chapter 6: sampled data systems and the z-transform 1
TRANSCRIPT
Chapter 6:
Sampled Data Systemsand the z-Transform
1
Sampled-data system
A sampled-data system operates on discrete-time rather than continuous-time signals.
2
6.1 The sampling process
A sampler is basically a switch that closes every T seconds.
3
r(t): continuous signal
r*(t): sampled signal
• q: amount of time the switch is closed
4
In practice, q T , and the pulses can be approximated by flat-topped rectangles.
5
If q is neglected, the operation is called “ideal sampling”
6
Ideal sampling• Ideal sampling of a continuous signal can be considered as a
multiplication of the continuous signal, r(t), with a pulse train, P(t).
• The pulse train:
7;)()(
n
nTttP
Ideal sampling
• Let, r(t) = 0, for t<0, then,
8
.)()()(*
n
nTtnTrtr
.)()()(*
n
nTttrtr
),()()(* trtPtr
.)()()(0
*
n
nTtnTrtr
The z-transform
• Taking Laplace transform of the sampled signal r*(t) gives:
• Let us define z = esT. Then,
• This is the definition of the z-transform.9
.)()(0
n
nznTrzR
.)()(0
*
n
nTsenTrsR
The z-transform
Notice that the z-transform consists of an infinite series in the complex variable z,
i.e. the r(nT ) are the coefficients of this power series at different sampling instants.
10
...,)3()2()()0()( 321 zTrzTrzTrrzR
The z-transform
• The z-transform is used in sampled data systems just as the Laplace transform is used in continuous-time systems.
• We will look at how we can find the z-transforms of some commonly used functions.
• We first give a closer look at the D/A converter.
11
D/A converter as a zero-order hold (ZOH)
• D/A converter converts the sampled signal r∗(t) into a continuous signal y(t).
• D/A can be approximated by a ZOH circuit.
• The ZOH remembers the last information until a new sample is obtained, i.e. it takes the value r(nT) and holds it constant for nT ≤ t < (n + 1)T , and the value r (nT) is used during the sampling period.
12
The zero-order hold (ZOH)• The impulse response of a ZOH:
• The transfer function of ZOH is
13
.11
)(s
e
s
e
ssG
TsTs
The zero-order hold (ZOH)
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A sampler and zero-order hold can accurately follow the input signal if the sampling time T is small compared to the transient changes in the signal.
Example 6.1Figure 6.10 shows an ideal sampler followed by a ZOH. Assuming the input signal r (t) is as shown in the figure, show the waveforms after the sampler and also after the ZOH.
Answer
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The z-transform of some common functions
16
Unit Step Function
17.1||,1
1
1
...1)()(
1
321
00
zz
zz
zzzzznTrzRn
n
n
n
.0,1
,0,0)(
n
nnTr
Useful closed-form series summations
18
xxxxx
n
n
1
11 321
0
232
0 )1(32
x
xxxxnx
n
n
x
xxxxx
NN
N
n
n
1
11
121
0
Unit Ramp Function
19
.0,
,0,0)(
nnT
nnTr
.1,)1(
...32)()(
2
321
00
zz
Tz
TzTzTznTzznTrzRn
n
n
n
Exponential Function
20
.0,
,0,0)(
ne
nnTr anT
.0,
,0,0)(
te
ttr at
.1
1
...1)()(
1
221
00
aTaT
aTaT
n
nanT
n
n
ez
z
ze
zezezeznTrzR
General Exponential Function
21
.0,
,0,0)(
np
nnr n
.
1
1
...1)()(
1
221
00
pz
z
pz
zppzzpznTrzRn
nn
n
n
Sine Function
22
.0,sin
,0,0)(
nTn
nnr
.1cos2
sin
1)(
)(
2
1
2
1)(
).(2
1
2sin
2
2
Tzz
Tz
eezz
eez
jez
z
ez
z
jzR
eejj
eeTn
TjTj
TjTj
TjTj
TjnTjnTjnTjn
Discrete Impulse Function
23
.0,0
,0,1)()(
n
nnnr
.1)()(0
n
nznTrzR
Delayed Discrete Impulse Function
24
.,0
,0,1)()(
kn
knknnr
.)()(0
k
n
n zznTrzR
25
The z-Transform of a Function Expressed as a Laplace Transform
• Given a function G(s), find G(z) which denotes the z-transform equivalent of G(s).
• It is important to realize that G(z) is not obtained by simply substituting z for s in G(s)!
26
Example 6.2
Given
Determine G(z).
27
.65
1)(
2
sssG
Answer: Using Inverse Laplace transform
• Partial fraction
• Inverse Laplace transform
• Substitute t = nT gives
• Finally,
28
.3
1
2
1
)3)(2(
1
65
1)(
2
sssssssG
.)}({)( 321 tt eesGLtg
.)( 32 nTnT eenTg
.))((
)()(
32
32
32 TT
TT
TT ezez
eez
ez
z
ez
zzG
Method 2: Laplace to z-transform table
• From table in Appendix A
• So,
29
.3
1
2
1
)3)(2(
1
65
1)(
2
sssssssG
.))((
)()(
32
32
32 TT
TT
TT ezez
eez
ez
z
ez
zzG