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Analogue and digital techniques in closed loop regulation applications Digital systems Sampling of analogue signals Sample-and-hold Parseval’s theorem

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Sampling process Continuous Function f(t) Sampler (t) Series of samples TT Reconstruction ?

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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 Y(f) 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 Linear System b( ) Fourier series Fourier transform

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= Complex Fourier Coefficients TT 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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To be minimal 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 ( ) Decomposition in Fourier series Periodic function n=0n=1n=-1 0

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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 The spectra are overlapping (Folding) 0 Primary components Fundamental components Complementary components Complementary components 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 NB:If the continuous signal and its n derivatives are sampled at the same rate then the sampling time may be: highest frequency component

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Can we reconstruct f(t) ? Sampler Filter f(t) f’(t) f °(t) In the frequency domain Window

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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 t f(t) Interpolation functions nTnT (n+1) T(n+2) T(n+3) T

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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 f[k T] is called a Zero-order extrapolator or zero-order-hold

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Sample-and-Hold devices Source

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t Sample mode Hold mode Input signal Output signal Droop = Acquisition time= Aperture time = Settling time Sample-and-hold circuit

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Laplace transform of output

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Transfer function F(s) Impulse response F(s)=L[h(t)] h(t) t (t) t k=0

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Transfer function

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Phase of F( )

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Parseval’s theorem x(t) and y(t) have Fourier transform X(f) and Y(f) respectively Convolution f’=0 y(t) y*(t)

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Parseval’s theorem x(t) y(t) This expression suggests that the energy of a signal Is distributed in time with a density Or is distributed in frequency with density

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