华侨大学：《信号与系统》课程教学资源（PPT课件讲稿，机电版）DTF周期性证明

DFT周期性证明 X(k)=∑x(m7),e"N X(k±N)=∑xm)e7 2丌 ∑[x(n)e"N,eFm2] ∑x(m)e"N=X(k) n=0 X(k±N)=X(k)

X 第 1 页 DFT周期性证明 1 2 0 ( ) ( ) N jnk N s n X k x nT e −  −  = =   1 2 0 2 [ ( ) ] N jn N n jn x n e e    − −  = =    1 2 ( ) 0 ( ) ( ) N jn k N N n X k N x n e −  −   =  =   1 2 0 ( ) ( ) N jn N n x n e X k −  −  = =  =    = X k N X k ( ) ( )