1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform The...
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Transcript of 1 G. Baribaud/AB-BDI Digital Signal Processing-2003 6 March 2003 DISP-2003 The z-transform The...
1G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
The z-transform
The sampling process
The definition and the properties
2G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
The z-transform
• Classification of signals• Sampling of continuous signals• The z-transform: definition • The z-transform: properties• Inverse z-transform• Application to systems• Comments on stability
3G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Convolution
)z(F)z(F])TnTk(f)Tn(f[Z)]t(f)t(f[Z 21
k
0n2111
])TnTk(f)Tn(f[Zk
0n21
]z)TnTk(f)Tn(f[])TnTk(f)Tn(f[Z k
0k
k
0n21
k
0n21
]z)TnTk(f)Tn(f[])TnTk(f)Tn(f[Z k
0k 0n21
k
0n21
nkm
m
0m2
0n
n1
k
0n21 z)Tm(fz)Tn(f])TnTk(f)Tn(f[Z
Analogous to Laplace convolution theorem
)z(F)t(f 11 )z(F)t(f 22
4G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
0k
k** z)k(f)z(F
0k
k1
0k
kk* )ze(ze)z(F
ez
z
ze1
1)k(F
1*
k* e)k(f
0k,0)k(f *
0
k
1
Apply z-transform
5G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
2
ee)kcos()k(f
kjkj*
)ez
z
ez
z(
2
1)z(F
jj*
]1)(zCos2z
)(zCos2z2[
2
z]
1)ee(zz
)ee(z2[
2
z)z(F
2jj2
jj*
1cosz2z
)cosz(z)z(F
2*
Discrete Cosine
ez
z]e[Z k
])ez)(ez(
)ez()ez([
2
z)z(F
jj
jj*
6G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Another approach jke)ksin(j)kcos()k(y
sinjcosz
z
ez
z)z(Y
j
)sinjcosz)(sinjcosz(
)sinjcosz(z)z(Y
1cosz2z
)sinjz)cosz(z)z(Y
2
1cosz2z
)cosz(z[cos]Z
2
1cosz2z
)sin(z[sin]Z
2
7G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Dirac function
1)]t([F
)t(
1)0(z)t()]t([F0k
Ts
8G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Sampled step function
t
u(t)
T T2 T3 T4 T5
1
0
1z
z
z1
1)z(U
e1
1e)s(U
1
Ts0k
Tsk
1z
z
z1
1z)z(U
e)s(U
10k
k
0k
Tsk
NB: Equivalent to Exp(-k) as 0
9G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
t
k
T )Tkt(
t
T
T
0k
ke
Ts
0k
T)k(se
'e z)T)k[(xee)T)k[(x),s(X
k
e'e ]T)k(t[)t(xx
Delayed pulse train
10G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Complete z-transform
0k
k
0k
k z),k(fz]T)k[(f),z(F
Example:exponential function
1T,0,e),k(f )k(
e
ez
zzeeze),z(F
0k
kk
0k
k)k(
eez
z),z(F
11G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Addition and substraction
k
0k222
k
0k111
z),Tk(f)],k(f[Z),z(F
z),Tk(f)],k(f[Z),z(F
),z(F),z(F)],k(f),k(f[Z 2121
12G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Multiplication by a constant
ttanconsa
z),Tk(f)],k(f[Z),z(F k
0k
),z(aF)],k(af[Z
13G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
-Linearity
),k(F),,k(F),k(f),,k(f 2121
tstanconsKK 2,1
),z(FK),z(FK)],k(fK),k(fK[Z 22112211
Application
1K
2K+
1f
2ff 1K
2K+
1f
2f
f
14G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Right shifting theorem
t
),k(f ),nk(f
0k
)nk(
0k
nk z),nk(fzz),nk(f)],nk(f[Z
)],k(f[Zzz),m(fz)],nk(f[Z0m
nmn
)],k(f[Zz)],nk(f[Z n 0n
15G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Right shifting theorem
Application
Unit step function which is delayed by one sampling period
1z
1)
1z
z(z)]Tt(u[Z 1
16G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Left shifting theorem
0k
)nk(
0k
nk z),nk(fzz),nk(f)],nk(f[Z
0m
1n
0m
mnmn z),m(f)],k(f[Zzz),m(fz)],nk(f[Z
t
),k(f
),nk(f
1n
0m
mn z),m(f)],k(f[Zz)],nk(f[Z
