华科奥本海姆讲义4.ppt

CHAPTER 4 THE CONTINUOUS-TIME FOURIER TRANSFORM INTRODUCTION ? Represent continuous-time aperiodic signals as binations plex exponentials. ? Fourier transform and inverse Fourier transform. (傅立叶变换) (傅立叶逆变换) ? Use Fourier methods to analyze and understand signals and LTI systems. REPRESENTATION OF APERIODIC SIGNALS: THE CONTINUOUS-TIME FOURIER TRANSFORM ……-2 T -T -T 1 T 1 T 2T t )( ~tx???? 2/2/ 0)( ~ 1 TT t jk kdtetxT a ? x(t) -T 1 T 1t As , ?? T???? 2/2/ 0)( ~ TT t jk kdt etx Ta ??????????????? dtetx dt etx Ta tj t jk k ??)()( 0 Use X(jω) to denote this integral, then we have: ???????dtetxjX tj??)()( Since ??,2 0??? kka Ta jX?? spectrum of x(t ) Although X(jω) is often abbreviated as “ spectrum ”, it is different from a k , which is the spectrum of periodic signals. Fourier transform of x(t) X(jω) is in fact spectrum-density function( 频谱密度函数) ??????? k t jkkeatx 0)( ~ ? As , ?? T?????, ),()( ~ 0??dtxtx And the part in the brackets is X(jω). Thus, we obtain ??????????dejXtx tj)(2 1)( Inverse Fourier transform ???????????????? k t jk TT t jkedtetxT 0 02/2/)( ~ 1 ?????????????????? k t jk TT t jkedtetx 0 02/2/ 0)( ~2 ???????????dtetxjX tj??)()(??????????dejXtx tj)(2 1)( ka djX????2 )( Fourier transform pair An useful relationship: 0)( 1 ??? k kjXT a ?? where X(jω) is the Fourier transform of x(t ), a k is the Fourier coefficients of . x(t ) is one period of the periodic signal )( ~tx)( ~tx ? Convergence of Fourier Transforms Dirichlet conditions: (t ) is absolutely integrable ; that is ???????dttx)( 2. x(t ) have a finite number of maxima and minima within any finite interval. 3. x(t ) have a finite number of discontinuities within any finite interval. Furthermore, each of these discontinuities must be finite. If impulse functions are permitted in the transform, some signals which are