This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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.
 Fall '11
 lee

Click to edit the document details