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