17G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Complex translation or damping
f(t) is multiplied in continuous domain by Exp(-t)And then sampled at the rate T
)ze(F]e)t(f[Z)z(G Tt
Laplace transform
dte)t(f)s(F0
st
)s(Fdte)t(fdtee)t(f)s(G0
t)s(
0
stt
T1TTsT)s( ezeee
18G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Application
Find the z-transform of sampled at T knowing that)tsin(e at
Ta2
zez,1)Tcos(z2z
)Tsin(z[sin]Z
1)Tcos(ze2ez
)Tsin(zesin]e[Z
TaTa22
Taat
Ta2Ta2
Taat
e)Tcos(ze2z
)Tsin(zesin]e[Z
19G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
k
0n
),n(f),k(s
t
S,f
),k(Sz),k(F),k(S 1
),1k(f),k(f),n(f),k(f),k(s1k
0n
),k(F1z
z),k(S
Sum of a function
20G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Difference equation
),1k(f),k(f),k(f
),z(Fz),z(F)],k(f[Z 1
),z(F)z1()],k(f[Z 1
),z(Fz
1z)],k(f[Z
21G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
),1k(f),k(f),k(f
kt
kt
kt
Example step function
22G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
]z1)[,z(F)],1k(f[Z)],k(f[Z)],k(f[Z 1
),z(Fz
1z)],k(f[Z
kt
11z
z
z
1z)z(V
u(t)
-u(t-T)
V(t)=u(t)-U(t-T)
23G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
0k
kz),k(f),z(F
Initial-value theorem
If f(t) has a z-transform F(z) and if lim F(z) as z exists
),z(Flim),0(f),k(flim z0k
2
0k
1k z)T2(fz)T(f)0(fz)k(f)z(F
24G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Final-value theorem
),z(F)1zlim(),k(flim
),z(Fz
1zlim),z(F)z1lim(),k(flim 1
k 1z
25G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Application:example
)208.0z416.0z)(1z(
z792.0)z(F
2
2
Initial value 0z
z792.0)z(F
3
2
Final value 1)208.0416.01(
792.0)k(f
Expanding F(z)
....z99.0z989.0z983.0z01.1z091.1z12.1z792.0)z(F 7654321
26G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
The z-transform
• Classification of signals• Sampling of continuous signals• The z-transform: definition • The z-transform: properties• Inverse z-transform• Application to systems• Comments on stability
27G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
kz),k(f
),k(f ),z(F
?
-Reference to tables-Practical identification-Analytic methods-Decomposition-Numerical inversion
Inverse
28G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
)z(G1z
z
ez
z
1z
z)z(F
Discrete exponential g(k)
Practical identification
Sum of a function
29G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
0k
kz),k(f),z(F
dzz),z(Fj2
1),k(f 1k
x
xx
xx
xo Re z
Im zLaurent series Cauchy theorem
Analytic method
Enclosing all singularities of F(z,)
30G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
Partial fraction expansion
With Laplace transform
....cs
C
bs
B
as
A)s(F
....CeBeAe)t(f ctbtat
With z-transform no such an expansion, one looks for terms like:
TakTa
Aeez
Az
The function F(z)/z is developed by partial-fraction expansion
31G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
The power series method
...z)T3(fz)T2(fz)T(f)0(f)z(F 321
The coefficients of the series expansion represent the values of f(t)(usually a series of numerical values)
TaTa2
Ta
Ta
Ta
ez)e1(z
z)e1(
)ez)(1z(
z)e1()z(F
....z)e1(z)e1()z(F 2Ta21Ta
)Tkt()e1()t('f0k
Tak
32G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
The z-transform
• Classification of signals• Sampling of continuous signals• The z-transform: definition • The z-transform: properties• Inverse z-transform• Application to systems• Comments on stability
