延安大学：《数字信号处理》课程PPT教学课件（DigitalSignal Processing，DSP）Chapter 2 The Discreete-Time Fourier Transform（DTFT）

2.2 The Discrete-Time Fourier transform The Fourier transform x(elo )can been represent as a sum of conjugate-sysmmetric part and conjugate-antisymmetric part Xe Xdleo+Xle Where X(e+Xle Xle

2.2 The Discrete-Time Fourier transform The Fourier transform can been represent as a sum of conjugate-sysmmetric part and conjugate-antisymmetric part ( ) j X e  ( ) ( ) ( ) j j j X e X e X e CS CA    = + where ( ) ( ) ( ) 1 * 2 j j j X e X e X e CS    − = +     ( ) ( ) ( ) 1 * 2 j j j X e X e X e CA    − = −    

2.2 The Discrete-Time Fourier transform Likely, the time-domain sequence xn] can be defined as a conjugate-symmetric and conjugate-antisymmetric sequence ofn if x[小]=-x[-n]

2.2 The Discrete-Time Fourier transform Likely, the time-domain sequence x[n] can be defined as a conjugate-symmetric and conjugate-antisymmetric sequence of n if     * x n x n = − −     * x n x n = −

2.2 The Discrete-Time Fourier transform The discrete-time sequence x n can been represent as a sum of conjugate-sysmmetric sequence and conjugate-antisymmetric sequence x[小]=xcs[n+xal[小 Where s团小=元x小+x] CA =x川 2

2.2 The Discrete-Time Fourier transform The discrete-time sequence x[n] can been represent as a sum of conjugate-sysmmetric sequence and conjugate-antisymmetric sequence x n x n x n   = + CS CA     where       1 * 2 CS x n x n x n = + −           1 * 2 CA x n x n x n = − −    

2.2 The Discrete-Time Fourier transform A complex-valued Fourier transform x[n], in general, can be expressed as a sum of a real part xreln] and a imaginary part Xim[n]. thus x[]=xm[小]+rxm[n

2.2 The Discrete-Time Fourier transform A complex-valued Fourier transform x[n], in general, can be expressed as a sum of a real part xre[n] and a imaginary part xim[n]. thus x n x n jx n   = + re im    

2.2 The Discrete-Time Fourier transform Symmetry relations We next derive the Fourier transforms of xre [n] and xim In the real and imaginary parts of the sequence xn respectively. Then F =X(e′ F

2.2 The Discrete-Time Fourier transform • Symmetry relations We next derive the Fourier transforms of and , the real and imaginary parts of the sequence x[n], respectively. Then x n re   x n im      ( ) j F x n X e re cs  =    ( ) j F jx n X e im ca  =