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In mathematics and signal processing , the advanced z-transform is an extension of the z-transform , to incorporate ideal delays that are not multiples of the sampling time . The advanced z-transform is widely applied, for example, to accurately model processing delays in digital control . It is also known as the modified z-transform .
It takes the form
F
(
z
,
m
)
=
∑
k
=
0
∞
f
(
k
T
+
m
)
z
−
k
{\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}
where
T is the sampling periodm (the "delay parameter") is a fraction of the sampling period
[
0
,
T
]
.
{\displaystyle [0,T].}
Properties
If the delay parameter, m , is considered fixed then all the properties of the z-transform hold for the advanced z-transform.
Linearity
Z
{
∑
k
=
1
n
c
k
f
k
(
t
)
}
=
∑
k
=
1
n
c
k
F
k
(
z
,
m
)
.
{\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F_{k}(z,m).}
Time shift
Z
{
u
(
t
−
n
T
)
f
(
t
−
n
T
)
}
=
z
−
n
F
(
z
,
m
)
.
{\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}
Damping
Z
{
f
(
t
)
e
−
a
t
}
=
e
−
a
m
F
(
e
a
T
z
,
m
)
.
{\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}
Time multiplication
Z
{
t
y
f
(
t
)
}
=
(
−
T
z
d
d
z
+
m
)
y
F
(
z
,
m
)
.
{\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}
Final value theorem
lim
k
→
∞
f
(
k
T
+
m
)
=
lim
z
→
1
(
1
−
z
−
1
)
F
(
z
,
m
)
.
{\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).}
Example
Consider the following example where
f
(
t
)
=
cos
(
ω
t
)
{\displaystyle f(t)=\cos(\omega t)}
F
(
z
,
m
)
=
Z
{
cos
(
ω
(
k
T
+
m
)
)
}
=
Z
{
cos
(
ω
k
T
)
cos
(
ω
m
)
−
sin
(
ω
k
T
)
sin
(
ω
m
)
}
=
cos
(
ω
m
)
Z
{
cos
(
ω
k
T
)
}
−
sin
(
ω
m
)
Z
{
sin
(
ω
k
T
)
}
=
cos
(
ω
m
)
z
(
z
−
cos
(
ω
T
)
)
z
2
−
2
z
cos
(
ω
T
)
+
1
−
sin
(
ω
m
)
z
sin
(
ω
T
)
z
2
−
2
z
cos
(
ω
T
)
+
1
=
z
2
cos
(
ω
m
)
−
z
cos
(
ω
(
T
−
m
)
)
z
2
−
2
z
cos
(
ω
T
)
+
1
.
{\displaystyle {\begin{aligned}F(z,m)&={\mathcal {Z}}\left\{\cos \left(\omega \left(kT+m\right)\right)\right\}\\&={\mathcal {Z}}\left\{\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right\}\\&=\cos(\omega m){\mathcal {Z}}\left\{\cos(\omega kT)\right\}-\sin(\omega m){\mathcal {Z}}\left\{\sin(\omega kT)\right\}\\&=\cos(\omega m){\frac {z\left(z-\cos(\omega T)\right)}{z^{2}-2z\cos(\omega T)+1}}-\sin(\omega m){\frac {z\sin(\omega T)}{z^{2}-2z\cos(\omega T)+1}}\\&={\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}.\end{aligned}}}
If
m
=
0
{\displaystyle m=0}
F
(
z
,
m
)
{\displaystyle F(z,m)}
F
(
z
,
0
)
=
z
2
−
z
cos
(
ω
T
)
z
2
−
2
z
cos
(
ω
T
)
+
1
,
{\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}},}
which is clearly just the z -transform of
f
(
t
)
{\displaystyle f(t)}
References
![{\displaystyle [0,T].}](/media/api/rest_v1/media/math/render/svg/d49a2b0474d5ee6d0e1967879a5489d3978f828c)
