3. Chapter-7-8-AKW

# 3. Chapter-7-8-AKW - ELEC211 Signals and Systems Lecture 19...

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1 AKW Fall 2010 Chapter 7-8 ELEC211: Signals and Systems Lecture 19 Chapter 7: Sampling Theorem Nyquist Sampling Theorem Reconstruction of CT Signals from Samples Sampling by Gating, Zero- and First-Order Hold Sampling Theorem Summary

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2 AKW Fall 2010 Chapter 7-8 Chapter 7: Sampling of Continuous-Time Signals In Chapters 3, 4, 5, we completed the study of four transforms: CTFS, DTFS, CTFT, DTFT. We went back-and-forth between continuous-time and discrete-time signals and systems Sampling bridges the gap between continuous-time signals and discrete-time signals - Continuous-time signal discrete-time signal • When can we represent a CT signal very well from its sampled values? Sampling Interpolation
3 AKW Fall 2010 Chapter 7-8 Representing CT Signals from Samples Do the following sampled points from the CT signal Do the following sampled points from the CT signal uniquely represent the CT signal? uniquely represent the CT signal? Answer: ?? Answer: ??

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4 AKW Fall 2010 Chapter 7-8 • In the above, we have three different signals that produce the same set of samples: but obviously • Intuitively, for the samples to represent the CT signal well, we expect that the samples must be taken so close together that the CT signal remains “smooth” between samples . How do we formalize this intuition based on the theories that we have learnt? 123 () x kT x x k = =∀ xt x t x t
5 AKW Fall 2010 Chapter 7-8 • The “smoothness” of a signal is determined by its bandwidth . The bandwidth of a signal is the frequency beyond which the energy in its spectrum becomes negligible. •A s ign a l x ( t ) with FT X ( j ω ) is call band-limited with unilateral bandwidth if Bandwidth of a Signal () 0 s . t . | | M Xj ω ωω = ∀> M - M M 1 X ( j ω ) Spectrum of a bandlimited signal The physical interpretation is that x ( t ) does not contain any sinusoid over any range of frequency larger than M The spectrum is zero for all frequencies | ω |> M

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6 AKW Fall 2010 Chapter 7-8 7.1 Sampling theorem In 1927, Harry Nyquist at Bell Labs made a surprising claim that theoretically, we can exactly reconstruct a CT signal from its samples if the signal is band-limited and the samples are taken at a high enough rate. •Le t x ( t ) be a band-limited signal with X ( j ω ) = 0 for | | > M . Then x ( t ) can be uniquely determined by its samples x ( nT ), n = 0, ± 1, ± 2, ± 3…, if s =2 π / T >2 M where s is the sampling frequency and T the sampling period/interval . Specifically: if 2 M < s ; M < c < s - M −∞ = = n c nT t nT t T nT x t x ) ( )) ( sin( ) ( ) ( ….(7.11)
7 AKW Fall 2010 Chapter 7-8 Consider p ( t ), the periodic impulse train with period T : Multiplying a CT signal x ( t ) with p ( t ) gives a CT signals which is a sequence of impulses (called a sampled impulse train ): ) ( ) ( nT t t p n = −∞ = δ −∞ = −∞ = = = = n n p nT t nT x nT t t x t p t x t x ) ( ) ( ) ( ) ( ) ( ) ( ) ( × ) ( t x t Interval = T ) ( t p t ) ( t x p t Proof of the Sampling Theorem When multiplying x ( t ) with p ( t ), we are throwing away all values of x ( t ) except for those values at t=nT . This is the essence of sampling!

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8 AKW Fall 2010 Chapter 7-8 - If x ( t
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3. Chapter-7-8-AKW - ELEC211 Signals and Systems Lecture 19...

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