Signals gnals & S & Systems ems Fourier Series (Part I) - PowerPoint PPT Presentation

Lecture 5 (Chapter 3) Signals gnals & S & Systems ems Fourier Series (Part I) Adapted from: Lecture notes from MIT Dr. Hamid R. Rabiee Fall 2013

Lecture 5 (Chapter 3) Transformation  General form:     ( ) ( ) x t a t i i  i Basis Function Coefficient Sharif University of Technology, Department of Computer Engineering, Signals & Systems 2

Lecture 5 (Chapter 3) Desirable Characteristics of a Set of “ Basic ” Signals  a) We can represent large and useful classes of signals using these building blocks.  b) The response of LTI systems to these basic signals is particularly simple , useful and insightful.  Previous focus: Unit samples and impulses  Focus now: Eigen functions of all LTI systems 3 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3)  k  ( t )  ( t ) System k k Eigen value Eigen function Eigenfunction in → Same function out with a “ gain ” 4 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) From the superposition property of LTI system        ( ) ( ) ( ) ( ) y t a t x t a t System k k k k k k k Now the task of finding response of LTI systems is to determine λ k 5 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Complex Exponentials as the Eigen functions of any LTI Systems    d     ( ) s t  ( ) ( ) y t h e st h(t) ( ) x t e            s st ( ) h e d e        st ( ) H s e Eigen value Eigen function 6 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Complex Exponentials as the Eigen functions of any LTI Systems s k t s t ( ) e H s e k k ( t ) ( t ) h(t) x y     st ( ) ( ) H s h t e dt        s t s t ( ) ( ) ( ) x t a e y t H s a e k k k k k k k 7 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Complex Exponentials as the Eigen functions of any LTI Systems     n m [ ] [ ] y n h m z  n h[n] [ ] x n z   m      m z  n [ ] h m z       m  n ( ) H z z Eigen value Eigen function 8 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Complex Exponentials as the Eigen functions of any LTI Systems n n z ( ) H z k z k k [ n ] [ n ] H[n] x y     n ( ) [ ] H z h n z        n n [ ] [ ] ( ) x n a z y n H z a z k k k k k k k 9 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) What kinds of signals can we represent as “ sums ” of complex exponentials? ? 10 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) What kinds of signals can we represent as “ sums ” of complex exponentials? For Now: Focus on restricted sets of complex exponentials s=j ω CT: signals of the form e j ω t Z=e j ω DT: signals of the form e j ω n 11 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Fourier Series Representation of CT Periodic Signals x(t) = x(t+T) for all t  Smallest such T is the fundamental period  2   0  is the fundamental frequency T x(t) … … t 0 -2T -T T 2T 12 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Fourier Series Representation of CT Periodic Signals x(t) = e j ω t Periodic with period T    2   jk t        jk t ( ) T k x t a e a e 0 0 k k      Periodic with period T  {a k } are the Fourier (series) Coefficients  k=0: DC  |k|=1: First Harmonic  |k|=2: Second Harmonic 13 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Question #1: How do we find the Fourier coefficients?     Example 1 : ( ) cos 4 2 sin 8 x t t t ? 14 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Question #1: How do we find the Fourier coefficients?     Example 1 : ( ) cos 4 2 sin 8 x t t t Euler' s Realtion 1 2           4 4 8 8 j t j t j t j t ( ) [ ] [ ] x t e e e e 2 2 j   2 2 1        4 T   0 4 2 0 15 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Fourier Series Representation of CT Periodic Signals For real periodic signals, there are two other commonly used forms for CT Fourier series:          ( ) [ cos sin ] x t a k t k t 0 0 0 k k  1 k or         ( ) [ cos( )] x t a k t 0 0 k k  1 k  Because of the Eigen function property of e j ω t , we will usually use the complex exponential form in this course  A consequence of this is that we need to include terms for both positive  0 ,   jk t jk t e e and negative frequencies: 0 16 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Question #1: How do we find the Fourier coefficients?     jk t ( ) x t a e 0 k  k             jn t jk t jn t ( ) x t e dt  a e  e dt 0 0 0 k    k T T          Here denotes integral over interval j k ( n ) t   a e dt 0 k   T  k of length T (one period). T   , T k n         ( ) j k n t  [ ] e dt T k n 0   0 , k n T 17 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Question #1: How do we find the Fourier coefficients?                  ( ) jn t  j k n t  ( ) . [ ] x t e dt a e dt a T k n 0 0 k k     T T     jn t ( ) x t e dt a T 0 n T     jk t ( ) Synthesis Equation x t a e 0 k   1     jk t ( ) Analysis Equation a x t e dt 0 k T T 18 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Example #2: Periodic Square Wave x(t) … … t 0 -T -T/2 -T 1 T 1 T/2 T 19 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Example #2: Periodic Square Wave x(t) … … t 0 -T -T/2 -T 1 T 1 T/2 T T 1 2 T    1 2 ( ) a x t dt For k = 0 0 T  T T 2 DC component is just the average 20 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Example #2: Periodic Square Wave T 1 1   T       1 jk t jk t ( ) 2 a x t e dt e dt 0 0 For k ≠ 0 k T   T T T 1 2  1 sin  k T 2        jk t T 0 1 | e ( ) 0 1  0   T T jk T 1 k 0 21 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Convergence of CT Fourier Series  How can the Fourier series for the square wave possibly make sense?      jk t ( ) ( ) e t x t a e  The key is: What do we mean by ? 0 k   One useful notion for engineers  There is no energy in the difference      jk t  ( ) 2 x t a e 0 | ( ) | 0 e t dt k  T  Just need x(t) to have finite energy per period.    2 | ( ) | x t dt T 22 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 5 (Chapter 3) Dirichlet Conditions  The Dirichlet conditions are sufficient conditions for a real- valued, periodic function f ( x ) to be equal to the sum of its Fourier series at each point where f is continuous.  The behaviour of the Fourier series at points of discontinuity is determined as well, by these conditions.  These conditions are named after Johann Peter Gustav Lejeune Dirichlet. 23 Sharif University of Technology, Department of Computer Engineering, Signals & Systems