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高级Z变换 .
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高级Z转换 (英语:Advanced z-transform,或 modified z-transform)是Z转换 的延伸,是数学 及信号处理 领域中的工具,它将不是取样周期整数倍的延迟考虑进去。具有以下形式
F
(
z
,
m
)
=
∑
k
=
0
∞
f
(
k
T
+
m
)
z
−
k
{\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k))
其中
m为延迟参数(delay parameter),
0
≤
m
<
T
{\displaystyle 0\leq m<T}
性质
如果延迟参数m固定,则Z转换具有的性质在高级Z转换也都成立。
线性
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)}
时移
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)}
Z域的尺度性质
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)}
微分
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)}
终值定理
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)}
参考文献
Eliahu Ibrahim Jury, Theory and Application of the z-Transform Method , Krieger Pub Co, 1973. ISBN 0-88275-122-0 .
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