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 )

3 Fourier Series Representation of Periodic Signals joo, e aot: Fundamental components e/. e 120ol: Second harmonic components JNOot D- jNOot: Nth harmonic components So, arbitrary periodic signal can be represented as ∞e (Fourier series Example 3.2

3 Fourier Series Representation of Periodic Signals So, arbitrary periodic signal can be represented as j t j t e e 0 0 ,  −  : Fundamental components j t j t e e 0 0 2 2 ,  −  : Second harmonic components jN t jN t e e 0 0 ,  −  : Nth harmonic components  + =− = k j k t x t ak e 0 ( )  ( Fourier series ) Example 3.2

3 Fourier Series Representation of Periodic Signals (2) Representation for Real Signal Real periodic signal: X(t=X(t 在￠ So ak-a k x()=a+∑ koot+a-k ∑ 2 Relate] Let()ak=Ake aaOt (koot+Bk) x(t)=ao+>2Ak cos(koot+0k)

3 Fourier Series Representation of Periodic Signals (2) Representation for Real Signal Real periodic signal: x(t)=x*(t) So a*k=a-k  + =− = k j k t x t ak e 0 ( )    + = − − + = = + = + + 1 0 1 0 2Re[ ] ( ) [ ] 0 0 0 k j k t k j k t k k j k t k a a e x t a a e a e    Let (A) ( ) 0 0 , k k j k t k j k t k j k k a A e a e A e    + = =  + =  = + + 1 0 0 ( ) 2 cos( ) k k k x t a A k t 

3 Fourier Series Representation of Periodic Signals Let(A)ak=Ager, aR e koof=Age(kooftR) x(t)=ao+>2Ak cos(koot+8%) (B)ak=B+jCk x(t)=a0+2>[Bk cos koot-Ck sin koot]

3 Fourier Series Representation of Periodic Signals Let (A) ( ) 0 0 , k k j k t k j k t k j k k a A e a e A e    + = =  + =  = + + 1 0 0 ( ) 2 cos( ) k k k x t a A k t  (B) k k k a = B + jC ( ) 2 [ cos sin ] 0 1 0 0 x t a B k t C k t k  k  k  + =  = + −

3 Fourier Series Representation of Periodic Signals 3.3.2 Determination of the Fourier series Representation of a Continuous-time periodic Signal koot dk(t)=e2x)y,k=0,±1,+2 Orthogonal function set Determining the coefficient by orthogonality Multiply two sides by e jnogt x()em=∑ake(k-n)o

3 Fourier Series Representation of Periodic Signals k (t) = e j k(2 /T )t , k = 0,1,2 3.3.2 Determination of the Fourier Series Representation of a Continuous-time Periodic Signal  + =− = k j k t x t ak e 0 ( )  ( Orthogonal function set ) Determining the coefficient by orthogonality: ( Multiply two sides by )  + =− − − = k j k n t k j n t x t e a e 0 0 ( ) ( )   jn t e − 0

3 Fourier Series Representation of Periodic Signals k-n) T. k O.k≠n 「x()emdh=∑ ak je/ck-moola 「,x()e Fourier Series Representation x(t ( Synthesis equation TJrx(r)ejkoo' dt(Analysis equation)

3 Fourier Series Representation of Periodic Signals Fourier Series Representation:     = =  − k n T k n e dt T j k n t 0, , 0 ( ) x t e dt a e dt ak T k T j k n t k T j n t  =  =   + =− − 0 − 0 ( ) ( )    −  = T j n t n x t e dt T a 0 ( ) 1         = =   − + =− T j k t k k j k t k x t e dt Analysis equation T a x t a e Synthesis equation ( ) ( ) 1 ( ) ( ) 0 0  