信号与系统（英文版）_3 Fourier Series Representation of Periodic Signals

3 Fourier Series Representation of Periodic Signals 3. Fourier Series Representation of Periodic Signal 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 Fourier Series Representation of Periodic Signals 3.Fourier Series Representation of Periodic Signal 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 Fourier Series Representation of Periodic Signals 3. 2 The Response of LTI Systems to Complex Exponentials (1 Continuous time LTI system x(t=es y(t=H(s)est h (t y(t)=x(1)*h(t Tn(TaT on es(n(t az=estf+oo h(red H(S H(s)=h(r)e dr system function

3 Fourier Series Representation of Periodic Signals 3.2 The Response of LTI Systems to Complex Exponentials (1) Continuous time LTI system h(t) x(t)=est y(t)=H(s)est ( ) ( ) ( ) ( ) ( )* ( ) ( ) ( ) ( ) e H s e h d e h e d y t x t h t x t h d s t s t s t s = = = = = −    + − − + − − + −           + − − =    H s h e d s ( ) ( ) ( system function )

3 Fourier Series Representation of Periodic Signals (2)Discrete time LTI system yIn]=H(zzn hnI y=*小=∑xn一 ∑=k]="∑ "H(=) H()=∑hkF system function

3 Fourier Series Representation of Periodic Signals (2) Discrete time LTI system h[n] x[n]=zn y[n]=H(z)zn ( ) [ ] [ ] [ ] [ ]* [ ] [ ] [ ] ( ) z H z z h k z z h k y n x n h n x n k h k n k n k k n k k = = = = = −    + =− − + =− − + =− k k H z h k z − + =− ( ) =  [ ] ( system function )

3 Fourier Series Representation of Periodic Signals (3)Input as a combination of Complex Exponentials Continuous time Lti system x()=∑ae y()=∑akH(Sk)e Discrete time LTI system x{m]=∑ yn=∑akH(=k)=k EXample 3.1

3 Fourier Series Representation of Periodic Signals (3) Input as a combination of Complex Exponentials Continuous time LTI system:        = =   = = N k s t k k N k s t k k k y t a H s e x t a e 1 1 ( ) ( ) ( ) Discrete time LTI system:        = =   = = N k n k k k N k n k k y n a H z z x n a z 1 1 [ ] ( ) [ ] Example 3.1

3 Fourier Series Representation of Periodic Signals 3.3 Fourier Series Representation of Continuous-time Periodic Signals 3.3. 1 Linear Combinations of harmonically Related Complex exponentials (1) General Form The set of harmonically related complex exponentials ΦA(t) ik(2T/T) k=0±1±2 Fundamental period: T( common period

3 Fourier Series Representation of Periodic Signals 3.3 Fourier Series Representation of Continuous-time Periodic Signals (1) General Form k (t) = e j k0 t = e j k(2 /T )t , k = 0,1,2 3.3.1 Linear Combinations of Harmonically Related Complex Exponentials The set of harmonically related complex exponentials: Fundamental period: T ( common period )