华侨大学：《信号与系统》课程教学资源（PPT课件讲稿，机电版）证明3-3 圆周卷积特性证明

圆周卷积特性证明x(m)③h(m)、 X(k),H(k)乘 若:x(1) DY(k)M()D→H(k) DET DET 设:Y(k)=X(k)H(k ∑Y(k),e 则:y(m)=DF((k)-Np N ∑X(k)H(,e ∑∑x(m):enH(k) ikon k=0m=0 ∑x(mi∑eAm,H(k)A y(n)=>x(m).h(n-m)RN(n)=x(n)@h(n)

X 第 1 圆周卷积特性证明 页 1 0 ( ) ( ) (( )) ( ) N N N m y n x m h n m R n − = =  −  Y k X k H k ( ) ( ) ( ) = 0 1 0 1 ( ) N jk n k Y N k e −  = =   0 0 1 1 0 0 1 [ ] ( ) ( ) N jk m m N jk n k m e H k e N x − − = −  =  =     0 1 0 ( ) ( ) 1 N jk n k k e N X H k −  = =    y n IDF ( ) ( ) = T Y k( ) ( ) ( ) DFT IDFT x t X k ⎯⎯⎯ ⎯⎯⎯→ ( ) ( ) DFT IDFT 若： h t H k ⎯⎯⎯ ⎯⎯⎯→ 0 0 1 1 0 0 1 ( )[ ( )] N N jk m jk n m k x m e H k e N − − −   = = =    设： 则： =  x n h n ( ) ( ) ( ) ( ) ( ) ( ) DFT    x n h n X k H k ( ) ( ) ( ) ( ) DFT x n h n X k H k   0 (( )) ( ) ( ) DFT N N I j mk DFT x n m R n e X k −  −