Elementary signals
Dirac
The Dirac impulse plays an important role for analog signal sampling, but also for filters characterization. If the discrete definition does not pose any problems, the rigorous analog definition relies on the mathematical notion of distribution.
Numerical Dirac impulse\
The Dirac impulse, denoted by $\delta_k$, is the sequence defined by:
$$
\delta_k[n] = \begin{cases} 1 & \text{ if } n = k\\0 & \text{ otherwise} \end{cases}
$$
In the analog case, we often found the physicist definition: $$ \delta_0(t) = \begin{cases}+\infty & \text{ if } t = 0\\0 & \text{ otherwise} \end{cases} $$ such that $$ \int \delta_0(t) \mathrm{d} t = 1\ . $$ However, this definition is mathematically a a nonsensen the previous integrale is always equal to $0$. But it can be sometimes convenient to use such a definition in order to (carefully) manipulate the Dirac impulse as “a function”\ldots
Without given all the details, the Dirac distribution $\delta_0$ is a linear form defined on the Test function space $\mathcal{D}$, such that for $f\in\mathcal{D}$: $$ \langle \delta_0, f\rangle = \int f(x)\delta_0(x) \mathrm{d} x = f(0)\ . $$ And more generally, we can define $\delta_t$ the distribution such that for $f\in\mathcal{D}$ $$ \langle \delta_t, f\rangle = \int f(x)\delta_t(x) \mathrm{d} x = f(t)\ . $$
Heaviside
The Heaviside function is also known as the unit step function.
Heaviside sequence\
The Heaviside sequence denoted by $\Theta$ is given by
$$
\Theta[n] = \begin{cases}1 & \text{ if } n \geq 0\\0 & \text{ otherwise}
\end{cases}
$$
In the analog case, the definition can be directly adated
Fonction de Heaviside\
La fonction de Heaviside est la fonction $\Theta:\mathbb{R}\rightarrow\mathbb{R}$ définie par
$$
\Theta(t) = \begin{cases} 1 & \text{ if } t \geq 0\\0 & \text{ otherwise}\end{cases}
$$
Next figure gives a representation of the Heaviside function
Rectangular window
The rectangular window will play an important role for ideal filtering.
Rectangular window\
The rectangular window function is given by $\Pi:\mathbb{R}\rightarrow\mathbb{R}$ such that
$$
\Pi(t) = \begin{cases} 1 & \text{ if } -\frac{1}{2}\leq t \leq \frac{1}{2}\\0 & \text{ otherwise}\end{cases}
$$
Monochromatic signal
Finally, the monochromatic signal is simply the sine function. This elementary periodic signal is the heart of the Fourier transforms.
