信号与系统（英文版）_4 The continuous time Fourier transform

4 The continuous time Fourier transform 4. The Continuous time Fourier Transform 4.1 Representation of aperiodic signals The Continuous time fourier transform 4.1.1 Development of the Fourier transform representation of the continuous time Fourier transform

4 The continuous time Fourier transform 4.1 Representation of Aperiodic signals: The Continuous time Fourier Transform 4.1.1 Development of the Fourier transform representation of the continuous time Fourier transform 4. The Continuous time Fourier Transform

4 The continuous time Fourier transform (1)Example(From Fourier series to Fourier transform) x(t) 口∏,,∏ T 2 A 4o0

4 The continuous time Fourier transform (1) Example ( From Fourier series to Fourier transform )

4 The continuous time Fourier transform 2)Fourier transform representation of Aperiodic sIgnal For periodic signal x(t) ae TJr x(te-jkootdt For aperiodic signal x(t x()=1im()或X()-120>x()

4 The continuous time Fourier transform (2) Fourier transform representation of Aperiodic signal        = =   − + =− T j k t k k j k t k x t e dt T a x t a e 0 0 ( ) 1 ~ ( ) ~   For periodic signal : ( ) ~ x t For aperiodic signal x(t) : ( ) ( ) ~ ( ) ~ ( ) x t limx t x t x t T T = ⎯ → ⎯ → → 或

4 The continuous time Fourier transform T 2T-7T10T1T

4 The continuous time Fourier transform T→

4 The continuous time Fourier transform When t>∞,x()-32,x() 丌 do T→∞ So a,T (t)e o dt=X(o (t)=lin foot T→>∞ lim Xoko roOt ∑X(kOo)e/m 7 Xoe

4 The continuous time Fourier transform When T→ ,      ⎯ ⎯→ = ⎯ ⎯→ ⎯ ⎯→ → → → T T T k d T x t x t 0 0 2 ( ) ( ) ~ So ( ) ( )  a T x t e dt X j j t k = =  + − −     + − + =− → + =− → + =− → = = = =             X j e d X j k e e T X j k x t a e j t k j k t k j k t T k j k t k T ( ) 2 1 2 lim ( ) ( ) lim ( ) lim 0 0 0 0 0 0 0