From Wikipedia, the free encyclopedia
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 . 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 timem (the "delay parameter") is a fraction of the sampling time
[
0
,
T
)
.
{\displaystyle [0,T).}
It is also known as the modified Z-transform .
The advanced Z-transform is widely applied, for example to model accurately processing delays in digital control .
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
m
c
k
f
k
(
t
)
]
=
∑
k
=
1
m
c
k
F
(
z
,
m
)
.
{\displaystyle Z\left[\sum _{k=1}^{m}c_{k}f_{k}(t)\right]=\sum _{k=1}^{m}c_{k}F(z,m).}
Time shift
Z
[
u
(
t
−
n
T
)
f
(
t
−
n
T
)
]
=
z
−
n
F
(
z
,
m
)
.
{\displaystyle Z\left[u(t-nT)f(t-nT)\right]=z^{-n}F(z,m).}
Dampening
Z
[
f
(
t
)
e
−
a
t
]
=
e
−
a
m
F
(
e
a
T
z
,
m
)
.
{\displaystyle 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 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
k
=
1
+
F
(
z
,
m
)
.
{\displaystyle \lim _{k=\infty }f(kT+m)=\lim _{k=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
)
)
]
{\displaystyle F(z,m)=Z\left[\cos \left(\omega \left(kT+m\right)\right)\right]}
F
(
z
,
m
)
=
Z
[
cos
(
ω
k
T
)
cos
(
ω
m
)
−
sin
(
ω
k
T
)
sin
(
ω
m
)
]
{\displaystyle F(z,m)=Z\left[\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right]}
F
(
z
,
m
)
=
cos
(
ω
m
)
Z
[
cos
(
ω
k
T
)
]
−
sin
(
ω
m
)
Z
[
sin
(
ω
k
T
)
]
{\displaystyle F(z,m)=\cos(\omega m)Z\left[\cos(\omega kT)\right]-\sin(\omega m)Z\left[\sin(\omega kT)\right]}
F
(
z
,
m
)
=
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
{\displaystyle F(z,m)=\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}}}
F
(
z
,
m
)
=
z
2
cos
(
ω
m
)
−
z
cos
(
ω
(
T
−
m
)
)
z
2
−
2
z
cos
(
ω
T
)
+
1
.
{\displaystyle F(z,m)={\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}.}
If
m
=
0
{\displaystyle m=0}
F
(
z
,
m
)
{\displaystyle F(z,m)}
Z-transform
F
(
z
,
m
)
=
z
2
−
z
cos
(
ω
T
)
z
2
−
2
z
cos
(
ω
T
)
+
1
{\displaystyle F(z,m)={\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).}
See also
![{\displaystyle Z\left[\sum _{k=1}^{m}c_{k}f_{k}(t)\right]=\sum _{k=1}^{m}c_{k}F(z,m).}](/media/api/rest_v1/media/math/render/svg/b31265da3c0cf3461ecec601e8320964deffc1fd)
![{\displaystyle Z\left[u(t-nT)f(t-nT)\right]=z^{-n}F(z,m).}](/media/api/rest_v1/media/math/render/svg/25a2e13ad79318a8d8787e738817c372a6122d53)
![{\displaystyle Z\left[f(t)e^{-a\,t}\right]=e^{-a\,m}F(e^{a\,T}z,m).}](/media/api/rest_v1/media/math/render/svg/cb734c6c1932d5d7939212e5e25bb8a9b9740d41)
![{\displaystyle Z\left[t^{y}f(t)\right]=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}](/media/api/rest_v1/media/math/render/svg/98deb6d6979b8bbddd256c7c1b129e27f79164a8)
![{\displaystyle F(z,m)=Z\left[\cos \left(\omega \left(kT+m\right)\right)\right]}](/media/api/rest_v1/media/math/render/svg/26c7b515ce7a27f7919ba1915c293a397f0fdc88)
![{\displaystyle F(z,m)=Z\left[\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right]}](/media/api/rest_v1/media/math/render/svg/2caa4618bfec763cf1666fa5c092f4258663f9f0)
![{\displaystyle F(z,m)=\cos(\omega m)Z\left[\cos(\omega kT)\right]-\sin(\omega m)Z\left[\sin(\omega kT)\right]}](/media/api/rest_v1/media/math/render/svg/a1f57d62ed1048ede83f55e265ac717c5bf6af83)
