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Engineering Handbook/Mathematics/Z Transformation
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Engineering Handbook
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Mathematics
Z Transformation Properties
edit
Time domain
Z-domain
ROC
Notation
x
[
n
]
=
Z
−
1
{
X
(
z
)
}
{\displaystyle x[n]={\mathcal {Z}}^{-1}\{X(z)\}}
X
(
z
)
=
Z
{
x
[
n
]
}
{\displaystyle X(z)={\mathcal {Z}}\{x[n]\}}
ROC:
r
2
<
|
z
|
<
r
1
{\displaystyle r_{2}<|z|<r_{1}\ }
Linearity
a
1
x
1
[
n
]
+
a
2
x
2
[
n
]
{\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }
a
1
X
1
(
z
)
+
a
2
X
2
(
z
)
{\displaystyle a_{1}X_{1}(z)+a_{2}X_{2}(z)\ }
At least the intersection of ROC
1
and ROC
2
Time shifting
x
[
n
−
k
]
{\displaystyle x[n-k]\ }
z
−
k
X
(
z
)
{\displaystyle z^{-k}X(z)\ }
ROC, except
z
=
0
{\displaystyle z=0\ }
if
k
>
0
{\displaystyle k>0\,}
and
z
=
∞
{\displaystyle z=\infty }
if
k
<
0
{\displaystyle k<0\ }
Scaling in the z-domain
a
n
x
[
n
]
{\displaystyle a^{n}x[n]\ }
X
(
a
−
1
z
)
{\displaystyle X(a^{-1}z)\ }
|
a
|
r
2
<
|
z
|
<
|
a
|
r
1
{\displaystyle |a|r_{2}<|z|<|a|r_{1}\ }
Time reversal
x
[
−
n
]
{\displaystyle x[-n]\ }
X
(
z
−
1
)
{\displaystyle X(z^{-1})\ }
1
r
2
<
|
z
|
<
1
r
1
{\displaystyle {\frac {1}{r_{2}}}<|z|<{\frac {1}{r_{1}}}\ }
Conjugation
x
∗
[
n
]
{\displaystyle x^{*}[n]\ }
X
∗
(
z
∗
)
{\displaystyle X^{*}(z^{*})\ }
ROC
Real part
Re
{
x
[
n
]
}
{\displaystyle \operatorname {Re} \{x[n]\}\ }
1
2
[
X
(
z
)
+
X
∗
(
z
∗
)
]
{\displaystyle {\frac {1}{2}}\left[X(z)+X^{*}(z^{*})\right]}
ROC
Imaginary part
Im
{
x
[
n
]
}
{\displaystyle \operatorname {Im} \{x[n]\}\ }
1
2
j
[
X
(
z
)
−
X
∗
(
z
∗
)
]
{\displaystyle {\frac {1}{2j}}\left[X(z)-X^{*}(z^{*})\right]}
ROC
Differentiation
n
x
[
n
]
{\displaystyle nx[n]\ }
−
z
d
X
(
z
)
d
z
{\displaystyle -z{\frac {\mathrm {d} X(z)}{\mathrm {d} z}}}
ROC
Convolution
x
1
[
n
]
∗
x
2
[
n
]
{\displaystyle x_{1}[n]*x_{2}[n]\ }
X
1
(
z
)
X
2
(
z
)
{\displaystyle X_{1}(z)X_{2}(z)\ }
At least the intersection of ROC
1
and ROC
2
Correlation
r
x
1
,
x
2
(
l
)
=
x
1
[
l
]
∗
x
2
[
−
l
]
{\displaystyle 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
)
{\displaystyle R_{x_{1},x_{2}}(z)=X_{1}(z)X_{2}(z^{-1})\ }
At least the intersection of ROC of X
1
(z) and X
2
(
z
−
1
{\displaystyle z^{-1}}
)
Multiplication
x
1
[
n
]
x
2
[
n
]
{\displaystyle x_{1}[n]x_{2}[n]\ }
1
j
2
π
∮
C
X
1
(
v
)
X
2
(
z
v
)
v
−
1
d
v
{\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}({\frac {z}{v}})v^{-1}\mathrm {d} v\ }
At least
r
1
l
r
2
l
<
|
z
|
<
r
1
u
r
2
u
{\displaystyle r_{1l}r_{2l}<|z|<r_{1u}r_{2u}\ }
Parseval's relation
∑
∞
x
1
[
n
]
x
2
∗
[
n
]
{\displaystyle \sum ^{\infty }x_{1}[n]x_{2}^{*}[n]\ }
1
j
2
π
∮
C
X
1
(
v
)
X
2
∗
(
1
v
∗
)
v
−
1
d
v
