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 − = + + + −
Chapter 3 Fourier series 53.3.2 Determination of Fourier Series Representation x()=∑ Synthesis equation 综合公式 k T J<To x(t e kobo dt Analysis equation 0 分析公式 a k Fourier series coefficients Spectral Coefficients 6
6 Chapter 3 Fourier Series §3.3.2 Determination of Fourier Series Representation ( ) jk t k k x t a e 0 + =− = ( ) 0 0 0 1 jk t k T a x t e dt T − = Synthesis equation 综合公式 Analysis equation 分析公式 k a ——Fourier Series Coefficients Spectral Coefficients
Chapter 3 Fourier series Example 3.5 Periodic square wave defined over one period as xlt < x 0T<t<T/2 TT/2-T10T1T/2 T +T/2 2T, x(t d T 2 ak=sinhoo k≠0 Defining sinc(x) SInd 2T sinc(koF) T 7
7 Chapter 3 Fourier Series Example 3.5 Periodic square wave defined over one period as ( ) = 0 T t / 2 1 t 1 1 T T x t 1 x(t) -T -T/2 –T1 0 T1 T/2 T t x(t)dt T a T T + = / 2 - /2 0 1 Defining ( ) x x c x sin sin = ( ) 0 1 1 sin 2 c k T T T ak = 0 1 sin 0 k k T a k k = T T1 2 =
Chapter 3 Fourier series 08 sin cloT 丌 05 sin c(oT) 0.4 17;2z Isin (oT) 0.2 0 -0.2 -0.4 -1.5 0.5 0.5 T固定,a的包络2Tsnc7定 8
8 Chapter 3 Fourier Series ( ) 1 1 sin T = c T ( ) 2 1 1 sin T = c T ( ) 3 1 1 sin T = c T T1固定，Tak 的包络 2T1 sin c(T1 )固定
Chapter 3 Fourier series Figure 4.2 2 4 (b)T=8T TIIDeanID MIID T=16T T↑→an=2丌/Tψ谱线变密
9 Chapter 3 Fourier Series T 0 = 2 /T 谱线变密 Figure 4.2 ( ) 4 1 a T = T ( ) 8 1 b T = T ( ) 16 1 c T = T
Chapter 3 Fourier series Example Periodic Impulse Trains(周期冲激串) x()=∑8(t-k7 k=-∞ 2T-TL0T 2T T/2T/2 k k=0,±1,±2,… T k 元 x()=∑e"r T k: 0002 ()=∑H/2n)m e
10 Chapter 3 Fourier Series Example Periodic Impulse Trains (周期冲激串) x(t) − 2T −T 0 T 2T t (1) ( ) 2 1 jk t T k x t e T + =− = ak T 1 -ω0 0 ω0 2ω0 −T / 2 T/ 2 0, 1, 2, 1 = k = T ak ( ) 2 1 2 jk t T k y t H jk e T T + =− = ( ) ( ) k x t t kT + =− = −