8.1 Representation of Periodic Sequence: the discrete Fourier series 2丌 (k+mn)n 2丌m ,k=0,1,2,…,N-1 Due to periodicity, we only need N complex exponentials for Fourier series representation of x[n] 2兀k 刘n=∑X[eN k Ankle N k=0 The Fourier series representation of periodic sequence x[n]8 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  =  ( ) 2 j k N N m n e  + ( ) 2 2 j kn N j mn e e   = 2 j kn N e  = , 0,1,2, , 1 k N = −