{\displaystyle {\frac {1}{j2\pi }}\oint _{C}X_{1}(v)X_{2}^{*}({\frac {1}{v^{*}}})v^{-1}\mathrm {d} v\ }
Initial value theorem
x
[
0
]
=
lim
z
→
∞
X
(
z
)
{\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }
, If
x
[
n
]
{\displaystyle x[n]\,}
causal
Final value theorem
x
[
∞
]
=
lim
z
→
1
(
z
−
1
)
X
(
z
)
{\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }
, Only if poles of
(
z
−
1
)
X
(
z
)
{\displaystyle (z-1)X(z)\ }
are inside unit circle
Z Transformation of Functions
edit
Here:
u
[
n
]
=
1
{\displaystyle u[n]=1}
for
n
>=
0
{\displaystyle n>=0}
,
u
[
n
]
=
0
{\displaystyle u[n]=0}
for
n
<
0
{\displaystyle n<0}
δ
[
n
]
=
1
{\displaystyle \delta [n]=1}
for
n
=
0
{\displaystyle n=0}
,
δ
[
n
]
=
0
{\displaystyle \delta [n]=0}
otherwise
Signal,
x
[
n
]
{\displaystyle x[n]}
Z-transform,
X
(
z
)
{\displaystyle X(z)}
ROC
1
δ
[
n
]
{\displaystyle \delta [n]\,}
1
{\displaystyle 1\,}
all
z
{\displaystyle {\mbox{all }}z\,}
2
δ
[
n
−
n
0
]
{\displaystyle \delta [n-n_{0}]\,}
z
−
n
0
{\displaystyle z^{-n_{0}}\,}
z
≠
0
{\displaystyle z\neq 0\,}
3
u
[
n
]
{\displaystyle u[n]\,}
1
1
−
z
−
1
{\displaystyle {\frac {1}{1-z^{-1}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
4
−
u
[
−
n
−
1
]
{\displaystyle -u[-n-1]\,}
1
1
−
z
−
1
{\displaystyle {\frac {1}{1-z^{-1}}}}
|
z
|
<
1
{\displaystyle |z|<1\,}
5
n
u
[
n
]
{\displaystyle nu[n]\,}
z
−
1
(
1
−
z
−
1
)
2
{\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
6
−
n
u
[
−
n
−
1
]
{\displaystyle -nu[-n-1]\,}
z
−
1
(
1
−
z
−
1
)
2
{\displaystyle {\frac {z^{-1}}{(1-z^{-1})^{2}}}}
|
z
|
<
1
{\displaystyle |z|<1\,}
7
n
2
u
[
n
]
{\displaystyle n^{2}u[n]\,}
z
−
1
(
1
+
z
−
1
)
(
1
−
z
−
1
)
3
{\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
8
−
n
2
u
[
−
n
−
1
]
{\displaystyle -n^{2}u[-n-1]\,}
z
−
1
(
1
+
z
−
1
)
(
1
−
z
−
1
)
3
{\displaystyle {\frac {z^{-1}(1+z^{-1})}{(1-z^{-1})^{3}}}}
|
z
|
<
1
{\displaystyle |z|<1\,}
9
n
3
u
[
n
]
{\displaystyle n^{3}u[n]\,}
z
−
1
(
1
+
4
z
−
1
+
z
−
2
)
(
1
−
z
−
1
)
4
{\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
10
−
n
3
u
[
−
n
−
1
]
{\displaystyle -n^{3}u[-n-1]\,}
z
−
1
(
1
+
4
z
−
1
+
z
−
2
)
(
1
−
z
−
1
)
4
{\displaystyle {\frac {z^{-1}(1+4z^{-1}+z^{-2})}{(1-z^{-1})^{4}}}}
|
z
|
<
1
{\displaystyle |z|<1\,}
11
a
n
u
[
n
]
{\displaystyle a^{n}u[n]\,}
1
1
−
a
z
−
1
{\displaystyle {\frac {1}{1-az^{-1}}}}
|
z
|
>
|
a
|
{\displaystyle |z|>|a|\,}
12
−
a
n
u
[
−
n
−
1
]
{\displaystyle -a^{n}u[-n-1]\,}
1
1
−
a
z
−
1
{\displaystyle {\frac {1}{1-az^{-1}}}}
|
z
|
<
|
a
|
{\displaystyle |z|<|a|\,}
13
n
a
n
u
[
n
]
{\displaystyle na^{n}u[n]\,}
a
z
−
1
(
1
−
a
z
−
1
)
2
{\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}
|
z
|
>
|
a
|
{\displaystyle |z|>|a|\,}
14
−
n
a
n
u
[
−
n
−
1
]
{\displaystyle -na^{n}u[-n-1]\,}
a
z
−
1
(
1
−
a
z
−
1
)
2
