Alternative definition

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

This is a gaussian distribution with spread 0.

Energy

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

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 ${\displaystyle \infty }$!

Convolution

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

Energy

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

Convolution

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