第08章 DFT(下)

第08章 DFT(下)

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11 2 2 [] , [] DFT x n X kx n X n Xk ↔↔ 8.6 properties of the Discrete Fourier transform 12 1 2 ax n bx n aX k bX k +↔ + 2. circular shift (循环或圆周移位) of a sequence [(( )) ] [ ] km NN N xn m R n WX k −↔ [ ( ( ) )] [] nl N Wx n l R k ↔− Assume: 1. linearity
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Figure 8.12 EXAMPLE. 图示循 环移位 ] [ ] ))) ( [(( ] [ ] )) [(( ] [ 1 n R m N n x n R m n x n x N N N N + = = Definition of circular shift of a sequence
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8.42 EXAMPLE. ] [ 1 n h ] [ 2 n h 8points DFT | ] [ | 1 k H | ] [ | 2 k H 1024 points DFT | ) ( | 1 ω j e H | ) ( | 2 j e H 当成 8 点信号是循环 移位关系,当成 1024 点则不是。
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) 2 . 0 cos( ] [ n n n x π = | ]} [ { | | ] [ | n x DFT k X = ]} [ { n X DFT EXAMPLE. [ ], 1,. ., 1 [ ] [(( )) ] [ ] [(( )) ] [ ] [0], 0 DFT NN Nx N k k N Xn N x k R k N x N k Nx k =− ↔− = = = 3. Duality (对偶性)
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] [ ] [ ] )) [(( ] [ ] )) [(( k N X k R k N X k R k X N N N N = = ] [ k X '[ ] [ ] Xk X N k =− 近似写法
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** * * [ ] [(( )) ] [ ] [ ] [ ] DFT NN x nX k R k X N k R k X N k −= * [(( )) ] [ ] [ ] [ ] DFT x nR n x N n X k 11 [ ] ( [ ] *[ ]) ( [ ] *[ ]) Re{ [ ]} 22 DFT ep xn x n x Nn X k X k X k =+ + = Re{ [ ]} ( [ ] *[ ]) ( [ ] ]) [ ] DFT ep x nx n x n X k X N k X k =+↔ + = Im{ [ ]} ( [ ] *[ ]) ( [ ] ]) [ ] DFT op jx n x n x n X k X N k X k =− = [] ([] * [ ] ) ( [] * ) Im { [] } DFT op x n x N n X k X k j X k = 4. Symmetry properties
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Here, we define: : the periodic conjugate-symmetric components (圆周(周期)共轭对称分量) : the periodic conjugate-antisymmetric components (圆周(周期)共轭反对称分量) ] [ k X ep ] [ k X op ] [ ]) [ ] [ ( ] [ ] [ ]) [ ] [ ( ] [ ] [ ] [ ] [ k N X k N X k X 2 1 k X k N X k N X k X 2 1 k X where, k X k X k X * op * op * ep * ep op ep = = = + = + = Any finite-length sequence can be decomposed as: The length of the three sequences are all N.
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] [ * ] [ k N X k X = | [ ]| | [ ]| [] [ ] Xk XN k = = −− (( ]} [ Im{ ]} [ Im{ ]} [ Re{ ]} [ Re{ k N X k X k N X k X = = ] [ * ] [ n x n x = 5. for a real sequence 圆周(周期)共轭对称:周期延拓后共轭对称
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N=10 EXAMPLE. 实序列 DFT 9 ... 0 ), 5 . 0 cos( 5 . 0 ] [ = + = n n n x n π Real{X[k]} Imag{X[k]} |X[k]| arg{X[k]}
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N=9 (奇数点) |X[k]| Arg{X[k]}
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时域 FT DFT FT 的取样) 相对原点共轭对称 因果相对 a 共轭对称 周期共轭对称 因果矩形序列 不存在 FT DFT 均为实的序列。 FT DFT 的对称性质比较(习题 8.46 需要) 函数(广义线性相位) 广义线性相位函数 广义线性相位序列 非线性相位复数函数 序列(广义线性相位) 广义线性相位函数 实序列(广义线性相位)
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性质 4 5 思考题 A B C 是实序列 D 1 )序列 ,傅立叶变换和 5 DFT
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This note was uploaded on 02/29/2012 for the course EE 325 taught by Professor Lilili during the Fall '11 term at BYU.