impulse sampling and data hold -...
TRANSCRIPT
Impulse Sampling and Data Hold
• Transfer function of First-Order Hold (cont.) – Derivation of the transfer function (cont.)
0
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1
1)(1)(
kTs
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eekTsX
The transfer function
T
Ts
s
e
Ts
Tse
sX
sHsG
TsTs
h
1111
)(
)()(
2
2
2
*1
Reconstructing Original Signals from Sampled Signals
*
1ln
0
*
02
2
*
( ) | ( ) ( )
( ) ( ) ( )
( ) ( )
1 1 1( )
1( )
1( ) ( )
s
s
s
s
K
s ZKT
n
jnW t
T n
n n
TjnW t j
Tn T
jnW t
T
n
jnW t
n
X s x X KT Z X Z
x t x t t nT
t t nT C e
C t e dt eT T T
t eT
X s x t eT
Reconstructing Original Signals from Sampled Signals
( )
0
1( ) sS jnW t
n
x t e dtT
* 1( ) ( )s
n
X s X S jnWT
*
0
1( ) ( ) sjnW t st
n
X s x t e e dtT
Reconstructing Original Signals from Sampled Signals
• Sampling Theorem – If the sampling frequency is sufficiently high compared with the
highest-frequency component involved in the continuous-time signal, the amplitude characteristics of the continuous-time signal may be preserved in the envelope of the sampled signal.
– To reconstruct the original signal from a sampled signal, there is a certain minimum frequency that the sampling operation must satisfy.
– We assume that x(t) does not contain any frequency components above 1 rad/sec.
Reconstructing Original Signals from Sampled Signals
• Sampling Theorem
– The frequency spectrum:
If s, defined as 2/T is greater than 21, where 1 is the
highest-frequency component present in the continuous-time
signal x(t), then the signal x(t) can be reconstructed
completely from the sampled signal x*(t).
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ss
k
k
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Reconstructing Original Signals from Sampled Signals
Reconstructing Original Signals from Sampled Signals
• Ideal Low-pass filter – The ideal filter attenuates all complementary components to zero
and will pass only the primary component.
– If the sampling frequency is less than twice the highest-frequency component of the original continuous-time signal, even the ideal filter cannot reconstruct the original continuous-time signal.
Reconstructing Original Signals from Sampled Signals
• Ideal Low-pass filter Is NOT Physically Realizable – For the ideal filter an output is required prior to the application of
the input to the filter – physically not realizable.
elsewhere
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Reconstructing Original Signals from Sampled Signals
• Frequency-Response Characteristics of the ZOH
s
esG
Ts
h
1)(0Transfer function of ZOH
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22sin
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2/1
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h
Reconstructing Original Signals from Sampled Signals
• Frequency-Response Characteristics of the ZOH(cont.)
Reconstructing Original Signals from Sampled Signals
• Frequency-Response Characteristics of the ZOH(cont.) – The comparison of the ideal filter and the ZOH.
– ZOH is a low-pass filter, although its function is not quite good.
– The accuracy of the ZOH as an extrapolator depends on the sampling frequency.
Reconstructing Original Signals from Sampled Signals
• Folding – The phenomenon of the overlap in the frequency spectra.
– The folding frequency (Nyquist frequency): N
– In practice, signals in control systems have high-frequency components, and some folding effect will almost always exist.
TsN
2
1
Reconstructing Original Signals from Sampled Signals
• Aliasing – The phenomenon that the frequency component ns 2 shows
up at frequency 2 when the signal x(t) is sampled.
– To avoid aliasing, we must either choose the sampling frequency high enough or use a prefilter ahead of the sampler to reshape the frequency spectrum of the signal before the signal is sampled.
Reconstructing Original Signals from Sampled Signals
• Hidden Oscillation – An oscillation existing in
x(t) between the sampling periods.
tttxtxtx 3sinsin)()()( 21
For example, if the signal
is sampled at t=0, 2/3, 4 /3,…, then the sampled signal will not show the frequency component with =3 rad/sec.