Dirac Delta Function

Dirac Function

Definition

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

$\delta (x)=\{\begin{array}{cc}\mathrm{\infty },& x=0\\ 0,& x\ne 0\end{array}$

Alternative definition

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

This is a gaussian distribution with spread 0.

Properties

Energy

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

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 $-\mathrm{\infty },\mathrm{\infty }$. The result will be quite surprising: it is $\mathrm{\infty }$!

Convolution

$y(t)*\delta (t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}y(\tau )\delta (t-\tau )d\tau =y(t)$

Kronecker Delta Function

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

$\delta (k)=\{\begin{array}{cc}1,& k=0\\ 0,& k\ne 0\end{array}$

Properties

Energy

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

Convolution

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