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102_1_lecture17_student - UCLA Fall 2011 Systems and...

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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 continuous-time signals Interpolation of band-limited signals (Sampling theorem) Processing of continuous-time signals using discrete-time 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 continuous-time 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 Final-Value Theorems Initial-Value 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 Final-Value 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 Final-Value Theorems Initial-Value Theorem: If x ( t ) = 0 for t < and x ( t ) contains no impulses or higher-order singularities at t = 0 , then x (0 + ) = lim s sX ( s ) Final-Value 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 continuous-time signal to discrete time, process it, then convert back to continuous time (using analog-to-digital, digital-to- 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.

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102_1_lecture17_student - UCLA Fall 2011 Systems and...

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