第十章讲义——Z变换

1 anu[n]←z→ 1 − az−1
z >a
Example 10.2
x[n] = −anu[−n − 1], determine z-transform
X ( z) = − ∑ anu[−n − 1]z−n
n=−∞ −1 +∞
= − ∑ az
n=−∞
(
−1
)
n
= −∑ a z
−1 n=1
+∞
(
)
n
if the X (z) is convergence so then
n
a-1z < 1 ⇒ z < a 1 z −a-1z X ( z) = = = 1- a-1z 1- az-1 z - a z <a
1 −a u[−n − 1]← → 1 − az−1
z
z <a
Example 10.3
Property 7 —— if the Z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity Property 8 —— if the Z-transform X(z) of x[n] is rational, and if x[n] is right sided, the ROC is the region in the z-plane outside the outermost pole ——i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). Furthermore,if x[n] is causal (i.e., if it is right sided and equal to 0 for n<0), then the ROC also includes z= ∞
Property 9 —— if the Z-transform X(z) of x[n] is rational, and if x[n] is left sided, the ROC is the region in the z-plane inside the innermost nonzero pole —— i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0. In particular,if x[n] is anticausal (i.e., if it is left sided and equal to 0 for n>0), then the ROC also includes z=0
Example 10.8
1 −1 (1 − z )(1 − 2z−1 ) 3 analysis the ROC of X (z) X ( z) = 1
Poles : Zeros :
z=
1 and z = 2 3 z1 = z2 = 0
Possible ROC: z >2 1 z< 3 1 < z <2 3
1 1 x[n] = 7 u[n] − 6 u[n], determine z-transform 3 2
n n
1 anu[n]←z→ 1 − az−1
1 1 z 3 u[n]← → 1 1 − z−1 3 n 1 1 z 2 u[n]← → 1 1 − z−1 2 7 6 X ( z) = − 1 −1 1 −1 1− z 1− z 3 2
Property 1 —— the ROC of X(z) consists of a ring in the z-plane centered about the origin
Property 2 —— the ROC does not contain any poles
Property 3 —— if x[n] is of finite duration, then the ROC is the entire z-plane, except possible z=0 and/or z= ∞
Fourier transform: h[ n] ←FT H(z) →
H(e ) =
jw +∞ n=−∞
z =e jw
h[n]e− jwn ∑
Z-transform:
X ( z) =
n=−∞
∑
+∞
x[n]z−n
x[n]←Z→ X (z)
The relationship between the Fourier transform and the Z-transform
F { x[n]} = X (z)
z =e jw
Fourier transform is a particular form of Z-transform If z = re jw
X (z) = X (re ) =
jw
n=−∞
∑
+∞
x[n](re jw )−n
=
n=−∞
{ x[n]r −n }e− jwn ∑
X ( z) = X ( re jw ) = F { x[n]r −n} =
n=−∞
∑
+∞
x[n]r −ne− jwn
⇒ x[n]r −n =
1 X (re jw )e jwndw 2π ∫2π n 1 jw jw x[n]= ⇒ ∫2π X ( re ) ( re ) dw 2π because dz = jre jwdw 1 so x[n] = X ( z)zn−1dz 2π j ∫
10.1 The Z-transform
The content of this section
Conception —— Z-transform
The relationship between Z-transform and Fourier transform
The representation of the Z-transform
n z
z >a
1 1 1 6 X ( z) = − 2 j 1 − 1 e jπ / 4 z−1 2 j 1 − 1 e− jπ / 4 z−1 3 3
1 z> 3
10.2 The region of convergence for the Z-transform
The characteristic of ROC The relationship between the ROC and the signals
10.3 The inverse Z-transform
The representation of the inverse Z-transform The two usual methods of determine the inverse Z-transform
The representation of the inverse Z-transform
Example 10.5
δ [n]← → ∑ δ [n]z−n = 1 ROC is entire z-plane
z n=−∞ +∞
z → δ [n − 1]← ∑ δ [n − 1]z−n = z−1 ROC is entire Z-plane exluding Z=0 n=−∞
+∞
δ [n + 1]← ∑ δ [n + 1]z−n = z ROC is entire Z-plane exluding Z=∞ →
2. Frequency-domain
Y (e jw ) = X e jw H e jw
( ) ( )
3. Z-domain
?
The contents of the chapter
The Z-transform and Inverse Z-transform √ The ROC for Z-transform √ Geometric evaluation of the Fourier transform from the Pole-Zero plot Properties of the Z-transform √ Analysis and characterization of LTI system √ using the Z-transform System function algebra and block diagram representations √ The unilateral Z-transform
z n=−∞
+∞
Property 4 —— if x[n] is right sided, and if the circle z = r0 is in the ROC, then all the finite values of z for which z > r0 will also be in the ROC Property 5 —— if x[n] is left sided, and if the circle z = r0 is in the ROC, then all the values of z for which 0 < z < r0 will also be in the ROC Property 6 —— if x[n] is two sided, and if the circle z = r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle z = r0