Unit3_FourierSeries - EE3210 Signal and Systems Unit 3...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
EE3210 Signal and Systems Unit 3 Fourier Series
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Unit 3: Fourier Series 2 Outline 1. Preliminaries: Complex Exponential Signals 2. Continuous-Time Fourier Series 3. Discrete-Time Fourier Series
Background image of page 2
Unit 3.1 Preliminaries: Complex Exponential Signals
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Unit 3: Fourier Series 4 Exponential Signal x(t) = e st (i) Real exponential positive s negative s (ii) Complex exponential (assume s is purely imaginary) x(t) = e j ω o t
Background image of page 4
Unit 3: Fourier Series 5 C.T. Complex Exponential Continuous-time: x(t) = e j ω o t Is it periodic? Use Euler’s formula : e j ω o t = cos( ω 0 t) + j sin( ω 0 t) fundamental period T 0 =2 π /| ω ο | How about the discrete-time case?
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Unit 3: Fourier Series 6 D. T. Complex Exponential Discrete-time: x[n] = e j ω o n Is it periodic? Use Euler’s formula : e j ω o n = cos( ω 0 n) + j sin( ω 0 n) It depends on whether 2 π /| ω ο | is rational or not. Details can be found in Unit 1.
Background image of page 6
Unit 3: Fourier Series 7 General Complex Exponential x(t) = e st where s=r+j ω ο ; (r, ω ο R) Note: Im{x(t)} takes the same form (with a phase shift). r>0 r<0 Decaying Sinusoids Re{x(t)} Growing Sinusoids
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Unit 3.2 Continuous-Time Fourier Series After studying this sub-unit, you will be able to 1. find the Fourier series representation of a given C.T. periodic signal 2. describe the properties of C.T. Fourier series Jean Baptiste Joseph Fourier (1768- 1830): a French mathematician
Background image of page 8
Unit 3: Fourier Series 9 Periodic Signals + < + = 0 2 0 | ) ( | . . ) ( ) ( T t t dt t x t s t T t x t x finite energy over one period T 0 0 0 2 T π ω= We focus on the following class of periodic signals: Signals of this class can be represented by Fourier series .
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Unit 3: Fourier Series 10 −∞ = = k t jk k e a t x 0 ) ( ω (sum of complex exponentials whose frequencies are multiples of ω 0 , the “fundamental frequency”.) The coefficients, a k , can be obtained by = 0 0 ) ( 1 0 T t jk k dt e t x T a Fourier series coefficient Fourier Series Representation Theorem: Let x(t) be a periodic signal with finite energy over one period. It can be expressed as a Fourier series: Memorize it!
Background image of page 10
Unit 3: Fourier Series 11 Harmonics L L + + + + + + = t j t j t j t j e a e a a e a e a t x 0 0 0 0 2 2 1 0 1 2 2 ) ( ω d.c. component (a constant) 1 st harmonic 2 nd harmonic 1 st harmonic: fundamental period = T 0 . 2 nd harmonic: fundamental period = T 0 / 2.
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/22/2009 for the course ELECTRONIC EE3210 taught by Professor Sungchiwan during the Spring '09 term at École Normale Supérieure.

Page1 / 37

Unit3_FourierSeries - EE3210 Signal and Systems Unit 3...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online