{\displaystyle {\frac {az^{-1}}{(1-az^{-1})^{2}}}}
|
z
|
<
|
a
|
{\displaystyle |z|<|a|\,}
15
n
2
a
n
u
[
n
]
{\displaystyle n^{2}a^{n}u[n]\,}
a
z
−
1
(
1
+
a
z
−
1
)
(
1
−
a
z
−
1
)
3
{\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}
|
z
|
>
|
a
|
{\displaystyle |z|>|a|\,}
16
−
n
2
a
n
u
[
−
n
−
1
]
{\displaystyle -n^{2}a^{n}u[-n-1]\,}
a
z
−
1
(
1
+
a
z
−
1
)
(
1
−
a
z
−
1
)
3
{\displaystyle {\frac {az^{-1}(1+az^{-1})}{(1-az^{-1})^{3}}}}
|
z
|
<
|
a
|
{\displaystyle |z|<|a|\,}
17
cos
(
ω
0
n
)
u
[
n
]
{\displaystyle \cos(\omega _{0}n)u[n]\,}
1
−
z
−
1
cos
(
ω
0
)
1
−
2
z
−
1
cos
(
ω
0
)
+
z
−
2
{\displaystyle {\frac {1-z^{-1}\cos(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
18
sin
(
ω
0
n
)
u
[
n
]
{\displaystyle \sin(\omega _{0}n)u[n]\,}
z
−
1
sin
(
ω
0
)
1
−
2
z
−
1
cos
(
ω
0
)
+
z
−
2
{\displaystyle {\frac {z^{-1}\sin(\omega _{0})}{1-2z^{-1}\cos(\omega _{0})+z^{-2}}}}
|
z
|
>
1
{\displaystyle |z|>1\,}
19
a
n
cos
(
ω
0
n
)
u
[
n
]
{\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}
1
−
a
z
−
1
cos
(
ω
0
)
1
−
2
a
z
−
1
cos
(
ω
0
)
+
a
2
z
−
2
{\displaystyle {\frac {1-az^{-1}\cos(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}
|
z
|
>
|
a
|
{\displaystyle |z|>|a|\,}
20
a
n
sin
(
ω
0
n
)
u
[
n
]
{\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}
a
z
−
1
sin
(
ω
0
)
1
−
2
a
z
−
1
cos
(
ω
0
)
+
a
2
z
−
2
{\displaystyle {\frac {az^{-1}\sin(\omega _{0})}{1-2az^{-1}\cos(\omega _{0})+a^{2}z^{-2}}}}
|
z
|
>
|
a
|
{\displaystyle |z|>|a|\,}
![{\displaystyle x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\displaystyle \delta [n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71bf83f188a8d5c544e4565e7690b4029baed424)
![{\displaystyle \delta [n-n_{0}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37420bd43ae13a5c7f300b7e74cae6f08d38929a)
![{\displaystyle u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e693a2911b29e6c8d440d97e46d27760559af7c5)
![{\displaystyle -u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffe6af41d5e73ca723916ce158e69325ab37f0f)
![{\displaystyle nu[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/666189dd08672bcd9366962001fdeb012ef48db6)
![{\displaystyle -nu[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c7bfd00539cf805ba91e15a60b73576194dbd1)
![{\displaystyle n^{2}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556ff0a974378872385484fbeadf8bee8da04e97)
![{\displaystyle -n^{2}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc9d247970a92c7c6a69da9b5a272190dadcd24)
![{\displaystyle n^{3}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0458b9d3eaa2c032ea05c85f0b2dc59809e61815)
![{\displaystyle -n^{3}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66cd02f765990cd569d2cc47dd44cc75bfad0a3)
