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）

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2.2 The Discrete-Time Fourier transform ◆ Symmetry relations We describe here some additional properties of the Fourier transform that are based on the symmetry relations. These properties can simplify the computational complexity and are often useful in digital signal processing applications

2.2 The Discrete-Time Fourier transform ◆ Symmetry relations We describe here some additional properties of the Fourier transform that are based on the symmetry relations. These properties can simplify the computational complexity and are often useful in digital signal processing applications

2.2 The Discrete-Time Fourier transform For a given sequence x[n] with a Fourier transform x(eo),it is easy to determine the fourier transform of its time reversed sequence xI-n and the complex conjugate sequence xInI F{x[-m}=∑xem=∑x[mem=x(e") F{[=∑x小em=∑ xInle X e n=-00 x[-m}=x(e)

2.2 The Discrete-Time Fourier transform For a given sequence x[n] with a Fourier transform ,it is easy to determine the Fourier transform of its time￾reversed sequence x[-n] and the complex conjugate sequence x*[n]. ( ) j X e         ( ) j n j m j n m F x n x n e x m e X e     − − − =− = − = − = =          ( ) * * * * j n j n j n m F x n x n e x n e X e     − − − =− =   = = =          ( ) * * j F x n X e  − =

2.2 The Discrete-Time Fourier transform A Fourier transform x(e)is defined a conjugate symmetric function of w if X(eo)=x(e) That is Xre(elo)=Xre(e jo X Xle

2.2 The Discrete-Time Fourier transform A Fourier transform is defined a conjugate￾symmetric function of ω if ( ) j X e  ( ) ( ) j j * X e X e   − = That is, ( ) ( ) j j X e X e re re   − = ( ) ( ) j j X e X e im im   − = −

2.2 The Discrete-Time Fourier transform The Fourier transform x(eo) is a conjugate-antisymmetric function ofo if Xle! x e J That is X(e0)=-X Xe=Xe

2.2 The Discrete-Time Fourier transform The Fourier transform is a conjugate-antisymmetric function of ω if ( ) j X e  That is, ( ) ( ) j j * X e X e   − = − ( ) ( ) j j X e X e re re   − = − ( ) ( ) j j X e X e im im   − =