Dirac Delta Function

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

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

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

This is a gaussian distribution with spread 0.

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

NB: ${\displaystyle \delta (t)^{2}}$ has no mathematical meaning, as ${\displaystyle \delta (t)}$ isn't an ordinary function but a distribution. The special nature of ${\displaystyle \delta (t)}$ 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 ${\displaystyle -\infty ,\infty }$. The result will be quite surprising: it is ${\displaystyle \infty }$!

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

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

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

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

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