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? Subscribe to view the full document.

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 The Overall System: How Does Relate to 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 10 (a) (c) (b) (c) (b) (a) Subscribe to view the full document.

Block (a) 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 11 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). Subscribe to view the full document.

The Equivalent DT and CT LTI Systems 9-Apr-19 EIE3001 Sig & Sys, Spring 2019 13 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 Subscribe to view the full document.

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) 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). Subscribe to view the full document.

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) 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 . Subscribe to view the full document. • Fall '13

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