Freq_anal_2_text

Freq_anal_2_text - 2. Sampling and reconstruction....

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2. Sampling and reconstruction. Continuous time signal discrete-time signal The sample values of x(t) are equal to the values x(nT), n=0, ± 1, ± 2, …. . Represent x(t) by x(nT) – under what condition the representation is unique? For analysis, we construct a continuous-time signal x s (t) from x(nT): () ( ) ( ) ( ) () () s nn x t x nT t nT x t t nT x t p t δδ ∞∞ =−∞ =− = = ∑∑ where p(t) is the impulse train given by −∞ = = n nT t t p ) ( ) ( δ To determine the Fourier transform of x(t)p(t), first express p(t) using FS. −∞ = = k t jk k s e c t p ω ) ( where T s π 2 = is the sampling frequency. k c = Fourier transform of p(t) is P = ( ) s xt pt X ↔= Let X(t) be a band-limited signal, X( ω ) = 0 , for | ω | > B If ω s > 2B there is no overlap between X( ω -k ω s ) Sampling Reconstruction 0 ω X( ω ) -B B
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x(t H ( ω ) x r (t) = n nT t t p ) ( ) ( δ x s (t)= x(t)p(t) X If ω s < 2B there is overlap When there is no overlap, x( ω ) can be recovered from X s ( ω ) by a lowpass filter.
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This note was uploaded on 10/15/2010 for the course BIM BIM108 taught by Professor Qiu during the Winter '09 term at UC Davis.

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Freq_anal_2_text - 2. Sampling and reconstruction....

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