Consider a sampling system as illustrated in the following block diagram The

Consider a sampling system as illustrated in the

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Consider a sampling system as illustrated in the following block diagram. The input continuous-time signal has the form x c ( t ² FRV± t ), and the sampling frequency is 1000 Hz. Suppose we know that the output discrete-time sequence is given by x [ n @ FRV±ʌ n /4). Determine the frequency of the input signal x c ( t ). 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 8 $V ZH NQRZ³ ʋ /4. Therefore, = ´7 µ ¶··· +] ¸¹· ʋ Hz. Q: Is this correct?
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Anti-aliasing Filter to Guarantee No Aliasing ¾ Anti-aliasing filter to bandlimit the input signal at at least half the sampling frequency (i.e., the pass band has a cutoff frequency at most half of the sampling frequency) 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 9
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The Overall System: How Does Relate to 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 10 (a) (c) (b) (c) (b) (a)
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Block (a) 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 11
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Blocks (b) and (c) 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 12 In principle, the D/C process can be analytically understood from the two-step process: relabeling the sequence to an impulse train, and ideal low pass filtering (although in practice, different interpolations may be used).
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The Equivalent DT and CT LTI Systems 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 13
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Discussion ¾ Is the overall (CT) system time-invariant? ¾ No, it is not LTI for arbitrary inputs (when there is no ideal anti- aliasing filter). Example: rectangular pulse train with pulse with less than T (sampling period). ¾ It is an LTI given band limited inputs. 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 14
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Example: Half-Sample Delay ¾ Consider the overall system (a)-(b)-(c) we have discussed so far. Suppose we know that y c ( t ² x c ( t T /2), where T is the sampling period in block (a). Design the equivalent frequency response and for the CT system and the DT system, respectively. Find the DT impulse response h [ n ]. Find the band limited interpolation formula to reconstruct y c ( t ) form x [ n ]. 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 15 (c) (b) (a)
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Example: Half-Sample Delay ¾ Time shift property of the FT: which implies ¾ Using the CT to DT conversion relation 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 16 (c) (b) (a) y c ( t ) = x c ( t T /2).
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Example: Half-Sample Delay ¾ The impulse response can be found from inverse Fourier transform: 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 17 (c) (b) (a)
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Example: Half-Sample Delay ¾ NOTE: The above discussion implicitly assumed that the input signal is band limited with frequency at most half of the sampling frequency 1/ T .
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