Pagina originale Conversione della tabella presente nella pagina in sezioni

Per la proprietà di linearità la trasformata di ${\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]}$ è ${\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)}$.

Infatti:

${\displaystyle {\begin{aligned}X(z)&=\sum _{n=-\infty }^{\infty }(a_{1}x_{1}(n)+a_{2}x_{2}(n))z^{-n}\\&=a_{1}\sum _{n=-\infty }^{\infty }(x_{1}(n))z^{-n}+a_{2}\sum _{n=-\infty }^{\infty }(x_{2}(n))z^{-n}\\&=a_{1}X_{1}(z)+a_{2}X_{2}(z)\end{aligned}}}$

La regione di convergenza è almeno la regione di intersezione di ROC₁ e ROC₂

Per la proprietà di espansione temporale si ha che la trasformata di

${\displaystyle x_{(k)}[n]={\begin{cases}x[r],&n=rk\\0,&n\not =rk\end{cases}}}$

con ${\displaystyle r}$ intero è:

${\displaystyle X(z^{k})}$

Infatti:

${\displaystyle {\begin{aligned}X_{k}(z)&=\sum _{n=-\infty }^{\infty }x_{k}(n)z^{-n}=\\&=\sum _{r=-\infty }^{\infty }x(r)z^{-rk}=\\&=\sum _{r=-\infty }^{\infty }x(r)(z^{k})^{-r}=\\&=X(z^{k})\end{aligned}}}$

La regione di convergenza è: ${\displaystyle r^{1/k}}$.

${\displaystyle x[n-k]}$ ${\displaystyle z^{-k}X(z)}$ ${\displaystyle Z\{x[n-k]\}=\sum _{n=0}^{\infty }x[n-k]z^{-n}}$
Posto ${\displaystyle j=n-k}$ si ha:
${\displaystyle \sum _{n=0}^{\infty }x[n-k]z^{-n}=\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}=\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}}$ ${\displaystyle =z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}}$ ${\displaystyle =z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}}$
essendo ${\displaystyle x[\beta ]=0}$ se ${\displaystyle \beta <0}$. Da cui:
${\displaystyle z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}=z^{-k}X(z)}$ ROC, eccetto ${\displaystyle z=0\ }$ se ${\displaystyle k>0\,}$ e ${\displaystyle z=\infty }$ se ${\displaystyle k<0\ }$

${\displaystyle a^{n}x[n]\ }$ ${\displaystyle X(a^{-1}z)\ }$ ${\displaystyle {\begin{array}{lcl}Z\{a^{n}x[n]\}=&\\\sum _{n=-\infty }^{\infty }a^{n}x(n)z^{-n}&\\=\sum _{n=-\infty }^{\infty }x(n)(a^{-1}z)^{-n}&\\=X(a^{-1}z)&\\\end{array}}}$ ${\displaystyle |a|r_{2}<|z|<|a|r_{1}\ }$

${\displaystyle x[-n]\ }$ ${\displaystyle X(z^{-1})\ }$ ${\displaystyle {\begin{array}{lcl}{\mathcal {Z}}\{x(-n)\}=&\\\sum _{n=-\infty }^{\infty }x(-n)z^{-n}\ &\\=\sum _{m=-\infty }^{\infty }x(m)z^{m}\ &\\=\sum _{m=-\infty }^{\infty }x(m){(z^{-1})}^{-m}\ &\\=X(z^{-1})&\\\end{array}}}$ ${\displaystyle {\frac {1}{r_{1}}}<|z|<{\frac {1}{r_{2}}}\ }$

${\displaystyle x^{*}[n]\ }$ ${\displaystyle X^{*}(z^{*})\ }$ ${\displaystyle {\begin{array}{lcl}Z\{x^{*}(n)\}=&\\\sum _{n=-\infty }^{\infty }x^{*}(n)z^{-n}\ &\\=\sum _{n=-\infty }^{\infty }[x(n)(z^{*})^{-n}]^{*}\ &\\=[\sum _{n=-\infty }^{\infty }x(n)(z^{*})^{-n}\ ]^{*}&\\=X^{*}(z^{*})&\\\end{array}}}$ ROC

${\displaystyle \operatorname {Re} \{x[n]\}\ }$ ${\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}$ ROC

${\displaystyle \operatorname {Im} \{x[n]\}\ }$ ${\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}$ ROC

${\displaystyle nx[n]\ }$ ${\displaystyle -z{\frac {dX(z)}{dz}}}$ ${\displaystyle {\begin{array}{lcl}Z\{nx(n)\}=&\\\sum _{n=-\infty }^{\infty }nx(n)z^{-n}\ &\\=z\sum _{n=-\infty }^{\infty }nx(n)z^{-n-1}\ &\\=-z\sum _{n=-\infty }^{\infty }x(n)(-nz^{-n-1})\ &\\=-z\sum _{n=-\infty }^{\infty }x(n){\frac {d}{dz}}(z^{-n})\ &\\=-z{\frac {dX(z)}{dz}}&\\\end{array}}}$

ROC

${\displaystyle x_{1}[n]*x_{2}[n]\ }$ ${\displaystyle X_{1}(z)X_{2}(z)\ }$ ${\displaystyle {\begin{array}{lcl}{\mathcal {Z}}\{x_{1}(n)*x_{2}(n)\}=&\\{\mathcal {Z}}\{\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)\}\ &\\=\sum _{n=-\infty }^{\infty }[\sum _{l=-\infty }^{\infty }x_{1}(l)x_{2}(n-l)]z^{-n}\ &\\=\sum _{l=-\infty }^{\infty }x_{1}(l)\sum _{n=-\infty }^{\infty }x_{2}(n-l)z^{-n}]\ &\\=[\sum _{l=-\infty }^{\infty }x_{1}(l)z^{-l}][\sum _{n=-\infty }^{\infty }x_{2}(n)z^{-n}]\ &\\=X_{1}(z)X_{2}(z)&\\\end{array}}}$ Almeno la regione di intersezione di ROC₁ e ROC₂

${\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]\ }$ ${\displaystyle R_{x_{1},x_{2}}(z)=X_{1}^{*}(1/z^{*})X_{2}(z)\ }$ Almeno la regione di intersezione di ROC of ${\displaystyle X_{1}(1/z^{*})}$ e ${\displaystyle X_{2}(z)}$

${\displaystyle x[n]-x[n-1]\ }$ ${\displaystyle (1-z^{-1})X(z)\ }$ Almeno la regione di intersezione di ROC of X₁(z) e ${\displaystyle |z|>0}$

${\displaystyle \sum _{k=-\infty }^{n}x[k]\ }$

${\displaystyle {\frac {1}{1-z^{-1}}}X(z)}$

${\displaystyle {\begin{array}{lcl}\sum _{n=-\infty }^{\infty }\sum _{k=-\infty }^{n}x[k]\cdot z^{-n}\\=sum_{n=-\infty }^{\infty }(x[n]+x[n-1]+\\x[n-2]\cdots x[-\infty ])z^{-n}\\=X[z](1+z^{-1}+z^{-2}+z^{-3}\cdots )\\=X[z]\sum _{j=0}^{\infty }z^{-j}\\=X[z]{\frac {1}{1-z^{-1}}}\end{array}}}$

${\displaystyle x_{1}[n]x_{2}[n]\ }$

${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\frac {z}{v}})v^{-1}\mathrm {d} v\ }$

Almeno ${\displaystyle r_{1l}r_{2l}<|z|<r_{1u}r_{2u}\ }$ |-

${\displaystyle \sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\ }$

${\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\frac {1}{v^{*}}})v^{-1}\mathrm {d} v\ }$