sampling - Sampling and Reconstruction Until now we have considered continuous-time signals systems separately from discrete-time signals systems We

sampling - Sampling and Reconstruction Until now we have...

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Sampling and Reconstruction: Until now, we have considered continuous-time signals & systems separately from discrete-time signals & systems. We will now connect them through uniform sampling : x [ n ] = x ( nT ) for n Z , where T (in seconds/sample) is the sampling interval and so 1 /T is the sampling rate . x ( t ) x [ n ] t = nT Key question: Do we always lose information in going from { x ( t ) } t to { x [ n ] } n = -∞ ? Under what conditions can we reconstruct { x ( t ) } t from { x [ n ] } n ? 1
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Aliasing — Nyquist zones: So far, we have seen that a sampled sinusoid x [ n ] can be associated with many continuous-time sinusoids x ( t ) . Can visualize this for x ( t ) = e j Ω k t via “Nyquist zones”: Defn: The k th Nyquist zone is Ω ( 2 πk - π T , 2 πk + π T ] . Ω Ω Ω ω 2 π 2 π T 2 π T 2 π T 0 0 0 0 2 π 2 π T 2 π T 2 π T ··· ··· | X c ( j Ω) | | X c ( j Ω) | | X c ( j Ω) | | X ( e ) | ω - 1 ω 0 ω 1 Ω - 1 Ω 0 Ω 1 ω k = Ω k T zone -1 zone -1 zone -1 zone 0 zone 0 zone 0 zone 1 zone 1 zone 1 1 /T -rate sampling Recall the DTFT is 2 π -periodic! Due to the linearity of sampling, the same phenomenon applies to generic signals: . Ω Ω Ω ω 2 π 2 π T 2 π T 2 π T 0 0 0 0 2 π 2 π T 2 π T 2 π T ··· ··· | X c ( j Ω) | | X c ( j Ω) | | X c ( j Ω) | | X ( e ) | zone -1 zone -1 zone -1 zone 0 zone 0 zone 0 zone 1 zone 1 zone 1 1 /T -rate sampling Spectral overlap due to “aliasing” will prevent perfect reconstruction! Nyquist zone locations determined by sampling rate 1 T ! 3
Nyquist sampling: As we have seen, sampling can introduce ambiguity: Can’t distinguish one Nyquist zone from another! How can we avoid this ambiguity? Sample in such a way that the CT signal is contained within a particular Nyquist zone. There are two basic ways to do this: 1. Nyquist sampling : Ensure that all frequency components of x ( t ) are in the 0 th Nyquist zone. Equivalently. . . Sample more than twice as fast as the largest absolute signal frequency: 1 T > | Ω | max π = 2 | f | max . Ω ω 2 π 2 π T 0 0 2 π 2 π T ··· ··· | X c ( j Ω) | | X ( e ) | zone -1 zone 0 zone 1 | Ω | max −| Ω | max 1 /T -rate sampling reconstructtion 2. Bandpass sampling : Ensure than all frequency components of x ( t ) are contained within some Nyquist zone k , for k negationslash = 0 . Note: works only for bandpass x ( t ) . But how does one choose 1 T and k in this case? Reconstruction then boils down to optimal interpolation. Stay tuned for details... 4

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