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
4 The continuous time Fourier transform Fourier transform X(O x(te o di (t) X(oeo do 2元 or x(t>Xo) Relation between fourier series and Fourier transform X(lasko (Periodic signal X(o=T ak ko.so(Aperiodic signal
4 The continuous time Fourier transform Fourier transform: = = + − + − − x t X j e d X j x t e dt j t j t ( ) 2 1 ( ) ( ) ( ) Relation between Fourier series and Fourier transform: = = = = ( ) 0 0 ( ) | ( )| 1 Aperiodic signal (Periodic signal) k k k k X j T a X j T a or x(t) X( j) ⎯F →
4 The continuous time Fourier transform X T X(w) 2T T
4 The continuous time Fourier transform
4 The continuous time Fourier transform 4.1.2 Convergence of Fourier transform Dirichlet conditions: (1)x(t is absolutely integrable x(t dt <oo (2)x(t have a finite number of maxima and minima within any finite interval (3)x(t have a finite number of discontinuity within any finite interval. Furthermore, each of these discontinuities must be finite
4 The continuous time Fourier transform 4.1.2 Convergence of Fourier transform Dirichlet conditions: (1) x(t) is absolutely integrable. (2) x(t) have a finite number of maxima and minima within any finite interval. (3) x(t) have a finite number of discontinuity within any finite interval. Furthermore, each of these discontinuities must be finite. + − | x(t)| dt
4 The continuous time Fourier transform 4.1.3 Examples of Continuous time Fourier Transform EXample4.14243444.5 EXample(1) (1)=e(>X(j0)=2n6(0-00 Solution:x(t)=X()eiondo 2(O-Ooeloda 2元 EXample(2) x(t)=CosO>X(0)=7(0-O0)+x(O+O0
4 The continuous time Fourier transform 4.1.3 Examples of Continuous time Fourier Transform Example 4.1 4.2 4.3 4.4 4.5 Example (1) ( ) ( ) 2 ( )0 0 x t = e ⎯→ X j = − j t F Example (2) ( ) cos ( ) ( ) ( ) = 0 ⎯→ = −0 + +0 x t t X j F + − + − = − = e e d Solution x t X j e d j t j t j t 2 ( ) 2 1 ( ) 2 1 : ( ) 0 0
4 The continuous time Fourier transform 4. 2 The Fourier Transform for Periodic Signal Periodic signal x(t)= rEeked ko ”>2(O-k0) thus x(1)=∑ae")X(jo)=∑a2r(a-km) EXample 4.6 4.7 4.8
4 The continuous time Fourier transform 4.2 The Fourier Transform for Periodic Signal Periodic signal: 2 ( )0 0 e k j k t F ⎯→ − thus + =− = k j k t x t ak e 0 ( ) + =− + =− = ⎯→ = − k k F k j k t k x(t) a e X ( j ) a 2 ( k ) 0 0 Example 4.6 4.7 4.8
