Dirac Delta Function (Unit Impulse Function)

The Dirac delta function (also called $\delta$ distribution) is used to model a point source or point charge in electrical engineering, physics, and other fields.

It is a a signal that is infinitesimally short in time but has infinite magnitude.

Formally learned in MATH213.

Dirac Delta Function \delta(x)

$\delta (x)=\{\begin{array}{ll}+\infty & x=0\\ 0,& x\ne 0\end{array}$ with ${\int }_{-\infty }^{\infty }\delta (x)dx=1$

Some notes:

• not a true function, but rather a “generalized function” or “distribution.”
• Often used in mathematical modeling to represent a point of discontinuity or singularity in a system.
  • It is also used in the theory of Fourier Transform, where it serves as the kernel for the continuous-time Fourier transform

Laplace Transform of Unit Impulse Function

The Laplace Transform of the unit impulse function is 1: $\mathcal{L}\{\delta (x)\}=1$

• I don’t think the prof proved this, a proof is here

Actually, the prof used this property to prove it

Theorem 1

If $f$ is a “well-behaved” function defined at $t$ then

\int_{-\infty}^\infty f(\tau) \delta(t - \tau) d\tau & = \int_{-\infty}^\infty f(t) \delta(t - \tau)d\tau \ & = f(t) \int_{-\infty}^\infty \delta(t - \tau) d\tau \ &= f(t) \ \end{align}$$

Apparently used in the Particle Filter and Optimal Transport problem from here.

• https://stats.stackexchange.com/questions/226582/role-of-dirac-function-in-particle-filters
• “weighted sum of diracs”

This thing is very very useful, because if you want to figure out the transfer function of a system, you can simply use the Unit Impulse Function!