Duality in Discrete Y Fourier series [=M到=x[可 丌 N points xn N ∑x[k k=0 n -N-N+1….-2-1012 N-1NN+1N+2 N_ points QLkn i Xk X(k)=∑(n1eN k -NN+1……2-∠-1012 N-1NN+1N+2 N points ] k -N-N+1 01 2 N-1NN+1N+2 N points 19-N-N+1…-2 01 2 N-1NN+1N+2
19 k X k 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points 1 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points N Y k N N y n 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points 1 n 0 1 -N -N+1…… -2 -1 2 …… N-1 N N+1 N+2 …… N points 1 x n Duality in Discrete Fourier Series k n Y k = − Nx k, yn n = X 1 0 2 1 [ ] N k j kn N x n X k N e − = = 1 0 2 ( ) ( ) N n j kn N X k x n e − = − =
xample 8.3 The Discrete Fourier Series of a Periodic Rectangular Puise Train Periodic sequence with period N=10 10 012345678910 Solution 2丌 x[=∑W=∑ i(27/10 kn 1-Wis 5k 10 k 10 r/1 5k(x/10 j5k(x/10) e 56(x/0)+(zk/10)e e A4 k/1o) sin(I k j(x/10 jk(x/10) e e sin( k/10 20
20 Example 8.3 The Discrete Fourier Series of a Periodic Rectangular Pulse Train ◆Periodic sequence with period N=10 5 10 10 1 1 k k W W − = − 1 4 0 0 kn n X k W = = ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k k e 1 Solution: ( ) 4 0 2 10 n j kn e = − = ( ) ( ) ( ) ( ) ( ) ( ) 10 10 10 10 10 5 5 5 10 j j k k k j k k k j j j e e e e e − − + − − = − 10 2 10 j W e − =
X] magnitude 1012345678910 k Xkl=e /(4k/o) sin(zk/2 ≮X(1k 10 hase x denotes indeterminate phase T (magnitude =0) 21
21 ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k X k k e magnitude phase x denotes indeterminate phase (magnitude =0)
X(ejo. IXIkI magnitude △△ 2 4丌 k ase Xk=e -(4zk/o) sin(z k /2) ≮X(e),≮X[k sin(z k/10) NNAATA k 22
22 ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k X k k e magnitude phase
8.2 Properties of the Discrete Fourier series ◆82,1 Linearity For two periodic sequence, both with period N DES <>[,团区] DES ax[n]+bx2[n]>ar,[k +bX2k
23 8.2 Properties of the Discrete Fourier Series ◆8.2.1 Linearity ◆For two periodic sequence, both with period N: 1 1 , DFS x n X k x n X k DFS 2 2 ~ ~ 1 2 1 2 DFS ax n bx n aX k bX k + +