102_1_lecture17

102_1_lecture17 - UCLA Fall 2011 Systems and Signals...

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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: L [ e- at u ( t )] = Z e- at e- st dt =- e- ( s + a ) t s + a = 1 s + a , ROC: <{ s } >- a L [- e- at u (- t )] = Z-- e- at e- st dt = e- ( s + a ) t s + a- = 1 s + a , ROC: <{ s } <- a f 1 ( t ) and f 2 ( t ) have the same Laplace transform functions, but different ROCs. In other words, given a Laplace transform function, with different ROCs, we obtain different inverse Laplace transforms. 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: H ( s ) = 1 s 2- s- 2 = 1 / 3 s- 2- 1 / 3 s + 1 This system has two poles at s =- 1 and s = 2 . 1) An LTI system is stable if and only if the ROC of its system function H ( s ) includes the entire j-axis. Therefore, ROC of H ( s ) has to be- 1 < <{ s } < 2 . We obtain: h ( t ) =- 1 3 e 2 t u (- t )- 1 3 e- t u ( t ) 2) The ROC associated with the system function H ( s ) for a causal system is a right-half plane. For a system with a rational system function, causality of the system is equivalent to the ROC being the right-half plane to the right of the rightmost pole. Therefore, the ROC for H ( s ) has to be <{ s } > 2 . As a result, h ( t ) = 1 3 e 2 t u ( t )- 1 3 e- t u ( t ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6 3) If the system is neither stable nor causal, the ROC for H ( s ) is <{ s } <- 1 . Therefore, h ( t ) =- 1 3 e 2 t u (- t ) + 1 3 e- t u (- t ) 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:...
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This note was uploaded on 12/13/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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102_1_lecture17 - UCLA Fall 2011 Systems and Signals...

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