Analogue and digital techniques in

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Analogue and digital techniques in closed loop
regulation applications
Digital systems Sampling of analogue signals
Sample-and-hold Parsevals theorem
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From time domain to frequency domain
Fourier transform
y(t) is a real function of time We define the
Fourier transform Y(f) A complex function in
frequency domain f
Y(f) is the spectral or harmonic representation
of y(t) Frequency spectrum
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From time domain to frequency domain
Example of Fourier transform
y(t)
t
-T
T
Real even functions
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From time domain to frequency domain
NB Comments on unit with Fourier function Use of
?2?f instead of f
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Fourier series
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Periodic function
f(t)? a series of frequencies multiple of 1/T
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Fourier coefficients for real functions
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Principle
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Analysis in frequency domain
F?(?) Fourier transform of f(t) ? (?)
Fourier transform of ?(t) F?(?) Fourier
transform of f(t)
Convolution in the frequency domain
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Analysis of ?(?)
Periodic function
Decomposition in Fourier series
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Analysis of ?(?)
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Transform of f(t)
Convolution
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Transform of f(t)
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Aliasing
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Primary components Fundamental components
Complementary components
Complementary components
The spectra are overlapping (Folding)
Folding frequency
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Requirements for sampling frequency
The sampling frequency should be at least twice
as large as the highest frequency component
contained in the continuous signal being
sampled In practice several times since
physical signals found in the real world contain
components covering a wide frequency range
NBIf the continuous signal and its n
derivatives are sampled at the same rate then
the sampling time may be
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Can we reconstruct f(t) ?
f(t)
f (t)
f(t)
Sampler
Filter
In the frequency domain
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Back to time domain
Convolution
Window in the time domain
f(t) in the time domain
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Back to time domain
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Reconstruction
f(t)
t
n?T
(n1)?T
(n2)?T
(n3)?T
Interpolation functions
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Delayed pulse train
t
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Analogue and digital techniques in closed loop
regulation applications
Zero-order-hold
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Reconstruction of sampled data
To reconstruct the data we have a series of data
Approximation
A device which uses only the first term fk?T is
called a Zero-order extrapolator or
zero-order-hold
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Sample-and-Hold devices
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Droop
Sample-and-hold circuit
Input signal
t
Output signal
Hold mode
Hold mode
Sample mode
Settling time
Acquisition time
Aperture time
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Laplace transform of output
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Transfer function
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Transfer function
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Phase of F(?)
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Parsevals theorem
x(t) and y(t) have Fourier transform X(f) and
Y(f) respectively
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Parsevals theorem
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Thank you for your attention