33G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Continuous Systems in series with an ideal sampler at each input
)s(Ga )s(Gbex '
ex ix 'ix sx
T
)z(X),z(G),z(X 'eai
)z(X)0,z(G)0,z(X)z(X 'eai
'i
)z(X),z(G),z(X 'ibs
'e
'eabs X),z(G)z(X)0,z(G),z(G),z(X
)0,z(G),z(G),z(G ab
34G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
)s(Ga )s(Gbex '
ex ix sx
T
)z(X),z(G),z(X 'es
)s(G)s(G)s(G ba
)]s(G)s(G[Z)]s(G[Z),z(G ba
In general ),z(G),z(G),z(G ba
Continuous Systems in series with an ideal sampler at first input
35G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
)z(X),z(G),z(X 'ibs
)s(Ga )s(Gbex ix
sx
T
'ix
)0,z(X)z(X 'i
'i
)s(X)s(G)s(X eai
),z(Gb ),z(Xs given by and by )]s(X)s(G[Z ea
Continuous Systems in series with an ideal sampler at second input
36G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
)z(X)z(D)z(X *ea
*i
)z(D),z(G),z(G ab
)z(X),z(G),z(X 'ibs
)z(X)z(X *i
'i
)s(Da )s(Gbex *
ix sx
T
'ix*
ex
)z(X),z(G)z(X)z(D),z(G),z(X *e
*eabs
Discrete and continuous Systems in series with an ideal sampler
37G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Continuous and discrete Systems in series with an ideal sampler
)z(X),z(G),z(X 'eai
)0,z(G)z(D)z(G ab*
)z(X)z(D)z(X *ib
*s
)s(Db)s(Ga
ex ix *sx
T
*ix'
ex
)z(X)z(G)z(X)0,z(G)z(D)z(X 'e
*'eab
*s
)0,z(X)z(X i*i
38G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Discrete Systems in series with an ideal sampler
)z(X)z(D)z(X *ea
*i
)z(D)z(D)z(D ab
)z(X)z(D)z(X)z(D)z(D)z(X *e
*eab
*s
)s(Db)s(Da
ex *sx
T
*ix*
ex
39G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Continuous Systems in parallel with an ideal sampler
),z(G),z(G),z(G ba
),z(X),z(X),z(X bas
)s(Gb
)s(Ga
exsx
T
'ex
ax
bx
+'ex
)z(X),z(G)z(X),z(G),z(X 'eb
'eas
)z(X),z(G)z(X)],z(G),z(G[),z(X 'e
'ebas
40G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Discrete Systems in parallel with an ideal sampler
)z(D)z(D)z(D ba
)z(X)z(X)z(X *b
*a
*s
)z(X)z(D)z(X)z(D)z(X *eb
*ea
*s
)z(X)z(D)z(X)]z(D)z(D[)z(X *e
*eba
*s
)s(Db
)s(Da
ex*sx
T
*ex *
ax
*bx
+*ex
41G. Baribaud/AB-BDI
Digital Signal Processing-2003
13 March 2003DISP-2003
The z-transform
• Classification of signals• Sampling of continuous signals• The z-transform: definition • The z-transform: properties• Inverse z-transform• Application to systems• Comments on stability
42G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Continuous Systems in series with zero-order holdTransfer function via impulse response
)s(Gssxex
T
T
'ex ix
)t(x 0e 'ex
tix
0ex
T t
)]Tt(u)t(u[xx 0ei
43G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Laplace transform
s
e1
)s(X
(s)X(s)G
Ts
e
im
)]Tt(u)t(u[xx 0ei
)s
e1(x]
s
e
s
1[x)s(X
Ts
0e
Ts
0ei
44G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
s
)s(G)e1()s(G
s
e1
)s(X
(s)XG(s) sTs
s
Ts
e
s
Global transfer function
s
)s(G(s)G s
I Equal to G(s) with an integrator
Z-transform of G(s) )s(Ge)s(GG(s) ITs
I
),z(G)s(G II ),z(Gz)s(Ge I1
ITs
),z(Gz
1z),z(G)z1(),z(G II
1
45G. Baribaud/AB-BDI
Digital Signal Processing-2003
6 March 2003DISP-2003
Consequences on the behaviour
n
1i i
is ps
r)s(G
]ps
1
s
1[
)p(
r
)ps(s
r)s(G
i
n
1i i
in
1i i
iI
Z-transform1z
z
s
1
tp
tpi
i
ie
ez
z
ps
1
]eez
1z1[
)p(
r),z(G
z
1z),z(G tp
tp
n
1i i
iI
i
i
There are n poles of G(z,), they depend on n the poles
of the transfer function of the continuous system