Discrete Fourier Series Pair k-r)n 1,k-r=mN,mis an integer 10,otherwise Using orthogonality of the complex exponentials: =]=[k灯 n=0 k=0 0≤r,k≤W-1 0only when k-r=0 (m=0,k=r) Xk-岁ae n=0 D5之a coefficients The Discrete Fourier Series Periodic 10
0 only when k − =r 0 10 Discrete Fourier Series Pair 1 0 2 1 ( ) N n j k r n N N e − = − = = X X [ ] [ ] r k 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 mN mis aninteger othe e r rwis = = (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 = The Discrete Fourier Series coefficients Periodic Using orthogonality of the complex exponentials: DFS 0 1 − r k, N ( 0, ) m k = = r
8.1 Representation of Periodic Sequence:the Discrete Fourier Series a periodic sequence with period N, n =n+rN]for any integer r The Discrete Fourier Series: (defination) =∑[]e京 Synthesis N k=0 equation Coefficients:(is periodic with period N) k]=∑nle0u、 N一 Analysis equation n=0
11 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆a periodic sequence x n with period N, x n x n rN for any integer r = + ◆The Discrete Fourier Series: 1 0 2 [ ] [ ] , N n j kn X k x n N e − = − = 1 0 2 1 [ ] , N k j kn N x n X k N e − = = Synthesis equation Analysis equation ◆Coefficients: (defination) (is periodic with period N) (
8.1 Representation of Periodic Sequence:the Discrete Fourier Series X[]=∑[n]e2xa n=0 The sequence]is periodic with period N [o]=[],[]=[N+] k+N-立远ee n=( n=0 经e如eP和-[个 = 17
12 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆The sequence is X k periodic with period N X X X X 0 , 1 1 = = + N N ( ) 1 0 N 2 n j k n N N X k x n N e − = − + + = 1 0 2 N 2 n j kn N j n x n X k e e − = − − = = ( ) 1 0 2 N n j N kn X k x n e − = − = 1 0 N 2 2 n j k n n j N N N x n e e − = − − =
Discrete Fourier Series (DFS) []=∑n]e ◆Let Wv=e N ◆Analysis equation:[k]=∑[nW n=0 ◆Synthesis equation:]-=∑[m 三0 like CFS periodic discrete diserete periodicnk periodic discrete discrete like DTFT periodic 13
13 Discrete Fourier Series (DFS) ◆Let 2 N j W N e − = 1 0 N kn N n X k x n W − = = DFS x n X k ◆Analysis equation: 1 0 1 N kn N k x n X k W N − − = ◆Synthesis equation: = 1 0 N 2 n j kn N X k x n e − = − = periodic discrete like CFS discrete like DTFT periodic discrete periodic periodic discrete (2 / ) ( ) ( ) , j t k k T k x t x t a e T + =− + = =
Ex.8.1 determine the DFS of a impulse train Consider the periodic impulse train -之-则-6wm N points -N-N+1·-2-101 2·N-1NN+1N+2 Solution:X[附-∑sm%==for all k W-e彩 n 大名刚县*=大艺 orthogonality 14
14 Ex. 8.1 determine the DFS of a impulse train ◆Consider the periodic impulse train 1, , 0, r n rN r is any integer x n n rN otherwise =− = = − = n -N -N+1…… -2 -1 0 1 2 …… N-1 N N+1 N+2 …… N points xn ~ 1 1 0 0 1 , N kn N n N X k nW W for all k − = Solution: = = = 1 0 1 N kn N k x n X k W N − − = = 0 1 2 1 N j k N n k e N − = = orthogonality 2 N j W e N − = 1 0 1 N kn N k W N − − = =