離散傅立葉變換(本科)-文庫吧資料
2025-01-26 09:54本頁面
【正文】 限長序列： ( ) ( ) ( )o p o Nx n x n R n?*1 / 2 [ ( ( ) ) ( ( ) ) ] ( )N N Nx n x N n R n? ? ?圓周共軛反對稱序列： ( ) ( ) ( )ep e Nx n x n R n?*1 / 2 [ ( ( ) ) ( ( ) ) ] ( )N N Nx n x N n R n? ? ?圓周共軛對稱序列： 離散傅里葉變換的性質(zhì) 圓周共軛對稱序列滿足： *( ) ( ( ) ) ( )e p e p N Nx n x N n R n?? R e [ ( )] R e [ (( )) ( )]e p e p N Nx n x N n R n??實(shí)部圓周偶對稱 I m [ ( )] I m [ (( )) ( )]e p e p N Nx n x N n R n? ? ?虛部圓周奇對稱 ( ) ( ( ) ) ( )e p e p N Nx n x N n R n??幅度圓周偶對稱a rg [ ( )] a rg [ (( )) ( )]e p e p N Nx n x N n R n? ? ?幅角圓周奇對稱 離散傅里葉變換的性質(zhì) 離散傅里葉變換的性質(zhì) 圓周共軛反對稱序列滿足： *( ) ( ( ) ) ( )o p o p N Nx n x N n R n? ? ? R e [ ( )] R e [ (( )) ( )]o p o p N Nx n x N n R n? ? ?實(shí)部圓周奇對稱 I m [ ( )] I m [ (( )) ( )]o p o p N Nx n x N n R n??虛部圓周偶對稱 ( ) ( ( ) ) ( )op op N Nx n x N n R n??幅度圓周偶對稱 幅角沒有對稱性離散傅里葉變換的性質(zhì) *( ) ( ( ) ) ( )o p o p N NX k X N k R k? ? ?*1 / 2 [ ( ( ) ) ( ( ) ) ] ( )N N NX k X N k R k? ? ?( ) ( ) ( )e p o pX k X k X k??同理： *1 / 2 [ ( ( ) ) ( ( ) ) ] ( )N N NX k X N k R k? ? ?*( ) ( ( ) ) ( )e p e p N NX k X N k R k??其中： 離散傅里葉變換的性質(zhì) 序列 DFT 共軛對稱性 ( ) ( )x n X kR e [ ( )] ( )epx n X kI m [ ( )] ( )opj x n X k( ) R e [ ( ) ]epx n X k( ) I m [ ( ) ]opx n j X k離散傅里葉變換的性質(zhì) 序列 DFT 實(shí)數(shù)序列的共軛對稱性 R e [ ( )] ( ) ( )epx n X k X k?I m [ ( )] 0 ( ) 0opj x n X k??( ) R e [ ( ) ]epx n X k( ) I m [ ( )]opx n j X k離散傅里葉變換的性質(zhì) 純虛序列的共軛對稱性 序列 DFT R e[ ( )] 0 ( ) 0epx n X k??I m [ ( )] ( ) ( )opj x n X k X k?( ) R e [ ( ) ]epx n X k( ) I m [ ( )]opx n j X k離散傅里葉變換的性質(zhì) 例：設(shè) x1(n)和 x2(n)都是 N點(diǎn)的實(shí)數(shù)序列，試用一次 N點(diǎn) DFT運(yùn)算來計(jì)算它們各自的 DFT: 11[ ( ) ] ( )D F T x n X k? 22[ ( ) ] ( )D F T x n X k?解：利用兩序列構(gòu)成一個(gè)復(fù)序列12( ) ( ) ( )w n x n j x n??12( ) [ ( )] [ ( ) ( )]W k D F T w n D F T x n j x n? ? ?則12[ ( )] [ ( )]D F T x n j D F T x n??12( ) ( )X k j X k??離散傅里葉變換的性質(zhì) 1 ( ) R e[ ( )]x n w n?由得11( ) [ ( )] { R e [ ( )]} ( )epX k D F T x n D F T w n W k? ? ?*1 [ (( )) (( )) ] ( )2 N N NW k W N k R k? ? ?2 ( ) I m [ ( )]x n w n?由得221( ) [ ( ) ] { I m [ ( ) ] } ( )opX k D F T x n D F T w n W kj? ? ?*1 [ ( ( ) ) ( ( ) ) ] ( )2 N N NW k W N k R kj? ? ?離散傅里葉變換的性質(zhì) ( ) 2 DF T ( ) 2 DF T : ( )x n N Nx n N X k例：設(shè) 是 點(diǎn)實(shí)數(shù)序列，試用一次 點(diǎn)來計(jì)算 的 點(diǎn)()xn解：將 按奇偶分組，令12( ) ( 2 ) 0 , 1 , ..., 1( ) ( 2 1 ) 0 , 1 , ..., 1x n x n n Nx n x n n N? ? ?? ? ? ?12 ( ) ( ) ( )w n x n j x n??構(gòu)成一個(gè)復(fù)序列12( ) DF T( ) [ ( ) ] ( ) ( )w n NW k DF T w n X k jX k? ? ?對 進(jìn)行一次 點(diǎn) 運(yùn)算 得12( ) ( )1( ) ( )epopX k W kX k W kj?? D F TN均為 點(diǎn)( ) 2 D F TX k N而 是 點(diǎn)?離散傅里葉變換的性質(zhì) 復(fù)共軛序列 * * *[ ( ) ] ( ( ) ) ( ) ( ( ) ) ( )N N N ND F T x n X k R k X N k R k? ? ? ?1**0[ ( ) ] ( ) ( )NnkNNnD F T x n x n W R k??? ?證：*10( ) ( )NnkNNnx n W R k??????????? * (( )) ( )NNX k R k??*1()0( ) ( )NN k nNNnx n W R k???????????* (( )) ( )NNX N k R k??離散傅里葉變換的性質(zhì) ? ?? ? ? ? ? ?** NND FT x n R n X k??????1**0[ ( ( ) ) ( ) ] ( ( ) ) ( )NnkN N N N NnD F T x n R n x n R n W??? ? ??證：*10( ( ) )NnkNNnx n W????????????*10( ( ) )NmkNNmx m W???????????? ?? ?*10NnkNNnx n W??????????* ()Xk?mn??令? ?*10NnkNnx n W??????????離散傅里葉變換的性質(zhì) DFT形式下的 Parseval定理 11**001( ) ( ) ( ) ( )NNnkx n y n X k Y kN???????*1 1 1*0 0 01( ) ( ) ( ) ( )N N N nkNn n kx n y n x n Y k WN? ? ??? ? ????????? ? ?證： 11*001 ( ) ( )NN nkNknY k x n WN????? ??1*01 ( ) ( )NkX k Y kN??? ?離散傅里葉變換的性質(zhì) 11**001( ) ( ) ( ) ( )NNnkx n x n X k X kN???????11 22001( ) ( )NNnkx n X kN???????即： ( ) ( )y n x n?令 ，則離散傅里葉變換的性質(zhì) 圓周卷
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