defined by the properties 
>
> $$
> \begin{align*}
>   \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \\
>   \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),&
> \end{align*}
> $$
>
> 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 $f_s: \mathbb{R} \to \mathbb{R}$ 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
>
> $$
>   \delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s)
> $$
>
> for all $t \in \mathbb{R}$ with a sampling period $T_s \in \mathbb{R}$. 

Then integration works out since we have

$$
    \int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k],
$$

by definition.

## Convolutions

> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the convolution product is defined as 
>
> $$
> f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du
> $$
>
> for all $t \in \mathbb{R}$. 

<br>

> *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 $t_0 \in \mathbb{R}$ 
>
> $$
>   f(t) * \delta(t - t_0) = f(t - t_0)
> $$
>
> for all $t \in \mathbb{R}$.

??? note "*Proof*:"

    let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal 

    $$
        f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0),
    $$

    for all $t \in \mathbb{R}$. 

In particular $f(t) * \delta(t) = f(t)$ for all $t \in \mathbb{R}$; $\delta$ is the unity of the convolution.

The average value of a signal $f: \mathbb{R} \to \mathbb{R}$ for an interval $\varepsilon \in \mathbb{R}$ may be given by

$$
    f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du.
$$

For sampled/discrete signals we have a similar definition for the convolution product, given by

$$
    f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m],
$$

for all $k \in \mathbb{Z}$. 

## Correlations

> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the cross-correlation is defined as
>
> $$
>   f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du
> $$
>
> for all $t \in \mathbb{R}$. 

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. This is proved in the section [Fourier series](fourier-series.md). 

For sampled/discrete signals a similar definition exists given by

$$
    f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m],
$$

for all $k \in \mathbb{Z}$.