Discrete Fourier series pair ◆ The fourier series:=∑x[kle j(2T/N)kn k=0 To obtain the Fourier series coefficients we multiply both sides by e (2/ N)rn for osn<N-1 and then sum both the sides we obtain n 2T(k-r)n ∑ x(ne ∑∑X(k)eN n=0 N k=0 ∑X(k)∑e (k-r)n k=0 n=0 88 Discrete Fourier Series Pair ◆The Fourier series: 1 0 2 ( ) N n j n N r x n e −  = −  0 1 1 0 2 ( ) ( ) 1 N n N k j k r n N N X k 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 −  1 1 0 0 2 ( ) ( ) 1 n N k N j n N k r N X k e − −  = = − =     (2 ) 1 / 0 1 [ ] j N N k kn x n e X N k  − = = 