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Unformatted text preview: UCLA Fall 2010 Systems and Signals Lecture 7: ContinuousTime Fourier Series I October 18, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1 Midterm Nov. 01 2010 In class, 2 hours. 4 problems. Closed book, closed notes. You can bring 1 handwritten letter sheet. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2 Introduction Todays topics: Review: Convolution LTI system response to complex exponentials Frequency domain signal representation Continuous Time Fourier Series EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3 Review. Given x ( t ) (an impulse train) and the LTI system shown below, find y ( t ) . x ( t ) = X k = ( t 4 k ) + *(t1) *[(t)(t1)] *[(t)(t1)] *rect(t/2) x(t) y(t) 2 2 First try to use properties of convolution to simplify the system. Consider the subsystem on the right (after rect ( t/ 2) ). It implements 2 ( t 1) * ( t ) ( t 1) + ( ( t ) ( t 1))2 . This can be simplified to 2 ( t ) + ( t 1) * ( t ) ( t 1) . EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4 and finally to 2 ( t ) ( t 2) . + *(t1) *[(t)(t1)] *[(t)(t1)] 2 2 + *(t1) *[(t)(t1)] *[(t)(t1)] 2 2 *(t) identity element of convolution commutative property *[(t)(t1)] 2 *(t)+(t1) by using distributive property *2[(t)(t2)] simplified subsystem EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5 Then y ( t ) = 2 X k = ( t 4 k ) * rect ( t/ 2) * ( t ) ( t 2) This is a square waveform, which can be written as: y ( t ) = 2 X k = rect t 4 k 2 rect t 4 k 2 2 ! 1 212 t 2 y(t) This example should not be difficult if you understand convolution with delayed delta functions! EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6 Introduction Before: Studied LTI systems response through convolution Represented signals in terms of shifted, scaled functions Now: Alternative representation of signals and LTI systems Signal representation in terms of complex exponentials ( e jwt ) (Frequency domain signal representation) Revisit convolution and see that the new signal representation makes computation much simpler First focus on discrete and continuous time periodic signals EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7 Frequency domain representation of continuous time signals in general means a Fourier series or Fourier transform. Fourier series: time limited signals and periodic signals. Fourier transforms: any energy signal, many power signals. Applications of Fourier transforms Decomposes signals into fundamental or primitive components Shortcuts to the computation of sums and integrals, Reveals hidden structure in systems or signals Sparser representation of many signals (speech, images) which is useful for compression....
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 Fall '08
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