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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 7: ContinuousTime Fourier Series I October 17, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Midterm Oct. 26 2011 • In class, 2 hours. • 4 problems. • Closed book, closed notes. • You can bring 1 handwritten letter sheet. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 Introduction Today’s topics: • Review: Convolution • LTI system response to complex exponentials • Frequency domain signal representation • Continuous Time Fourier Series EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Review. Given x ( t ) (an impulse train) and the LTI system shown below, find y ( t ) . x ( t ) = ∞ X k =∞ δ ( t 4 k ) + *δ(t1) *[δ(t)δ(t1)] *[δ(t)δ(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 δ ( t 1) * δ ( t ) δ ( t 1) + ( δ ( t ) δ ( t 1))2 . This can be simplified to 2 δ ( t ) + δ ( t 1) * δ ( t ) δ ( t 1) . EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 and finally to 2 δ ( t ) δ ( t 2) . + *δ(t1) *[δ(t)δ(t1)] *[δ(t)δ(t1)] 2 2 + *δ(t1) *[δ(t)δ(t1)] *[δ(t)δ(t1)] 2 2 *δ(t) identity element of convolution commutative property *[δ(t)δ(t1)] 2 *δ(t)+δ(t1) by using distributive property *2[δ(t)δ(t2)] simplified subsystem EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 Then y ( t ) = 2 ∞ X k =∞ δ ( t 4 k ) * rect ( t/ 2) * δ ( t ) δ ( 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 2011, Jin Hyung Lee 6 Introduction Before: • Studied LTI systems response through convolution • Represented signals in terms of shifted, scaled δ 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 2011, 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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 Fourier Series, Jin Hyung Lee

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