Dirac Delta Function

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


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

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.

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 $$k = 0$$


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

Energy

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

Convolution

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