review for chapter two & three
DESCRIPTION
Review for chapter two & three. Review :periodicity of sequence. A sequence x [ n ] is defined to be periodic if and only if there is an integer N ≠0 such that x [ n ] = x [ n + N ] for all n . In such a case, N is called the period of the sequence. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/1.jpg)
Review for chapter two & three
![Page 2: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/2.jpg)
2
Review :periodicity of sequence• A sequence x[n] is defined to be periodic if and
only if there is an integer N≠0 such that x[n] = x[n + N] for all n. In such a case, N is called the period of the sequence.
• Note, not all discrete cosine functions are periodic.
– If 2π/ω is an integer (整数) or a rational number(有理数 ), this sequence will be periodic;
– If 2π/ω is an irrational number(无理数) , this cosine function will not be periodic at all.
kN
kωN
nNnn
2
2
)),(cos()cos(
Z
![Page 3: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/3.jpg)
3
Review: characteristics of discrete-time systemThe characteristics of the discrete-time system y(n) = H {x(n)} :
– Linearity: If y1(n)= H { x1(n)}, y2(n)= H { x2(n)},then H {ax(n)}=aH {x(n)} and H { x1(n)+ x2(n)}=H {x1(n)}+H {x2(n)} for any constants a and b.
– time invariance: If y (n)= H { x(n)},thenH { x(n-n0)}=y(n-n0)
– Causality: If, when x1(n) = x2(n) for n < n0, then H {x1(n)} = H {x2(n)}, for n < n0
– Stability: For every input limited in amplitude, the output signal is also limited in amplitude.
n
nh )(
![Page 4: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/4.jpg)
4
Review: LTI system• The output y(n) of a linear time-invariant
system can be expressed as
where h(n) = H {δ(n)} is the impulse response of the system.
• Two linear time-invariant systems in cascade form a linear time-invariant system with an impulse response which is the convolution sum of the two impulse responses.
)()()()()( nhnxknhkxnyk
)]()([)()( 21 nhnhnxny
![Page 5: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/5.jpg)
5
Review: FIR & IIR systems• A nonrecursive system such as
are often referred to as finite-duration impulse-response (FIR) filters.
• A recursive digital system such as
(at least one ai≠0)are often referred to as infinite-duration impulse-response (IIR) filters.
M
l
l lnxbny0
)()(
M
l
l
N
i
i lnxbinyany01
)()()(
![Page 6: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/6.jpg)
6
Review: z-transform• The z transform X(z) of a sequence x(n) is
defined as
where z is a complex variable. The z transform given by this equation is referred to as the two-sided z transform.
n
nznxnxzX ][]}[{)( Z
![Page 7: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/7.jpg)
7
Review: Convergence of the z transform
21
2121
02
02
01
01
0
1
10
][)(
0 0
0 0 ][)( 左边边序
0
0 ][)( 右边边序
0 0
0 0
0,0n 0
][)(
0
0
1
0
rr
rrrzrznxzX
nrz
nrzznxzX
nzr
nzrznxzX
nz
nz
nz
znxzX
n
n
n
n
n
nn
n
n
nn
n
无收敛域收敛域为双边序列
收敛域为
收敛域为
收敛域为有限长序列
![Page 8: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/8.jpg)
8
Review: H(z) for a stable causal system• For a causal system, its ROC should be |z|>r1.
• For a stable system, its ROC should include unit circle.
• So the convergence circle’s radius of the z transform of the impulse response of a stable causal system should be smaller than unit.
i.e. 11 r
![Page 9: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/9.jpg)
9
Review: Zeros and poles• An important class of z transforms are those for
which X(z) is a ratio of polynomials in z, that is
• The roots of the numerator polynomial (分子多项式) N(z) are those values of z for which X(z) is zero and are referred to as the zeros (零点) of X(z).
• Values of z for which X(z) is infinite are referred to as the poles (极点) of X(z). The poles of X(z) are the roots of the denominator polynomial(分母多项式) D(z).
)(
)()(
zD
zNzX
![Page 10: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/10.jpg)
10
Review: PolesX(z) is also can be expressed as
K
k
mk
kpz
zNzX
1
)(
)()( 因为 X(z) 在极点处无
意义,所以其收敛域一定不包括极点。
因为 X(z) 在极点处无意义,所以其收敛域一定不包括极点。
• Right-handed: The convergence region is |z| > r1, so all the poles must be inside the circle |z| = r1, r1 = max{|pk|}(figure (a);
• Left-handed: The convergence region is |z| < r2, all the poles must be outside the circle |z| = r2, r2 = min{|pk|} (figure (b);
• Two-sided sequences: The convergence region is r1 < |z| < r2, some poles are inside the circle |z| = r1 and the others outside the circle |z| = r2 (figure (c).
![Page 11: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/11.jpg)
11
Review: Figure 2.2 (a)
r1
Im{z}
Re{z}
![Page 12: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/12.jpg)
12
Review: Figure 2.2 (b)
r2
Im{z}
Re{z}
![Page 13: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/13.jpg)
13
Review: Figure 2.2 (c)
r1r2
Re{z}
Im{z}
![Page 14: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/14.jpg)
14
Review: The z transform of basic sequences
x(n) X(z) ROC
1
u(n)
anu(n)
anu(n-1)
nu(n)
nanu(n)
)(n z0
1z
z z1
az
z
za
az
a
2)1( z
z z1
2)( az
az
za
za
![Page 15: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/15.jpg)
15
Review: inverse z transform• For rational z transform, a partial-fraction expansion is
carried out firstly, and then the inverse z transforms of the simple terms are identified.
