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# 102_1_lecture17_student - UCLA Fall 2011 Systems and...

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UCLA Fall 2011 Systems and Signals Lecture 17: Sampling Theorem I November 28, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1

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Agenda Today’s 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

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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

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Solution: EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 6
Solution (cont.): EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 7

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The Initial- and Final-Value Theorems Initial-Value Theorem: It has been shown in previous lectures that L [ f 0 ( t )] = sF ( s ) - f (0) . If f ( t ) doesn’t have discontinuity at t = 0 , then f (0) = f (0 - ) = f (0 + ) . We have: Z 0 e - st f 0 ( 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 0 f 0 ( t ) dt = f ( ) - f (0) We also have: Z 0 f 0 ( t ) dt = lim s 0 Z 0 e - st f 0 ( t ) dt = lim s 0 sF ( s ) - f (0) Compare the above equations, we obtain: f ( ) = lim t →∞ f ( t ) = lim s 0 sF ( s ) EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 9

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The Initial- and Final-Value Theorems Initial-Value Theorem: If x ( t ) = 0 for t < 0 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 < 0 and x ( t ) has a finite limit as t → ∞ , then lim t →∞ x ( t ) = lim s 0 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

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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... Analog to digital conversions are understood in the framework of sampling. In this course, we aim to understand sampling and process of reconstructing continuous-time signals from their samples.
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102_1_lecture17_student - UCLA Fall 2011 Systems and...

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