02/04/1439 Chapter 3 - PDF document

02/04/1439 Chapter 3 Fourier Series Le Lectures es on on Signals & system ems Eng nginee eering • A signal can be represented as a linear combination of basic signals. Designed and Presented by • The response of LTI to any input consisting of linear Dr. Ayman Elshenawy Elsefy combination of basic signals is the linear combination of the individual responses to each of the basic signals. • Dept. of Systems & Computer Eng. Al-Azhar University Convolution Sum & Convolution Integral represent a Email : eaymanelshenawy@yahoo.com signal as linear combination of shifted impulses. • Fourier series and transform uses a complex exponential signals with different frequencies will be used instead of shifted impulses ( Delayed or advanced). Chapter 3 Fourier Series Representations of Periodic Signals 2 Chapter 3 Fourier Series Chapter 3 Fourier Series Jean Baptiste Joseph Fourier, born in 1768, in France. 1807,periodic signal could be represented by sinusoidal series. 1829,Dirichlet provided precise conditions. 1960s,Cooley and Tukey discovered fast Fourier transform. 3 4 1

02/04/1439 Chapter 3 Fourier Series Chapter 3 Fourier Series The Response of LTI Systems to Complex Exponentials LTI • Importance of complex exponentials in LTI   st e system : 1. Continuous-time system y t   h t • The response of an LTI system to a complex         y ( t ) x ( t ) * h ( t ) x ( t ) h ( ) d   exponential input is the same complex exponential        s ( t ) with only a change in amplitude. e h ( ) d          st s e e h ( ) d     e st H ( s ) x ( t ) H ( s ) —— Eigenvalue         st H s h t e dt • The complex amplitude factor H(s) or H(z) is a   Eigen function st e st  H s e function of the complex variable s or z. 5 6 Chapter 3 Fourier Series Chapter 3 Fourier Series The Response of LTI Systems to Complex Exponentials LTI For the continuous LTI systems consider the input : 2. Discrete-time system   n y n z     h n    y [ n ] x [ n ] * h [ n ] x [ n k ] h [ k ] From the eignfunction property, the response to each part is:    k    n k z h [ n ]   k     n k z z h [ n ]   k   z n H ( z ) x [ n ] H ( z ) —— Eigenvalue And from the superposition property:  Eigen function          n n z n H z h n z H z z   n 7 8 2

02/04/1439 Chapter 3 Fourier Series Chapter 3 Fourier Series Example 3.1 (3) Input as a combination of Complex Exponentials       Consider an LTI system : X(t) y t x t 3 Continuous time LTI system: Impulse response   N  s t  x ( t ) a e k       j 2 t k 1 x t e   k 1 𝐼 𝑡 = 𝑓 −3𝑡 N    s t y ( t ) a H ( s ) e k  k k   k 1 Discrete time LTI system:  N   n  x [ n ] a z  k k   k 1 N    n y [ n ] a H ( z ) z  k k k   k 1 9 10 Chapter 3 Fourier Series Chapter 3 Fourier Series Example 3.1 Example :       y t x t 3 Consider an LTI system : X(t)   1  1   Consider an LTI system for which the input x t cos 2 t Impulse response: X(t) 2               𝑰 𝒕 = 𝒇 −𝟒𝒕 and the impulse response determine the output t y t x t t t h t e u t 2 cos4 cos7 𝑰 𝒕 𝒇 𝒌𝟖𝒖 𝑰 𝒕 𝒇 𝒌𝟓𝒖 𝒇 −𝒌𝟑𝟐 𝒇 𝒌𝟖𝒖 𝒇 −𝒌𝟐𝟑 𝒇 𝒌𝟓𝒖 Try to solve it 𝟐 𝟐 𝟑 𝒇 −𝒌𝟑𝟐 𝒇 𝒌𝟖𝒖 𝟑 𝒇 −𝒌𝟐𝟑 𝒇 𝒌𝟓𝒖 1 1 𝟐 𝟑 𝒇 −𝒌𝟐𝟑 𝒇 𝒌𝟓𝒖 + 𝟐 𝟐 𝟑 𝒇 −𝒌𝟑𝟐 𝒇 𝒌𝟖𝒖 + 𝟐 𝟑 𝒇 −𝒌𝟐𝟑 𝒇 −𝒌𝟓𝒖 + 𝟑 𝒇 −𝒌𝟑𝟐 𝒇 −𝒌𝟖𝒖   𝒛 𝒖 = 4 4       y t 1 e j 2 t e j 2 t     1 j 2 1 j 2               y t cos4 t 3 cos7 t 3 x t 3 12 11 3

