Discrete fourier series pair the Fourier series representation of a periodic sequence =∑[k]e 2丌/Nkn To obtain the Fourier series coefficients we multiply both sides by e-/(2 N)rn for0≤n≤N-1 and then sum both the sides we obtain 2丌 N-1 ∑(n)eN"=∑∑X(k)eN k-r)n 0 =0 2丌 2丌 ∑ xnen ∑X(k)∑eN (k-r)n n=0 k=0 n=0 88 Discrete Fourier Series Pair   ( ) 1 0 1 2 / [ ] N k j N kn x n X k N e  − = =  ◆The Fourier series representation of a periodic sequence: 1 1 1 0 0 0 2 ( ) 2 1 ( ) ( ) N N N n n k j n k N N r j r n x n X k N e e − − −   = = = − −    = 1 1 0 0 1 0 2 ( ) 2 1 ( ) ( ) N N n k N n j r j k r n N n N x n X k N e e − −   = = − = − −  =   ◆To obtain the Fourier series coefficients we multiply both sides by for 0nN-1 and then sum both the sides , we obtain j n (2 / ) N r e − 