2.1a.1 Dirac Delta

Dirac Delta

The Dirac delta function, denoted as  $\delta (t)$, is a mathematical construct used to model an idealized point impulse. Mathematically, it is defined by the piecewise equation

\begin{cases} \infty & \text{for } t = 0 \\ 0 & \text{for } t \neq 0 \end{cases}.$$ >[!warning] Note! >The Dirac delta is actually not a function. Rather it is a distribution, or a generalized function.

Properties

(Sampling) $x(t)\delta (t-t_0)=x(t_0)\delta (t-t_0)$

(Sifting) ${\int }_A^{B}x(t)\delta (t-t_0)dt=\{\begin{array}{ll}x(t_0)& \text{if }A<t_0<B\\ 0& \text{otherwise}\end{array}$

(Convolution) $y(t)=\alpha \delta (t-t_0)⟹(x\text{textasteriskcentered}y)(t)=\alpha x(t-t_0)$

We can easily prove this using the two previous properties:

(x \textasteriskcentered y)(t) &= \int^{\infty}_{-\infty}x(\tau)y(t-\tau)d\tau \\ &= \alpha\int^{\infty}_{-\infty}x(\tau)\delta(t-\tau-t_0)d\tau \\ \end{align}$$ Note here that $\delta(t-\tau-t_0) \neq 0 \iff \tau = t-t_0$, so $$\alpha\int^{\infty}_{-\infty}x(\tau)\delta(t-\tau-t_0)d\tau = \alpha x(t-t_0).$$ #### (Time-Scaling) $\delta(\alpha t) = \frac{1}{|\alpha|}\delta(t)$ This is a bit harder to prove, but since we know that the