|
| Time domain
| Z-domain
| ROC
|
|---|
| Notation
| ![{\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e642979d4bbea30a164bd3c3c0478dd4f42c2d)
| ![{\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3aefa942e18926dd24f0a75ca1f495002704e35f)
| ROC: 
|
| Linearity
| ![{\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb707cd1570b5004afa6306d8e349d36efffb58)
| 
| At least the intersection of ROC1 and ROC2
|
| Time shifting
| ![{\displaystyle x[n-k]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd4e132f4bd95a617de0e6eb00ba09e2801e1e0)
| 
| ROC, except  if  and  if 
|
| Scaling in the z-domain
| ![{\displaystyle a^{n}x[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/654e28509ebf86a5253c1e79e7a6b12fa1087466)
| 
| 
|
| Time reversal
| ![{\displaystyle x[-n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ca1d46f1a6942599a856f413f141deb2ab36cf)
| 
| 
|
| Conjugation
| ![{\displaystyle x^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/836731dd6ac0987320b601f404492523187f4dec)
| 
| ROC
|
| Real part
| ![{\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
|
| Imaginary part
| ![{\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
|
| Differentiation
| ![{\displaystyle nx[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9d129545d7bb8f225e3d3d6cf29828055b2aa0)
| 
| ROC
|
| Convolution
| ![{\displaystyle x_{1}[n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a88f085e632f498b5ad7cae9fc45379ab931ff7)
| 
| At least the intersection of ROC1 and ROC2
|
| Correlation
| ![{\displaystyle r_{x_{1},x_{2}}(l)=x_{1}[l]*x_{2}[-l]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/29460269c4171c649511cb92c01422b1b4e4dbac)
| 
| At least the intersection of ROC of X1(z) and X2( )
|
| Multiplication
| ![{\displaystyle x_{1}[n]x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b8ee2f74b87c0783a1e479136a290c947ff945)
| 
| At least 
|
| Parseval's relation
| ![{\displaystyle \sum ^{\infty }x_{1}[n]x_{2}^{*}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1016c0de1be930de1ddfd1b3ae279d6b1a443cfe)
| 
| |
![{\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/34e74d32b1013c9db6578a43b03066f7dd31202c)
![{\displaystyle x[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b575dce543a2cf10a5a3e108204b928c2c9aaa54)
![{\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71d6d17cc3b7ef5c69a58caf4ebc6d2acc2c8ee3)
