Chapter 3 Fourier Series Representations of Periodic signals
1 Chapter 3 Fourier Series Representations of Periodic Signals
Chapter 3 Fourier series 53.2 The Response of LTI Systems to Complex Exponentials LTI系统对复指数信号的响应 1. Continuous-time system st h() Eigenfunction 特征函数 H(ses H()=h()e" dt--Eigenvalue(特征值) 2. Discrete-time system n y Eigenfunction 特征函数 H(z)z H(z)=∑hlk"- Eigenvalue(特征值 2
2 Chapter 3 Fourier Series §3.2 The Response of LTI Systems to Complex Exponentials LTI 系统对复指数信号的响应 y(t) st e h(t) 1. Continuous-time system ( ) ( ) st H s h t e dt + − − = Eigenfunction 特征函数 ——Eigenvalue (特征值） hn yn n z 2. Discrete-time system Eigenfunction 特征函数 ( ) ——Eigenvalue (特征值） n n H z h n z − + =− = st e ( ) st H s e n z ( ) n H z z
Chapter 3 Fourier series Example 3.1 Consider an LTI system: h(0)=8(t-3) J x(t (1)x()=e2 y()=e 2(-3) =x(t-3 x(t=cos 4t + cos 7t y()=cos4(t-3)+c0s7(t-3)=x(t-3)
3 Chapter 3 Fourier Series Example 3.1 Consider an LTI system : h(t)= (t −3) y t x t ( ) = − ( 3) ( ) ( ) 2 1 j t x t e = ( ) ( ) ( ) 2 3 3 j t y t e x t − = = − (2 cos4 cos7 ) x t t t ( ) = + y t t t x t ( ) = − + − = − cos4 3 cos7 3 3 ( ) ( ) ( )
Chapter 3 Fourier series 533 Fourier Series Representation(傅立叶级数) of Continuous-time Periodic signals 5331 Linear combinations(线性组合) of Harmonically related Complex exponentials x(t)=∑ae Fourier ser k=-∞ a,, -Fourier series Coefficients Spectral Coefficients(频谱系数) 4
4 Chapter 3 Fourier Series §3.3 Fourier Series Representation（傅立叶级数） of Continuous-time Periodic Signals §3.3.1 Linear Combinations （线性组合） of Harmonically Related Complex Exponentials ( ) jk t k k x t a e 0 + =− = ——Fourier Series k a ——Fourier Series Coefficients Spectral Coefficients （频谱系数）
Chapter 3 Fourier series Example 3.2 ,水k2t k=-3 ao= a1,=1/4 2 1/2,a,=1/ ±3 3 Example Consider an LTI system for which the input x()=1+-cos2rt and the impulse response h(t=e" u(t)determine the outputy(t y()=1+ jeTt e e 1+i2丌 1-j2丌 5
5 Chapter 3 Fourier Series Example 3.2 ( ) jk t k k x t a e 2 3 3 + =− = = = = = 1/ 2 , 1/ 3 1 , 1/ 4 2 3 0 1 a a a a Example : Consider an LTI system for which the input and the impulse response determine the output x(t) cos 2t 2 1 =1+ h(t) e u(t) −t = y(t) ( ) 2 2 1 1 4 4 1 1 2 1 2 j t j t y t e e j j − = + + + −