Dirac Delta Function

Dirac Function

Definition

The Dirac function $\delta (t)$ is a "signal" with unit energy that is concentrated around $t=0$

$\delta (x)={\begin{cases}\infty ,&x=0\\0,&x\neq 0\end{cases}}$

Alternative definition

$\delta (t)=\lim _{\sigma \to 0}{\frac {1}{\sigma {\sqrt {(}}2\pi )}}\exp(-{\frac {t^{2}}{2\sigma ^{2}}})$

This is a gaussian distribution with spread 0.

Properties

Energy

$E=\int _{-\infty }^{\infty }\delta (t)^{2}dt=\infty$

NB: $\delta (t)^{2}$ has no mathematical meaning, as $\delta (t)$ isn't an ordinary function but a distribution. The special nature of $\delta (t)$ appears clearly e.g. when you try to square the same Gaussian distribution above and try to compute the same limit of the integral in $-\infty ,\infty$. The result will be quite surprising: it is $\infty$!

Convolution

$y(t)*\delta (t)=\int _{-\infty }^{\infty }y(\tau )\delta (t-\tau )d\tau =y(t)$

Kronecker Delta Function

The Kronecker delta function is the discrete analog of the Dirac function. It has Energy 1 and only a contribution at $k=0$

$\delta (k)={\begin{cases}1,&k=0\\0,&k\neq 0\end{cases}}$

Properties

Energy

$E=\sum _{k=-\infty }^{\infty }\delta (k)^{2}=1$

Convolution

$y(k)*\delta (k)=\sum _{m=-\infty }^{\infty }y(k)\delta (k-m)=y(k)$