Trasformata di un gradino

Definiamo la funzione gradino come g$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = $\left\{\vphantom{ \begin{array}{ccc} 1 & \hbox {per} & t>0\\ \frac{1}{2} & \hbox {per} & t=0\\ 0 & \hbox {per} & t<0 \end{array}}\right.$$\begin{array}{ccc} 1 & \hbox {per} & t>0\\ \frac{1}{2} & \hbox {per} & t=0\\ 0 & \hbox {per} & t<0 \end{array}$ che, fornendo $\int^{\infty }_{-\infty }$$\left\vert\vphantom{ g\left( t\right) }\right.$g$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ $\left.\vphantom{ g\left( t\right) }\right\vert$dt = $\infty$, non dovrebbe avere una trasformata. Come già visto per la costante, il calcolo può essere condotto a termine come limite a cui tende la trasformata di una diversa funzione, il cui limite tende al gradino. Scegliamo quindi g_$\alpha$$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = e^{- $\alpha$t} per t > 0, per la quale risulta $\lim_{\alpha \rightarrow 0}^{}$g_$\alpha$$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = g$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$, e troviamo

G_$\alpha$$$\left(\vphantom{ f}\right.$$f$$\left.\vphantom{ f}\right)$$ = $$\int^{\infty }_{0}$$e^{- $\alpha$t}e^-j2$\pi$ft = $$\left.\vphantom{ \frac{e^{-\left( \alpha +j2\pi f\right) t}}{-\left( \alpha +j2\pi f\right) }}\right.$$$${\frac{e^{-\left( \alpha +j2\pi f\right) t}}{-\left( \alpha +j2\pi f\right) }}$$ $$\left.\vphantom{ \frac{e^{-\left( \alpha +j2\pi f\right) t}}{-\left( \alpha +j2\pi f\right) }}\right\vert _{0}^{\infty }$$ = $${\frac{1}{\alpha +j2\pi f}}$$ = $${\frac{\alpha -j2\pi f}{\alpha ^{2}+\left( 2\pi f\right) ^{2}}}$$

Si può mostrare che $\lim_{\alpha \rightarrow 0}^{}$$\Re$$\left\{\vphantom{ G_{\alpha }\left( f\right) }\right.$G_$\alpha$$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ $\left.\vphantom{ G_{\alpha }\left( f\right) }\right\}$ = $\lim_{\alpha \rightarrow 0}^{}$${\frac{\alpha }{\alpha ^{2}+\left( 2\pi f\right) ^{2}}}$ = ${\frac{1}{2}}$$\delta$$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$, mentre risulta

$\lim_{\alpha \rightarrow 0}^{}$$\Im$$\left\{\vphantom{ G_{\alpha }\left( f\right) }\right.$G_$\alpha$$\left(\vphantom{ f}\right.$f$\left.\vphantom{ f}\right)$ $\left.\vphantom{ G_{\alpha }\left( f\right) }\right\}$ = $\lim_{\alpha \rightarrow 0}^{}$${\frac{-j2\pi f}{\alpha ^{2}+\left( 2\pi f\right) ^{2}}}$ = ${\frac{1}{j2\pi f}}$, ottenendo in definitiva

G$$\left(\vphantom{ f}\right.$$f$$\left.\vphantom{ f}\right)$$ = $$\mathcal {F}$$$$\left\{\vphantom{ g\left( t\right) }\right.$$g$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ $$\left.\vphantom{ g\left( t\right) }\right\}$$ = $${\textstyle\frac{1}{2}}$$$$\left(\vphantom{ \delta \left( t\right) -\frac{j}{\pi f}}\right.$$$$\delta$$$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ - $${\frac{j}{\pi f}}$$ $$\left.\vphantom{ \delta \left( t\right) -\frac{j}{\pi f}}\right)$$