8.1 Representation of Periodic Sequence: the Discrete Fourier Series )m) 2元 JN ,k=0,1,2,…,W-1 Due to periodicity,we only need N complex exponentials for Fourier series representation of [n]: 网∑[e祭n =之[]e: The Fourier series representation of periodic sequence [n].7 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆Due to periodicity,we only need N complex exponentials   1 0 2 1 [ ] j N N k kn x n X k e N −  = =    2 1 [ ] N k j kn x n X k e N  =  , 0,1,2, , 1 k N = − ( ) 2 j k N N m n e  + 2 j n N k e  = ( ) 2 2 j n j m k N n e e   =