Chapter3 Fourier series

Chapter3 Fourier series - ECSE-303A Signals and Systems I...

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LECTURE 13 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals ECSE-303A Signals and Systems I Friday, February 12, 2010

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Chapter 3: Fourier series of periodic signals 3.0 Introduction 3.1 Historical perspective 3.2 The response of LTI systems to complex exponentials 3.3 Fourier series representation of continuous-time periodic signals 3.4 Convergence of Fourier series 3.5 Properties of continuous-time Fourier series 3.8 Fourier series and LTI systems 3.9 Filtering
Q. How can you shape a rectangular profile from a combination of circles? ECSE303 Chap. 3 MR - 2/12/2010 3

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The Fourier series By analogy, could we plot a square function using sinusoidal functions? In the 1800’s, Jean Baptiste Joseph Fourier found that most periodic functions with finite average power could be represented by a sum of sines and cosines f ( t ) t () 0 00 0 0 cos sin cos sin jk t k k k k kk k xt ae ak t j k t aA k t B k t ω ωω +∞ =−∞ +∞ +∞ = = =+ + ECSE303 Chap. 3 MR - 2/12/2010 4
, Example Consider the following periodic square wave ECSE303 Chap. 3 MR - 2/12/2010 5

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= + + + . .. f 0 = 1Hz f 0 = 1Hz f 0 = 3Hz f 0 = 5Hz f 0 = 7, 9. ..Hz 6 ECSE303 Chap. 2 MR - 2/12/2010
With three sin(2 π f 0 t) components f 0 = 1, 3, 5 Hz With twenty- five sin(2 π f 0 t) components f 0 = 1, 3, 5. ..51 Hz 7 ECSE303 Chap. 2 MR - 2/12/2010

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To represent any arbitrary periodic function, what frequency component must be chosen? What is the amplitude coefficient of each term? ECSE303 Chap. 2 MR - 2/12/2010 8
Chapter 3: Fourier series of periodic signals 3.0 Introduction 3.1 Historical perspective 3.2 The response of LTI systems to complex exponentials 3.3 Fourier series representation of continuous-time periodic signals 3.4 Convergence of Fourier series 3.5 Properties of continuous-time Fourier series 3.8 Fourier series and LTI systems 3.9 Filtering

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Eigenfunctions of LTI Difference and Differential Systems We already know that the complex exponentials of the type Ae s t and Cz n are invariant under the action of time shifts (difference systems) and derivatives (differential systems). The response of an LTI system to a complex exponential input is the same complex exponential with only a change in (complex) amplitude. Continuous-time LTI system: Discrete-time LTI system: where the complex amplitude factors H ( s ), H ( z ) are functions of the complex variable s or z . () s ts t nn eH s e zH z z ECSE303 Chap. 3 MR - 2/12/2010 10
Input signals like x ( t )=e st and x [ n ]= z n for which the system output is a constant times the input signal are called eigenfunctions of the LTI system, and the complex gains are the system's eigenvalues . 11 ECSE303 Chap. 3 MR - 2/12/2010

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To show that x ( t )=e st is indeed an eigenfunction of any LTI system of impulse response h ( t ), we look at the following convolution integral: The system's response has the form y ( t )= H ( s )e st , where is an eigenvalue and e st is an eigenfunction () ( ) ( ) () st st s y th x t d he d ehe d τ ττ +∞ −∞ +∞ −∞ +∞ −∞ =− = = Hs h e d s = −∞ +∞ z 12 ECSE303 Chap. 3 MR - 2/12/2010
Example How would you calculate y ( t )?

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Chapter3 Fourier series - ECSE-303A Signals and Systems I...

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