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

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