![{\displaystyle a^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec08f4507612bf171a306727ec543e8b22909a8b)
![{\displaystyle -a^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf8f7594a9aa188a37767ad6cbbcd1c932f5408c)
![{\displaystyle na^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b15d6e4309b253c58fdf105b270217869762ad)
![{\displaystyle -na^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a7df946d3646eb5c94313eb414edf4c9070d782)
![{\displaystyle n^{2}a^{n}u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc9a83f632b9781ec66653e486b9f77fafe3bb0)
![{\displaystyle -n^{2}a^{n}u[-n-1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85083b09d1b7c73d32bdfcc120bd85c56504b66)
![{\displaystyle \cos(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07538580111cb3bee33a686fa3970039105fee57)
![{\displaystyle \sin(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a4e25fc75ace79158f75ab06339a9b87ce3665)
![{\displaystyle a^{n}\cos(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a451552ec134261501b282063fca173b531a0ba)
![{\displaystyle a^{n}\sin(\omega _{0}n)u[n]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7ec057f4c8c7c307f3a58a9091e6e03d91f01d)
![{\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)
![{\displaystyle a_{1}x_{1}[n]+a_{2}x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb707cd1570b5004afa6306d8e349d36efffb58)
![{\displaystyle x[n-k]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd4e132f4bd95a617de0e6eb00ba09e2801e1e0)
![{\displaystyle a^{n}x[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/654e28509ebf86a5253c1e79e7a6b12fa1087466)
![{\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 \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)
![{\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)
![{\displaystyle nx[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9d129545d7bb8f225e3d3d6cf29828055b2aa0)
![{\displaystyle x_{1}[n]*x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a88f085e632f498b5ad7cae9fc45379ab931ff7)
![{\displaystyle r_{x_{1},x_{2}}(l)=x_{1}[l]*x_{2}[-l]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/29460269c4171c649511cb92c01422b1b4e4dbac)
![{\displaystyle x_{1}[n]x_{2}[n]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b8ee2f74b87c0783a1e479136a290c947ff945)
![{\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)
![{\displaystyle u[n]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3405c534c1af1bf479ff017d87682729722321)
![{\displaystyle u[n]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75f0ec7d52e8703689ee40dc9655fbecb74265b2)
![{\displaystyle \delta [n]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4953a952218d3ccd27c2a5797e4b2d0c7eb08b39)
![{\displaystyle \delta [n]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2fb443c0e773350512b296fea00e317bbcf0d7d)