数字信号处理a(双语)chapter 6-z transform b

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6.4.3 Inverse z-Transform by Partial-Fraction Expansion
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6.4.3 Inverse z-Transform by Partial-Fraction Expansion
Solutions: Step 1-- Converting G(z) into the form of
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6.4.1 General Expression
●Recall that, for z= re jω, the z-transform G(z)
given by
G(z) G(re j ) g[n]r ne jn n
is merely the DTFT of the modified sequence g(n)r-n ● Accordingly, the inverse DTFT is thus given by
Chapter 6 z-Transform
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Part B: Inverse ZT and ZT Theorems
6.4 The Inverse z-Transform 6.5 z-Transform Theorems
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6.4 The Inverse z-Transform
proper fractions by long division Step 2-- Summing the inverse transform of
the individual simpler terms in the expansion
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6.4.3 Inverse z-Transform by Partial-Fraction Expansion
z=0 in the ROC of G(z)
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6.4.1 General Expression
逆z变换是一个对X(Z)Zn-1进行的围线积分，积 分路径C是一条在X(Z)收敛环域（Rx-，Rx+） 内以逆时针方向绕原点一周的单围线。
jI m[z]
|z |＝ R x ＋
c
|z |＝ R x －
g[n] Z 1[G(z)] 1 G(z)zn1dz
2j c
where c is a counterclockwise contour of integration
defined by |z|= r
But the integral remains unchanged when c is
replaced with any contour c’ encircling the point
• 一阶极点的留数
x (n ) N R e s [X (z )z n 1 ;z k ] N [(z z k )X (z )z n 1 ]
k 1
k 1
z z k
• N 阶极点的留数
R es[X (z)zn 1 ;zk](N 1 1 )!d d zN N 1 1[(z zk)N X (z)zn 1 ]
6.4.1 General Expression 6.4.2 Inverse z-Transform by Table Look-Up
Method 6.4.3 Inverse z-Transform by Partial-Fraction
Expansion 6.4.4 Partial-Fraction Using MATLAB 6.4.5 Inverse z-Transform via Long Division 6.4.6 Inverse z-Transform Using MATLAB
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6.4.1 General Expression
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6.4.1 General Expression
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6.4.2 Table Look-up Method
Example 6.12 H (z)z20 z.5 z0.25 ,z0.5
H(z)z 0.0 5.z521 00 .5 .5 zz1 12
Look up Table 6.1 on Page 253
nn[n]
z1
1z1
2
z
h[n]n0.5n[n]
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6.4.3 Inverse z-Transform by Partial-Fraction Expansion
A rational z-transform G(z) with a causal inverse transform g(n) has an ROC that is exterior to a circle
Here it is more convenient to express G(z) in a partial-fraction expansion form and then determine g(n) by summing the inverse transform of the individual simpler terms in the expansion
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1.1 General Expression
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6.4.1 General Expression
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6.4.1 General Expression
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6.4.1 General Expression
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6.4.1 General Expression
g[n]r n 1 G(re j )e jnd
2
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6.4.1 General Expression
By making a change of variable z re j , the
previous equation can be converted into a contour integral given by
o
R e [z]
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6.4.1 General Expression
直接计算围线积分比较麻烦，一般不采用此 法求逆z变换，求解逆z变换的常用方法有： 留数定律法 查表法 部分分式法★ 长除法
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6.4.1 General Expression
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利用留数定理计算围线积分