Esempi di Sviluppo in serie

Nello schema che segue, sono mostrate le ampiezze delle componenti armoniche X_n per alcuni segnali periodici di periodo T, di cui è fornita l'epressione nel tempo per $\left\vert\vphantom{ t}\right.$t$\left.\vphantom{ t}\right\vert$ < T/2.

Onda quadra simmetrica

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = $$\left\{\vphantom{ \begin{array}{cr} +1 & \left\vert t\right\vert <T/4\\ -1 & T/4\leq \left\vert t\right\vert <T/2 \end{array}}\right.$$$$\begin{array}{cr} +1 & \left\vert t\right\vert <T/4\\ -1 & T/4\leq \left\vert t\right\vert <T/2 \end{array}$$

X_n = $$\left\{\vphantom{ \begin{array}{cr} \hbox {sinc}\left( \frac{n}{2}\right) & n\neq 0\\ 0 & n=0 \end{array}}\right.$$$$\begin{array}{cr} \hbox {sinc}\left( \frac{n}{2}\right) & n\neq 0\\ 0 & n=0 \end{array}$$

 

Treno di impulsi rettangolari

 

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = $$\left\{\vphantom{ \begin{array}{cr} +1 & \left\vert t\right\vert <\tau /2\\ 0 & \tau /2\leq \left\vert t\right\vert <T/2 \end{array}}\right.$$$$\begin{array}{cr} +1 & \left\vert t\right\vert <\tau /2\\ 0 & \tau /2\leq \left\vert t\right\vert <T/2 \end{array}$$

X_n = $${\frac{\tau }{T}}$$sinc$$\left(\vphantom{ \frac{n\tau }{T}}\right.$$$${\frac{n\tau }{T}}$$ $$\left.\vphantom{ \frac{n\tau }{T}}\right)$$

 

Onda triangolare simmetrica

 

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = 1 - 4$${\frac{\left\vert t\right\vert }{T}}$$    $$\left\vert\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right\vert$$ < T/2

X_n = $$\left\{\vphantom{ \begin{array}{cr} \hbox {sinc}^{2}\left( \frac{n}{2}\right) & n\neq 0\\ 0 & n=0 \end{array}}\right.$$$$\begin{array}{cr} \hbox {sinc}^{2}\left( \frac{n}{2}\right) & n\neq 0\\ 0 & n=0 \end{array}$$

 

Dente di sega simmetrico

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = 2$${\frac{t}{\tau }}$$    $$\left\vert\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right\vert$$ < T/2

X_n = $$\left\{\vphantom{ \begin{array}{cr} j\frac{\left( -1\right) ^{n}}{n\pi } & n\neq 0\\ 0 & n=0 \end{array}}\right.$$$$\begin{array}{cr} j\frac{\left( -1\right) ^{n}}{n\pi } & n\neq 0\\ 0 & n=0 \end{array}$$

 

Rettificata a singola semionda

 

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = $$\left\{\vphantom{ \begin{array}{cr} \sin \omega _{0}t & 0\leq t<T/2\\ 0 & -T/2\leq t<0 \end{array}}\right.$$$$\begin{array}{cr} \sin \omega _{0}t & 0\leq t<T/2\\ 0 & -T/2\leq t<0 \end{array}$$

X_n = $$\left\{\vphantom{ \begin{array}{cl} \frac{1}{\pi \left( 1-n^{2}\r... ...ari}\\ -j\frac{1}{4} & n=\pm 1\\ 0 & \hbox {altrimenti} \end{array}}\right.$$$$\begin{array}{cl} \frac{1}{\pi \left( 1-n^{2}\right) } & n\: \hbox {pari}\\ -j\frac{1}{4} & n=\pm 1\\ 0 & \hbox {altrimenti} \end{array}$$

 

Rettificata a onda intera

x$$\left(\vphantom{ t}\right.$$t$$\left.\vphantom{ t}\right)$$ = $$\left\vert\vphantom{ \sin \omega _{0}t}\right.$$sin$$\omega_{0}^{}$$t$$\left.\vphantom{ \sin \omega _{0}t}\right\vert$$

X_n = $$\left\{\vphantom{ \begin{array}{cr} \frac{2}{\pi \left( 1-n^{2}\right) } & n\: \hbox {pari}\\ 0 & \hbox {altrimenti} \end{array}}\right.$$$$\begin{array}{cr} \frac{2}{\pi \left( 1-n^{2}\right) } & n\: \hbox {pari}\\ 0 & \hbox {altrimenti} \end{array}$$