Discrete Fourier Series (DFS 2丌 区]=∑小e ◆LetW w-e N n=0 ◆ Analysis equation:[小]=∑斗小] n=0 ◆ Synthesis equation:x{小=∑X[k]W like cfs p ergodic discrete DES F[n>X[k n discrete like dtft periodic
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
Ex 81 determine the dFs of a impulse train Consider the periodic impulse train x小]=∑[n-rN]= n=rN, r is any integer 0. otherwise n point xn -N-N+1 -2-10 2∴……N-1NN+1N+2 Solution ￥[小]=∑6[小]W=W=1 7=0
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 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points xn ~ 1 0 0 1 N kn N N n X k n W W − = = = = Solution:
Ex,8.1 DFS of a impulse train N points -N-N+1 2-101 2 N-1NN+1N+2 rN. r is any integer F三-00 0 otherwise N points N-N+1 2 0 2 N-1NN+1N+2 [k]=∑8[]W如=W=1,=eN [小=∑[k]平=∑ j(2T/N)kn N k=0 k=0 15 orthogonality
15 Ex. 8.1 DFS of a impulse train 1 0 0 1, N kn N N n X k n W W − = = = = 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points X k ~ k 1 0 1 N kn N k x n X k W N − − = = 1, , 0, n rN r is any integer otherwise = = r x n n rN =− = − 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points xn ~ ( ) 1 2 0 1 N j N kn k e N − = = 2 N j W N e − = orthogonality
Example 8.2 Duality in the Discrete Fourier series If the discrete fourier series coefficients is the periodic impulse train, determine the signal N points F[小]=∑N{k-mN]N [] N 2-1012 Solution 到]=∑=∑N{T如=W=1 k=0 k=0 计=对w的=∑小 v k=0 n=0
17 Example 8.2 Duality in the Discrete Fourier Series ◆If the Discrete Fourier Series coefficients is the periodic impulse train, determine the signal. r Y k k rN N =− = − 1 0 1 N kn N k y n Y k W N − − = = 1 0 1 N kn N k x n X k W N − − = = 0 1 2 … … N … -2 -1 … … -N … N points N N N Y k Solution: DFS 1 0 0 1 1 N kn N N k N k W W N − − = = = = 1 0 N kn N n X k x n W − = =
N points N-N+1…-2-1012……N1NN+1N+2 N points -N-N+1 2…N-1NN+1N+2 n 18
18 k n 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points Y k N y n 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points 1