02/04/1439 Chapter 3 Fourier Series Chapter 3 Fourier Series Fourier Series Representation of CT Periodic Signals • If the input to an LTI system is represented as a linear combination of complex exponential signal Then • The output also can be represented as linear combination of the same complex exponential signal • Output component = input component X eignvalue 14 Chapter 3 Fourier Series Chapter 3 Fourier Series Fourier Series Representation of CT Periodic Signals So, arbitrary periodic signal can be represented as +∞ +∞ 3.3.1 Linear Combinations of Harmonically Related 𝑏 𝑙 𝑓 𝑘𝑙𝑥 0 𝑢 = 2𝜌 𝑈)𝑢 ( Fourier series ) 𝑏 𝑙 𝑓 𝑘𝑙( 𝑦 𝑢 = Complex Exponentials 𝑙=−∞ 𝑙=−∞ (1) General Form The set of harmonically related complex exponentials: a —— Fourier Series Coefficients k Spectral Coefficients         jk t jk ( 2 / T ) t  ( t ) e e , k 0 , 1 , 2 0 x(t) is constant, DC component K=0 k    0 , : Fundamental or 1 st harmonic components j t j t K=1 e e 0    j 2 t j 2 t K=2 : 2 nd harmonic components e 0 , e 0 Fundamental period: T ( common period )  0 ,   jN t jN t K=N : Nth harmonic components e e 0 15 16 4

02/04/1439 Chapter 3 Fourier Series     a 1 , a 1 / 4   0 1 3     Example 3.2   jk 2 t x t a e     a 1 / 2 , a 1 / 3 k   2 3   k 3 17 18 Chapter 3 Fourier Series Determination of Fourier Series Representation       jk t Synthesis equation x t a e 0 k   k Analysis equation 1       jk t a x t e dt 0 k   T T 0 0 a —— Fourier Series Coefficients k Spectral Coefficients 19 5

02/04/1439 𝒚 𝒖 = 𝟐 + 𝐭𝐣𝐨𝝏 𝟏 𝒖 + 𝟑 𝐝𝐩𝐭 𝝏 𝟏 𝒖 + 𝒅𝒑𝒕(𝟑𝝏 𝟏 𝒖 + 𝝆 𝒚 𝒖 = 𝟐 + 𝐭𝐣𝐨𝝏 𝟏 𝒖 + 𝟑 𝐝𝐩𝐭 𝝏 𝟏 𝒖 + 𝒅𝒑𝒕(𝟑𝝏 𝟏 𝒖 + 𝝆 𝟓) 𝟓) 𝒃 𝟏 =1 𝒚 𝒖 = 𝟐 + 𝟐 𝟑𝒌 𝒇 𝒌𝝏 𝟏 𝒖 − 𝒇 −𝒌𝝏 𝟏 𝒖 + 𝒇 𝒌𝝏 𝟏 𝒖 − 𝒇 −𝒌𝝏 𝟏 𝒖 + 𝟐 𝝆 𝟓) − 𝒇 −𝒌(𝟑𝝏 𝟏 𝒖+ 𝒃 𝟑 = 𝟐 𝝆 𝟓) = 𝟑 𝟑 𝒇 𝒌(𝟑𝝏 𝟏 𝒖+ 𝝆 𝟓) 𝟑 𝒇 𝒌( 𝟓 (𝟐 + 𝒌) |𝒃 𝒍 | 𝒃 𝟐 = 𝟐 + 𝟐 𝟑𝒌 𝒃 −𝟑 = 𝟐 𝝆 𝟓) = 𝟑 𝟑 𝒇 −𝒌( 𝟓 (𝟐 − 𝒌) 𝒃 −𝟐 = 𝟐 − 𝟐 𝟐 𝟑𝒌 𝒇 −𝒌𝝏 𝟏 𝒖 + ( 𝟐 𝟐 𝟐 𝟑 𝒇 𝒌( 𝟑 𝒇 −𝒌( 𝟑𝒌 )𝒇 𝒌𝝏 𝟏 𝒖 + 𝟐 − 𝝆 𝟓) )𝒇 𝒌(𝟑𝝏 𝟏 𝒖) + ( 𝝆 𝟓) )𝒇 −𝒌(𝟑𝝏 𝟏 𝒖) 𝒚 𝒖 = 𝟐 + (𝟐 + 𝟑𝒌 𝒃 𝒍 =0 , |k|>2 𝒚 𝒖 = 𝒃 𝟏 + 𝒃 𝟐 𝒇 𝒌𝝏 𝟏 𝒖 + 𝒃 −𝟐 𝒇 −𝒌𝝏 𝟏 𝒖 + 𝒃 𝟑 𝒇 𝒌(𝟑𝝏 𝟏 𝒖) + 𝒃 −𝟑 𝒇 −𝒌(𝟑𝝏 𝟏 𝒖) 𝒃 𝟐 = 𝟐 + 𝟐 𝒃 𝟑 = 𝟐 𝝆 𝟓) = 𝟑 𝟑 𝒇 𝒌( 𝒃 𝟏 =1 𝟓 (𝟐 + 𝒌) < 𝒃 𝒍 𝟑𝒌 𝒃 −𝟐 = 𝟐 − 𝟐 𝒃 −𝟑 = 𝟐 𝟑 𝟑 𝒇 −𝒌( 𝝆 𝟓) = 𝟓 (𝟐 − 𝒌) 𝟑𝒌 Plots of the magnitude and phase of the Fourier Series of the signal 𝒃 𝒍 =0 , |k|>2 6

02/04/1439 Chapter 3 Fourier Series Chapter 3 Fourier Series Convergence of Fourier series   T  a 4 T 1   T  b 8 T 1   T  c 16 T 1      T 2 / T 25 26 0 Chapter 3 Fourier Series 7