# part7 - Part 7: Discrete-time Signal and System Sampling...

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Part 7: Discrete-time Signal and System Sampling z How to obtain a discrete time signal from a continuous time signal? z Sampler z It can be accomplished with a switch closes at t=nT , as shown above. x(t) nT x(nT) T 2T x(t)

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Sampling Function z Let p(t) be the sampling function, we have z p(t) is the periodic pulse train with period T (T=1/f s ), and can be represented by Fourier series where z Taking Fourier Transform of x s (t) () ( ) ( ) () s n n t nf f j n n t f j t f jn n n t f j n t f jn n s nf f X c dt e t x c dt e e t x c dt e e t x c f X s s s = = = = −∞ = −∞ = −∞ = −∞ = π 2 2 2 2 2 −∞ = = n t f jn n s e c t p 2 = 2 2 2 1 T T t f jn n dt e t p T c s () () () t p t x t x s = Sampling Function z Duration of sampler is very small z p(t) can be considered as an infinite train of impulse function of period T z Express in Fourier Series (as before), for all n z z Taking Fourier Transform s n f T c = = 1 −∞ = = n s s nf f X T f X 1 () ( ) −∞ = = n nT t t p δ () ( ) ( ) −∞ = = n s nT t nT x t x
Shannon’s Sampling Theorem z Given a finite bandwidth signal x(t) with a bandwidth [-f h , +f h ] z Note: f s is sampling frequency z If f s 2f h (called Nyquist rate), by using an ideal filter, x(t) can be recovered. z If f s < 2f h , alising appears and can’t recover x(t) correctly. f s -f h - f h f h f X s (f) f s C 0 C 1 C -1 . . . . . . bandpass filter - f h f h f X s (f) f s C 0 C 1 C -1 . . . . . . - f h f h f X(f) Zero-order Hold z How to obtain a continuous signal from a discrete-time signal? z Zero-order hold z The output x(kT) is used for the period kT<=t<(k+1)T z Impulse response of zero-order hold z Applying Laplace Transform x(t) T2 T 3T () sT e s s H = 1 1 ( )( ) T t u t u t h =

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Is ? Discrete Signals z Discrete impulse function z A time-delayed version z Discrete step function () = = 0 0 0 1 n n nT δ = = k n k n T k n 0 1 ( ) ( ) nT t t nT = = < = 0 0 0 1 n n nT u () () −∞ = = n k kT nT u ( ) ( )( ) nT T
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## This note was uploaded on 01/11/2011 for the course EE 3118 taught by Professor Kitsangtsang during the Spring '08 term at City University of Hong Kong.

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part7 - Part 7: Discrete-time Signal and System Sampling...

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