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Impulse Sampling and Data Hold
• Transfer function of First-Order Hold (cont.) – Derivation of the transfer function (cont.)
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The transfer function
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Reconstructing Original Signals from Sampled Signals
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Reconstructing Original Signals from Sampled Signals
( )
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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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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.
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Reconstructing Original Signals from Sampled Signals
• Frequency-Response Characteristics of the ZOH
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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.
