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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b97ce6ff93cf3ccb0258ad080057561fe1defb16)
è 
.
Infatti:
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9f9952a352bd13efc6e0310e925f44aeced1b5)
con 
intero è:
Infatti:
La regione di convergenza è: 
.
![{\displaystyle \operatorname {Re} \{x[n]\}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c4d81639a0f761dbc7855d7de10d03c2a42d6b)
![{\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbea203386496bdc11a51c6f14798cf0b7373ef7)
ROC
![{\displaystyle \operatorname {Im} \{x[n]\}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aaa2a8950fdcb31a68819605c826960e6c48c5c)
![{\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e00a599cca7e1c1a78f75d7f7710b7474c12da)
ROC
![{\displaystyle nx[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9d129545d7bb8f225e3d3d6cf29828055b2aa0)
ROC
![{\displaystyle \sum _{k=-\infty }^{n}x[k]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cfdbdd32d7420715b393ab382646f76dc6e3dc2)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/034b94ce21ddf238a7b7aba597565602e8abb3c4)
![{\displaystyle \sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/14b55a573a79d115fd4d071bd6f6cb3bbf5172a1)
![{\displaystyle x[n-k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd4fa5b96ade59fee1aa33657f28a6ed743fee0)
![{\displaystyle Z\{x[n-k]\}=\sum _{n=0}^{\infty }x[n-k]z^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa0de778b9f396d8a7ab2f4d69e4e026f898c38)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef278e01f625d3cfa3b6a84a0d1824592228f62)
![{\displaystyle =z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd6d5deda0382713e0c5a60145f5cc7bd18440bb)
![{\displaystyle =z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59013379113960990910b96410a4a8609687649)
![{\displaystyle x[\beta ]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a4d7c548e61faad05d8e4a8234782833fe654e)
![{\displaystyle z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}=z^{-k}X(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c29ec28fbf38d33c495cdfaeb53976857b1462b0)
![{\displaystyle a^{n}x[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/654e28509ebf86a5253c1e79e7a6b12fa1087466)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b968c04b8900e9e16c42ef0333062c3d385d4523)
![{\displaystyle x[-n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ca1d46f1a6942599a856f413f141deb2ab36cf)
![{\displaystyle x^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/836731dd6ac0987320b601f404492523187f4dec)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c974b68d4c86dd8624f0f5ef6fd71634e789cfac)
![{\displaystyle x_{1}[n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a88f085e632f498b5ad7cae9fc45379ab931ff7)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48ac821be34871813f81e8a3db121d9082cf83b0)
![{\displaystyle r_{x_{1},x_{2}}=x_{1}^{*}[-n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3137b2b225055345f857b5fa1262472fef95e7)
![{\displaystyle x[n]-x[n-1]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/64d1f6e2e3298702699bdb047de5c5241347dca2)
![{\displaystyle x_{1}[n]x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b8ee2f74b87c0783a1e479136a290c947ff945)