• If X(z) = N(z) / D(z) has K different poles pk, k = 1,…,K, each of multiplicity mk, then the partial-fraction expansion(部分分式展开法) of X(z) is
where M and L are the degrees of the numerator and denominator of X(z), respectively.
• The coefficients gl, l = 0,…, M – L, is the quotient (商)of polynomials N(z) and D(z). If M < L, then gl = 0 for any l .
K
k
m
ii
k
kiLM
l
ll
k
pz
czgzX
1 10 )()(
![Page 16: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/16.jpg)
16
Review: Partial-fraction expansion• The coefficients cki are given by
Particularly, in the case of a simple pole (单极点) , cki is given by
Since the z transform is linear and the inverse z transform of each of the terms is easy to compute, then the inverse z transform follows directly from the above equation
k
k
k
k
pz
mkm
m
kki zXpz
dz
d
mc
)]()[()!1(
1)1(
)1(
kpzkki zXpzc
)()(
![Page 17: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/17.jpg)
Review: long division• For X(z) = N(z) / D(z), we can perform dividing N(z) by D(z) and the quotient 商 is a power series of z.
• In the power series 幂级数 , the coefficient of the term involving z–n simply corresponds the sequence x[n].
2102 ]2[]1[]0[]1[]2[
][)(
zxzxzxzxzx
znxzXn
n
![Page 18: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/18.jpg)
18
Review: Time-shift theoremTime-shift theorem (时移定理)
Assume that x[n] ↔ X(z), then x[n + l] ↔ zl
X(z), where l is an integer.
If the ROC of X(z) is r1 < |z| < r2, then the ROC of Z{x(n + l)} is the same as the ROC of X(z).
If x[n] is right-handed or left hand, the ROC of Z{x[n + l]} is the same as the ROC of X(z), except for the possible inclusion or exclusion of the regions z=0 and |z| = ∞.
![Page 19: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/19.jpg)
19
Review: Transform-Domain Analysis of LTI Discrete-Time System
• A linear system can be characterized by a difference equation as follows
Applying the z transform on both sides, we get that
Applying the time-shift theorem, we obtain
M
ll
N
ii lnxbinya
00
][][
}][{]}[{00
M
ll
N
ii lnxbinya ZZ
M
l
ll
N
i
ii zXzbzYza
00
)()(
![Page 20: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/20.jpg)
20
Review: Transfer functionsMaking a0=1, we define
as the transfer function of the system relating the output Y(z) to the input X(z).
• From the convolution theorem, we have
therefore the transfer function of the system is the z transform of its impulse response h(n).
N
i
ii
M
l
ll
za
zb
zX
zYzH
1
0
1)(
)()(
][][][)()()( nhnxnyzHzXzY
![Page 21: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/21.jpg)
21
Review: Frequency-domain representation of discrete-time signals and systems
• The direct and inverse Fourier transforms of the discrete-time signal x(n) are defined as
– In fact, the Fourier transform X(ejω) is the z transform of the discrete-time signal x(n) at the unit circle.
– the Fourier transform X(ejω) is periodic with period 2π, therefore the Fourier transform of x(n) requires specification only for a range of 2π, for example, ω [-∈π,π] or ω [0, 2π].∈
deeXnx
enxeX
njj
n
njj
)(2
1][
][)(
![Page 22: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/22.jpg)
22
Review: Properties of the Fourier transform
Several properties:x(n) X(ejω)
realreal
imaginaryimaginary
realreal
imaginaryimaginary
conjugatesymmetric
conjugatesymmetric
conjugate antisymmetri
c
conjugate antisymmetri
c
conjugatesymmetric
conjugatesymmetric
conjugate antisymmetric
conjugate antisymmetric
![Page 23: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/23.jpg)
23
Review: Frequency response
H(z)(系统函数)
H(ejω)(频率响应)
jezk
kjj zHekhnheH
)()()}({)( F
( ) Z{ ( )} ( ) k
k
H z h n h k z
h(n) — 单位冲
激响应
)()(
)()()()()()( )()(
jnj
k
kjnj
k
knjenx
k
eHeekhe
khekhknxnhnxnynj
- 复正弦输入得到复正弦输出
)()(
)()()()()()()(
zHzzkhz
khzkhknxnhnxny
n
k
kn
k
knznx
k
n
- 指数输入得到指数输出
![Page 24: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/24.jpg)
Quiz OneMarch 21th, 2011
![Page 25: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/25.jpg)
25
1.Characterize the system below as linear/nonlinear, causal/noncausal and time invariant/time varying.
y(n)=(n+a)2x(n+4)
2.For the following discrete signal, determine whether it’s periodic or not. Calculate the fundmental period if it is periodic.
2 2( ) cos ( )
15x n n
![Page 26: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/26.jpg)
26
3. Compute the convolution sum of the following pairs of sequences.
4. Discuss the stability of the system described by the impulse response as below:
h(n)=0.5nu(n)-0.5nu(4-n)
therwise , 0
41 , )( and
therwise , 0
87 , 1
63 , 0
20 , 1
)(o
nnnh
o
n
n
n
nx
![Page 27: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/27.jpg)
5. Compute the Fourier transform of the following sequences:
6. Given x(n) as following, find X(z) and discuss its ROC.
( ) ( )nx n a u n
]2[3][2 nunx n
![Page 28: Review for chapter two & three](https://reader036.vdocument.in/reader036/viewer/2022081519/568134ef550346895d9c2fac/html5/thumbnails/28.jpg)
7. A LTI causal system can be described by the different equation:
1) Compute the transfer function H(z) of the system.
2) Compute the impulse response h[n] of system.
3) compute the frequency response H(ejw) of system.
4)Determine the system is stable or not. tips: if ROC of H(z) includes |z|=1
]1[]2[]1[][ nxnynyny