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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 17: Sampling Theorem I November 28, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Agenda Todays topics Sampling of continuoustime signals Interpolation of bandlimited signals (Sampling theorem) Processing of continuoustime signals using discretetime systems EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 Laplace Transform Review Example: Find the Laplace transform of the following functions: 1) f 1 ( t ) = e at u ( t ) 2) f 2 ( t ) = e at u ( t ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Solution: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 Example: Consider a continuoustime LTI system with below system function: H ( s ) = 1 s 2 s 2 Determine the impulse response function h ( t ) for each of the following cases: 1) The system is stable 2) The system is causal 3) The system is neither stable nor causal EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 Solution: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 Solution (cont.): EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7 The Initial and FinalValue Theorems InitialValue Theorem: It has been shown in previous lectures that L [ f ( t )] = sF ( s ) f (0) . If f ( t ) doesnt have discontinuity at t = 0 , then f (0) = f (0 ) = f (0 + ) . We have: Z e st f ( t ) dt = sF ( s ) f (0 + ) As s the left hand side of the above equation goes to zero, then f (0 + ) = lim s sF ( s ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 8 FinalValue Theorem: If f ( ) exists, then we have: Z f ( t ) dt = f ( ) f (0) We also have: Z f ( t ) dt = lim s Z e st f ( t ) dt = lim s sF ( s ) f (0) Compare the above equations, we obtain: f ( ) = lim t f ( t ) = lim s sF ( s ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 9 The Initial and FinalValue Theorems InitialValue Theorem: If x ( t ) = 0 for t < and x ( t ) contains no impulses or higherorder singularities at t = 0 , then x (0 + ) = lim s sX ( s ) FinalValue Theorem: If x ( t ) = 0 for t < and x ( t ) has a finite limit as t , then lim t x ( t ) = lim s sX ( s ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 10 Introduction Many signals around us are not continuous Movies: discrete sequence of frames Images: 2D array of discrete elements (pixels) Music: discrete sequence of signal amplitudes All perceived as continuous signals! Discreteness usually not apparent. Sampling bridges continuous and discrete time signals. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 11 Often useful to convert continuoustime signal to discrete time, process it, then convert back to continuous time (using analogtodigital, digitalto analog converters) Digital systems are more flexible, inexpensive, programmable......
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This note was uploaded on 03/30/2012 for the course ELEC ENGR 102 taught by Professor Lee during the Fall '11 term at UCLA.
 Fall '11
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