数字信号处理-2

b0 b1 z 1 X ( z) a0 a1 z 1
If M N , then
N bM z M Ak N 1 aN z 1 d z k 1 k
b0 b1 z 1 X ( z) a0 a1 z 1 BM N z
M N
ideal sampling
* Relationship between Laplace-transform and continuous time Fourier transform * Relationship between Z-transform and Laplace-transform
* Relationship between Z-transform and discrete time Fourier transform
2.4 Residues
x(n) Res[ X ( z ) z n 1 , ak ]
k 1 M
N
or
x(n) Res[ X ( z ) z n 1 , bk ] Res[ X ( z ) z n 1 , ]
k 1
Where poles ak (k=1,2, ... ,N) are inside the counterclockwise contour C, poles bk (K=1,2, … ,M) are outside the C. Example: Determine the inverse z-transform of the following z-transform.
2.1 Power series method
Example: Determine the inverse z-transform of the following z-transform.
z X ( z) , z 0.5
z 0.5
2.2 Partial fraction expansion
Discussion:
Example 2: Determine the z-transform and the ROC of the following sequence.
x(n) (n 1) 2 (n) (n 1) (n 3)
Solution:
Z [ x(n)] x(n) z n
(1) Finite length sequence
X ( z ) x ( n) z n
n n1
n2
when n1 0
0 z
when n2 0
0 z
when n1 0, n2 0 0 z
ROC : 0 | z |
(2) Causal sequence

n
n
a nu n z n

a n u n 1z n a z
n n n
a n z n
n 0
a 1 z
n
1


n
az 1
n 0


n
a 1 z
n 1


n
az 1
bM z M aN z N Ak B1 z B0 1 1 d z k 1 k
1 N

Example: Determine the inverse z-transform of the following z-transform.
z 2 0.5 z X ( z) , z 0.5
x(n m) z X ( z)
ROC:
m
(3) Convolution
Convolution in the time domain becomes multiplication in the z-domain.
Other properties of the z-transform
4 Relationships between Z-transform, Fouriertransform and Laplace-transform
z 2 0.5 z X ( z) , z 0.5
z 0.5
3 Properties of Z-transform
Basic properties of z-transform:
(1) Linearity The z-transform of a linear combination of signals is equal to the linear combination of z-transforms.
z 0.5
2.3 Long Division Method
The basic idea in this method is as follows: Given a z-transform X(z) with its ROC, we can expand, in the ROC, this function into a power series of the form
ax(n) by(n) aX ( z ) bY ( z )
ROC:
(2) Delay The delay property states that the effect of delaying a signal by m sampling units is equivalent to multiplying its z-transform by a factor z-m, where m is an integer.
The z-transform of the sampled signal
is
ze
sT
s j Assume j z r e
r e , then T
Z-S relationship Point mapping
T
4.2 Relationship transform (DTFT)
Chapter 2
Z-transform
* Definition and the region of convergence * Inverse z-transform * Properties of z-transform * Relationships between z-transform, Fouriertransform and Laplace-transform * Transfer function
between
Z-transform and
Fourier-
ze
j
5 Transfer function
5.1 Definition
H ( z ) Z [h(n)]
With zero initial condition
4.1 Relationship between Z-transform and Laplacetransform
In working with linear time-invariant continuous-time systems, we are accustomed to using the s-plane in investigating various system properties. It will be helpful in the study of discrete-time systems if those s-plane properties of continuous-time systems can be transformed directly to the z-plane. In our course, a class of digital filters is designed using such transformations between two planes.
z za 1 or z a
z za a 1 z 1 or z a
Without the knowledge of the ROC, there is no unique relationship between a sequence and its z-transform.
X ( z ) x ( n) z
n 0

n
ROC:
Rx | z |
(3) Anti-causal sequence
X ( z)
ROC:
n
n x ( n ) z
1
0 | z | Rx
(4) Others
*double-side sequence
causal sequence + anticausal sequence
It is the most practical approach. When X(z) is a rational function of z-1 , it can be expressed as a sum of simpler terms using the partial fraction expansion. Then the individual sequences corresponding to these terms can be written down using the z-transform table.
where the variable z is complex.
z re
j
1.2 Region of convergence (收敛域)
For a given sequence x(n), the set r of values of z for which its z-transform X(z) converges is called the region of convergence (ROC). The necessary and sufficient condition is:
n 1
3
z 2 z 1 z 3
ROC : 0 z
2 Inverse Z-transform
x(n) Z [ X ( z )]
1
1 2 j

C
X ( z ) z n 1dz
C is counterclockwise contour encircling the origin and lying in the ROC. Power series method (幂级数法) Partial fraction expansion (部分分式展开法) Residues（留数法）
Chapter 2
1.1 Definition
Z-transform
1 Definition and the region of convergence
The z-transform of a discrete sequence x(n), expressed as X(z), is defined as
Example:
x(n) a nu (n); x(n) a nu (n 1)
x(n) a nu(n);
X z
n n x n z
x(n) a nu(n 1)
X z
n
xn z
n 1
Solution:
X 1 z
n
x1 nz n
n n n 0

X 2 z
n 1
x nz
2

n
a z z za z a
b n z n
n
z za zb
X z Z[ x1 (n) x2 (n)]
*right-side sequence *left-side sequence
z-transform
Example 1: Determine the z-transform and the ROC of the following sequence.
an , n wenku.baidu.com 0 x ( n) n b , n 1