数字信号处理Ch3. The z-transform

M
∑∑ X ( z) =
P(z) Q(z)
=
bk z − k
k =0
N
ak z−k
k =0
(3.37)
Obtain an alternative expression for X(z) as a sum of simpler terms, each of which is tabulated.
The inverse z-transforms of each simple term is identified (from table 3.1) and summarized to get the desirable result x[n].
ROC of H(z) includes the unit circle (|z|= 1)
Causality: h[n] = 0, n < 0 (2.70) right-sided
ROC of H(z) : max{ pk } < z ≤ ∞
Stable and Causal:
ROC of H(z) : max{ pk }< z ≤ ∞, max{ pk }<1
Methods of determining the inverse z-transform z Inspection method z Partial fraction expansion z Power series expansion
15
3.3.1 Inspection method
The z-transform pairs in table 3.1 are invaluable in applying the inspection method
pi < z < p j ( pi < p j )
11
Stability, Causality and the ROC
H (z) = Z{h[n]} —— System Function
∞
Stability: ∑ h[n] < ∞ (2.65) n=−∞
The FT of h[n] exists.
09:06:41
Ch3. The z-transform
3.0 Introduction
z-transform for D-T signals
Laplace transform for C-T signals
The Fourier transform does not converge for all sequences, so it is useful to have a generalization of it.
7
ROC of Finite-duration sequences
x[n]
=
⎧≠ ⎨⎩=
0, 0,
N1 ≤ n ≤ N otherwise
2
N2
∑ X (z) = x[n]z−n n= N1
X(z) 是有限项级数和，只要级数的每一项有界，级 数就收敛，即要求：x[n] z −n < ∞ 由于x[n]有界，故要求：z −n < ∞
(3.8)
ROC: 0 ≤ Rg − < z < Rg + ≤ ∞
Rg + Rg −
5
Rational z-transform
X(z) is a rational function inside the ROC X (z) = P(z) Q(z)
zeros: the values of z for X(z)=0 poles: the values of z for X(z)=∞ The poles for finite values of z are the roots of Q(z). Poles may occur at z=0 or z=∞ . Important relationships exist between the locations
n=−∞
n= N1
n=−∞
正幂级数
0< z <∞
阿贝尔定理：存在一个收敛半径R，级数在以 原点为中心，以R为半径的圆内都绝对收敛。
0≤ z <R
0 < z < R ROC不能包含极点 0 < z < m in{ pk }
0≤
z
< min{ pk
}, N2
≤0 10
ROC of Two-sided sequences
An infinite-duration sequence extending from n = – ∞ to n = + ∞
two-sided sequence= right-sided sequence + left-sided sequence pi = max{ pRk } < z < ∞ 0 < z < min{ pLk } = pj
显然|z|在整个开域(0, ∞)都能满足以上条件，因此， 有限长序列的收敛域至少是除0和∞两个点以外的整 个z平面: 0 < z < ∞
若对N1, N2加以限制，收敛域可进一步扩大为包括0
点或∞点的半开域:
⎪⎧ 0 < ⎨⎪⎩0 ≤
z z
≤ ∞, N1 < ∞, N2
≥0 ≤0
8
ROC of Right-sided sequences
12
2
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Example 3.7
ROC: 1 < z < 2 2
—— The system is stable.
ROC: z > 2 —— The system is causal.
ROC: z < 1 2
—— The system is neither stable nor causal. For this specfic pole-zero plot, there is no ROC that
(3.39)
If M≥N, and the poles are all first order,
∑ ∑ X
(z)
=
M −N r=0
Br
z−r
+
N k =1
1−
Ak dk
z−1
(3.43)
Br ——Long division Ak = (1− dk z−1)X(z) z=dk (3.41)
3.3.2 Partial fraction expansion
R< z ≤∞
X(z)的极点
R < z < ∞ ROC不能包含极点
max{ pk } < z < ∞
max{
pk
}<
z
≤ ∞,
N1
≥0 9
ROC of Left-sided sequences
x[n]
=
⎧≠ ⎨⎩ =
0, 0,
n n
≤ >
N2 N2
N2
N2
N1 −1
∑ ∑ ∑ X (z) = x[n]z−n = x[n]z−n + x[n]z−n (N1 ≤ 0)
3.3.2 Partial fraction expansion
M
∏ X ( z ) = b0 ∏ a 0
(1 − ck z −1 )
k =1
N
(1 − d k z −1 )
k =1
(3.39)
If M<N, and the poles are all first order,
∑ X
(z)
=
N
k =1 1 −
For any given sequence, the set of values of z for which the ZT converges, is call the region of convergence (ROC).
ZT convergence condition:
∞
∑ X (z) = x[n] z −n < ∞ n = −∞
x[n]
=
⎧≠
⎨ ⎩
=
0, 0,
n n
≥ <
N1 N1
j Im(z)
z-plane
a b c Re (z)
∞
N2
∞
∑ ∑ ∑ X (z) = x[n]z−n = x[n]z−n +
x[n]z−n (N2 ≥ 0)
n = N1
n = N1
n=N2 +1
负幂级数
0< z <∞
阿贝尔定理：存在一个收敛半径R，级数在以 原点为中心，以R为半径的圆外都绝对收敛。
of poles and the ROC. p104, table 3.1: Some common z-transform pairs
6
1
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3.2 Properties of Region of Convergence
The ROC is a ring or disk in the z-plane centered at the origin: 0 ≤ Rg− < z < Rg+ ≤ ∞
would imply the system is both stable and causal. 13
Importance of ROC
ROC of ZT is an important concept. Without the knowledge of the ROC, there is no
n = −∞ ∞