Discrete fourier series pair 1 N1 2T(k-r)n 1,k-r=N, m an integer ∑ 0 n=0 otherwise orthogonality of the complex exponentials Problem 8.51 丌 rn 27(k-r)n x(ne =∑(k)∑ =0 X(r+mN) k=0 0 X(k)=∑ 2兀kn X(r r(new Periodic n=0 DFS coefficients=1∑[k]e2m k=0 The discrete Fourier series9 Discrete Fourier Series Pair 1 0 2 1 ( ) N n j k r n N N e −  = −  = X ( )r Problem 8.51 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 − −  = = − =  1, - , 0, k r mN m an integer otherwise  = =     (2 ) 1 / 0 1 [ ] j N N k kn x n e X N k  − = =  1 0 2 ( ) N n j n N k x n e −  = − X ( ) k = = X ( ) r + mN The Discrete Fourier Series coefficients Periodic orthogonality of the complex exponentials. DFS