Discrete Fourier Series(DFS X[]=∑x1 CLet W=c ◆ Analysis equation:[k]=∑] 0 Synthesis equation: xIn/Ir 「kW如 N k=0 like cfs periodic discrete DES discrete periodic x[n]> X[k] discrete periodic discrete like dte periodic 14
14 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 discrete periodic
Ex 8 1 determine the des of a impulse train Consider the periodic impulse train [n]=∑[n-mN n=rN. r is any integer otherwise points -N-N+1 -2 2 N-1NN+1N+2 Solution N X[k]=∑[川]W=W=1 0 15
15 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…2 012…N-1NN+1N+2 i[n]=2o1n-rsI, n=rN, ris any integer 0. otherwise n points X k -NN+1 012………N1NN+1N+2… 2丌 X[k]=∑[]WM=W=1 N e 2丌 =∑X[] N N ∑ orthogonality 16
16 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 ~ 0 1 2 1 N j k N n 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 y[l]∑N[k-內N]NNN N Solutions .-2-1012 到{=∑[小]W=1∑N[小]W=W=1 k=0 对小]=∑x[k]Wx DES X[k]=∑年]
18 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 − = =
Example 8.2 Duality in the Discrete Fourier Series N points Y[k -N-N+1 2 2 N-1NN+1N+2 N points f{] N-N+1 2-1012 N-1NN+1N+2∴n 19
19 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 Example 8.2 Duality in the Discrete Fourier Series