信号与系统-课件-(第三版)郑君里 (4)

n
The region of z for X(z) to converge in zplane be called ROC.
School of Computer Science and Information
NOTICE:
j n ROC z re at which | x( n) r | n
ROC : | z || e j | 1
School of Computer Science and Information
4.4 The calculation for inverse Z-transform
1. Compute by Fraction expansion
Example
z2 If X ( z ) 2 then x( n) ? ( ROC : | z | 1) z 1.5 z 0.5 Solution :
X ( z ) x ( n) z n
n 0
School of Computer Science and Information
4.2 Region of Convergence (ROC)
Condition of X(z) to converge:
n | x ( n ) z |
i
n 1 Res [ F ( z ) z , pi in encircle C ]
Res [ F ( z ) z n1 , pi out of encircle C ]
i
When n≥0 and n ＜ 0, the calculation should be done separately.
School of Computer Science and Information
4. If x(n) is a two-sided sequence
X (z)
n
x ( n) z

n

n
x ( n) z
1
n
x ( n) z
n 0

n
School of Computer Science and Information
When n<N1, x(n)=0. then:

X (z)
n N 1
n x ( n ) z
School of Computer Science and Information
When N1>0
n
| x ( n) z

n
| x ( n) | | n n N 1 | z |
Re s [ F ( z ) z n1 , pi ]z pi 1 d r n 1 [( z pi ) F ( z ) z ]z pi r 1 ( r 1)! dz
r 1
School of Computer Science and Information
Especially:
When the ROC of F(z) is a circle or cirque, There will be lef-sided sequence in f (n). Then:
2 j C
i
1
F ( z ) z n1dz Res [ F ( z ) z n1 , all pi ] 0
i
School of Computer Science and Information
When pi is 1 order pole:
Re s [F ( z ) z n1 , pi ]z pi [( z pi ) F ( z ) z n1 ]z pi
When pi is r order pole:

School of Computer Science and Information
Conclusion:
If the value of z is large enough, the condition is allowed.
School of Computer Science and Information
Theory:
For then f ( n) F ( z ) f ( n) 2 j C 1 F ( z ) z n1dz
The encircle C is inside the ROC and surround the origin.
So f ( n) Re s[ F ( z ) z n1 , pi ]z pi
School of Computer Science and Information
4.3 Familiar Z-Tranform And ROC
1 1. ( n) 0
ZT [ ( n)]
( n 0) ( n 0)
n
( n) z

n
1
ROC : all z plane
ROC : | z | a
School of Computer Science and Information
4. cos( n)u( n);
Biblioteka Baidu e
j n
sin( n)u( n)
u(n) cos( n)u(n) j sin( n)u(n)
jω n
ZT [e
u(n)]
z ze
X (z)
n N 1 n x ( n ) z N2
When N1＜0, N2＞0
School of Computer Science and Information
N 1 0 z n ||z|
N 2 0 z n ||z|0
Conclusion:
jω
z( z cos ω) z sin ω 2 j 2 z 2 z cos ω 1 z 2 z cos ω 1
ROC : | z || e j | 1
School of Computer Science and Information
z ( z cos ω) ZT [cos(ωn)u( n)] 2 z 2 z cos ω 1 z sin ω ZT [sin( ωn)u( n)] 2 z 2 z cos ω 1
School of Computer Science and Information
3. If x(n) is a left-sided sequence
When n>N2, x(n)=0. then:
N2
X (z)
n
n x ( n ) z
School of Computer Science and Information
z2 2z z X (z) ( z 1)( z 0.5) z 1 z 0.5
x( n) [2 0.5 ]u( n)
n
School of Computer Science and Information
2. Integration of complex function by equation
School of Computer Science and Information
1 2. u( n) 0
ZT [u( n)]
( n 0) ( n 0)
n
u(n) z
n 0

n
u( n) z
n
z z 1
ROC : | z | 1
School of Computer Science and Information
When N2>0
N2 0 z
n
||z| 0
School of Computer Science and Information
Conclusion:
If the value of z is small enough, the condition is allowed, except possibly at z=0.
ROC : | z |
School of Computer Science and Information
When N1≥0, N2>0
School of Computer Science and Information
N 1 0,
Conclusion:
N 2 0 z n ||z|0
If x(n) is of finite duration, then the ROC is the entire z-plane. Except possible at z=0 and/or z=∞.
ROC : 0 | z |
School of Computer Science and Information
School of Computer Science and Information
n a n 3. a u( n) 0
( n 0) ( n 0)
ZT [a u( n)]
n
n
a

n
u( n) z
1 n
n
z (a z ) za n 0
If x(n) is of finite duration, then the ROC is the entire z-plane. Except possible at z=0.
ROC : | z | 0
School of Computer Science and Information
2. If x(n) is a right-sided sequence
When N1＜0, N2≤0
School of Computer Science and Information
N 1 0, N 2 0 z n ||z|
Conclusion:
If x(n) is of finite duration, then the ROC is the entire z-plane. Except possible at z=∞.
Conclusion:
The first is left-sided sequence. Its ROC is |z|<R2 The second is right-sided sequence. Its ROC is |z|>R1 When R1<R2, then ROC is R1<|z|<R2 When R1>R2, then ROC is empty.
ROC depends only on r=|z| , just like the ROC in s-plane only on Re(s) .
School of Computer Science and Information
1. If x(n) is of finite duration
If x(n) has non-zero value while N1≤n≤N2, then:
School of Computer Science and Information
4.1 Definition of Z-Transform
Double-Sides Z-Transform:
X (z)
n
x ( n) z


n
Single-Side Z-Transform:
When N1<0
N1 0 z
n
||z|
School of Computer Science and Information
Conclusion:
If the value of z is large enough, the condition is allowed except possibly at z=∞.
When N2<0
n
n | x ( n ) z |

n
n | x ( n ) z |
N2
School of Computer Science and Information
Conclusion:
If the value of z is small enough, the condition is allowed.