Discrete Fourier series pair e点m(,kr=mN, n an intege 0. otherwise n=0 orthogonality of the complex exponentials Problem x(ne =∑(k)1 8.51 (k-r) 0 石N三X(+mN) (4)=∑(n)eNk=x() Periodic DFS N coefficients[n]=>X[k]e2r/n) k=0 The discrete Fourier series 1010 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 r→k