8.1 Representation of Periodic Sequence: the discrete Fourier series 丌 2丌 +mIn tMn e k=0,1,2,…,N-1 Due to the periodicity we only need N complex exponentials for discrete time Fourier series =∑]2x0 ∑[k]e k=0 The Fourier series representation of a periodic sequence.7 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆Due to the periodicity ,we only need N complex exponentials   (2 ) 1 0 1 / [ ] N k j N kn x n X k e N  − = =    1 (2 / ) [ ] 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 = − for discrete time Fourier series: The Fourier series representation of a periodic sequence