数字信号处理双语-Z变换.
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• In general, the region of convergence R of a ztransform for a sequence x[n] is an annular region环形区域 of the z-plane（z平面）. proof
R1 | z | R2, where 0 R1 R2
3
3.1 The Z transform Z变换
Considering a LTI system,when theinput is an
exponential function, x[n] zn , then theoutput is
y[n] x[n - k]h[k] z(n-k) h[k]
3.5 Summary
1
Homework
• pp. 127-131 • 3.1 b f g • 3.2 • 3.6 b c
• 3.8 • 3.19 b • 3.20 a
2
3.0 Introduction
• Advantages of Z transform – It suits for more sequence analysis than Fourier transform. For many cases, we could have Z transforms for sequences when their Fourier transfroms do not exsist. – It is more convenient than Fourier transform in many analytical problem.
n
Where z=Re{z}+jIm{z} is a complex variable.
• Notation:
X(z) exists only when the summation converges. So X(z) is only defined for the regions of the complex z plane in which the summation is on the right convergence.正确收敛.
10
• Alternately，it can be rewritten in factor form
• H(z) is called as the transfer function传递函数 of system.
5
Z transform of sequence
• For a given sequence x[n], its z-transform is
defined as
X (z) Z{x[n]} x[n]zn
k-
k-
来自百度文库
znz-kh[k] zn h[k]z-k x[n]H (z)
k-
k-
Where H(z) h[k]z-k.
k
We call H (z) as the Z transformfor sequenceh[n].
4
If theinput zn is a complexsinusoid with
7
Figure of ROC R1 < |z| < R2
z=Re{z}+jIm{z} Im{z}
R2 R1
Re{z}
8
Example 3.1.1
Compute the z transform and specify
the ROC of sequence x[n] [n].
• Solution:it is a finite length sequence. its z transform is
Z[n] [n]zn z0 1 n and its ROC is 0 z , or we can say all z plane is its ROC. 9
Zeros and Poles of Rational z-transforms 有理Z变换的零点和极点
• In the case of LTI discrete-time system with that we are concerned in this course, all pertinent相关 的 z-transforms are rational functions of z-1, we
call them as Rational z-transforms. A rational z-
transform could be written as a ratio of two polynomials多项式 in z-1:
X
(z)
N(z) D(z)
n0 d0
n1z1 ... nM zM d1z1 ... dN zN
6
• The set R of values of z for which its z-transform converges is called the Region Of Convergence (ROC)收敛域.
• X(z) converges if and only if
x[k]zk M
Chapter three the Z Transform Z 变换 3.0 Introduction
3.1 the Z transform Z变换 3.2 Properies of the Region Of Convergence for the Z transform收敛域 3.3 The Inverse z-Transform Z逆变换 3.4 Properties of the z-Transform Z变换的性质
frequency, that is, z e jω , then
H(z) H(e jω ) h[k ]e-jk H (e j ) e j () k -
y[n] H(z)x[n] H(e jω )e jn |H(e jω )|e j[n ()]
• Then we get Fourier transform H (e j ) of h[n]. • H (e j ) is system’s frequency response.
• In general, the region of convergence R of a ztransform for a sequence x[n] is an annular region环形区域 of the z-plane（z平面）. proof
R1 | z | R2, where 0 R1 R2
3
3.1 The Z transform Z变换
Considering a LTI system,when theinput is an
exponential function, x[n] zn , then theoutput is
y[n] x[n - k]h[k] z(n-k) h[k]
3.5 Summary
1
Homework
• pp. 127-131 • 3.1 b f g • 3.2 • 3.6 b c
• 3.8 • 3.19 b • 3.20 a
2
3.0 Introduction
• Advantages of Z transform – It suits for more sequence analysis than Fourier transform. For many cases, we could have Z transforms for sequences when their Fourier transfroms do not exsist. – It is more convenient than Fourier transform in many analytical problem.
n
Where z=Re{z}+jIm{z} is a complex variable.
• Notation:
X(z) exists only when the summation converges. So X(z) is only defined for the regions of the complex z plane in which the summation is on the right convergence.正确收敛.
10
• Alternately，it can be rewritten in factor form
• H(z) is called as the transfer function传递函数 of system.
5
Z transform of sequence
• For a given sequence x[n], its z-transform is
defined as
X (z) Z{x[n]} x[n]zn
k-
k-
来自百度文库
znz-kh[k] zn h[k]z-k x[n]H (z)
k-
k-
Where H(z) h[k]z-k.
k
We call H (z) as the Z transformfor sequenceh[n].
4
If theinput zn is a complexsinusoid with
7
Figure of ROC R1 < |z| < R2
z=Re{z}+jIm{z} Im{z}
R2 R1
Re{z}
8
Example 3.1.1
Compute the z transform and specify
the ROC of sequence x[n] [n].
• Solution:it is a finite length sequence. its z transform is
Z[n] [n]zn z0 1 n and its ROC is 0 z , or we can say all z plane is its ROC. 9
Zeros and Poles of Rational z-transforms 有理Z变换的零点和极点
• In the case of LTI discrete-time system with that we are concerned in this course, all pertinent相关 的 z-transforms are rational functions of z-1, we
call them as Rational z-transforms. A rational z-
transform could be written as a ratio of two polynomials多项式 in z-1:
X
(z)
N(z) D(z)
n0 d0
n1z1 ... nM zM d1z1 ... dN zN
6
• The set R of values of z for which its z-transform converges is called the Region Of Convergence (ROC)收敛域.
• X(z) converges if and only if
x[k]zk M
Chapter three the Z Transform Z 变换 3.0 Introduction
3.1 the Z transform Z变换 3.2 Properies of the Region Of Convergence for the Z transform收敛域 3.3 The Inverse z-Transform Z逆变换 3.4 Properties of the z-Transform Z变换的性质
frequency, that is, z e jω , then
H(z) H(e jω ) h[k ]e-jk H (e j ) e j () k -
y[n] H(z)x[n] H(e jω )e jn |H(e jω )|e j[n ()]
• Then we get Fourier transform H (e j ) of h[n]. • H (e j ) is system’s frequency response.