Z-dönüşümü
Özellikler

|   | zaman bölgesi 	 | Z- bölgesi 	 | İspat  	 | ROC   |
| --- | --- | --- | --- | --- |
| Notasyon 	 |  x[n]={\mathcal {Z}}^{-1}\{X(z)\}  |  X(z)={\mathcal {Z}}\{x[n]\}  |   | ROC:  r_{2}<|z|<r_{1}\   |
| Doğrusallık  |  a_{1}x_{1}[n]+a_{2}x_{2}[n]\   |  a_{1}X_{1}(z)+a_{2}X_{2}(z)\   |  {\begin{array}{lcl}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{array}}  | En azından ROC1 ve ROC2 bölgelerinin kesişimi   |
| Zamanda genişleme  |  x_{(k)}[n]={\begin{cases}x[r],&n=rk\\0,&n\not =rk\end{cases}}  r: tam sayı  |  X(z^{k})\   |   | R^{1/k}   |
| Zamanda kayma 	 |  x[n-k]\   |  z^{-k}X(z)\   |  {\begin{array}{lcl}Z\{x[n-k]\}&=&\sum _{n=0}^{\infty }x[n-k]z^{-n}{\text{ , let }}j=n-k\\&=&\sum _{j=-k}^{\infty }x[j]z^{-(j+k)}\\&=&\sum _{j=-k}^{\infty }x[j]z^{-j}z^{-k}\\&=&z^{-k}\sum _{j=-k}^{\infty }x[j]z^{-j}\\&=&z^{-k}\sum _{j=0}^{\infty }x[j]z^{-j}{\text{ , since }}x[\beta ]=0{\text{ if }}\beta <0\\&=&z^{-k}X(z)\\\end{array}}  | ROC, eğer  k>0\, ise dışında   z=0\   ve eğer  k<0\  ise  z=\infty   dışında   |
| Scaling in the z-domain 	 |  a^{n}x[n]\   |  X(a^{-1}z)\   |  {\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}}  |  |a|r_{2}<|z|<|a|r_{1}\   |
| Time reversal 	 |  x[-n]\   |  X(z^{-1})\   |  {\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}}  |  {\frac {1}{r_{1}}}<|z|<{\frac {1}{r_{2}}}\   |
| Karmaşık eşlenik  |  x^{*}[n]\   |  X^{*}(z^{*})\   |  {\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   |
| Reel kısım  |  \operatorname {Re} \{x[n]\}\   |  {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]  |   | ROC   |
| Imajiner kısım  |  \operatorname {Im} \{x[n]\}\   |  {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]  |   | ROC   |
| Türev 	 |  nx[n]\   |  -z{\frac {dX(z)}{dz}}  |  {\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   |
| Convolution  |  x_{1}[n]*x_{2}[n]\   |  X_{1}(z)X_{2}(z)\   |  {\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}}  | en azından ROC1 ve ROC2 keşisim kümesi   |
| Correlation  |  r_{x_{1},x_{2}}(l)=x_{1}[l]*x_{2}[-l]\   |  R_{x_{1},x_{2}}(z)=X_{1}(z)X_{2}(z^{-1})\   |   | en azından X1(z)'e ait ROC ve X2( z^{-1})'e ait ROC'un keşisimi.   |
| First Difference 	 |  x[n]-x[n-1]\   |  (1-z^{-1})X(z)\   |    | En azından X1(z) ve  |z|>0 keşisimi   |
| Accumulation 	 |  \sum _{k=-\infty }^{n}x[k]\   |  {\frac {1}{1-z^{-1}}}X(z)  |  {\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}}  |   |
| Çarpma  |  x_{1}[n]x_{2}[n]\   |  {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\frac {z}{v}})v^{-1}\mathrm {d} v\   |   | -   |
| Parseval teoremi  |  \sum _{n=-\infty }^{\infty }x_{1}[n]x_{2}^{*}[n]\   |  {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\frac {1}{v^{*}}})v^{-1}\mathrm {d} v\   |   |  |
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