Delta function approximation

Let narrow

\delta_\Delta(t)

, continuous unit step function

u_\Delta(t)

for analogy

The continuous-time unit impulse can be viewed as a very narrow boxcar function.

\delta(t) = \frac{d}{dt}u(t)

x(t)\delta(t - t_0) = lim_{\Delta \rightarrow0}x(t)\delta_\Delta(t - t_0) \approx x(t_0)\delta(t - t_0)

\delta(t) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{j\omega t}d\omega \newline \delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{j\omega t}dt