Duality in Discrete Fourier series y{n=x[n时,y[k Nx-k N points 刘m=∑[ke N k=0 N-N+1-2 2∴…N-1NN+1N+2 N_points X k R(k)=∑(n)eN n=0 k -1012 N-1NN+1N+2 N points 周期为N 2-101 2 N-1NN+1N+2 N points N-02 2-10 2 N-1NN+1N+2
20 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 Y k N 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 yn = X n时, Y k = − Nx k, 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 − = − = 周期为N
Example 8.3 The Discrete Fourier Series of a Periodic Rectangular pulse Train Periodic sequence with period n=10 x 012345678910 Solution: 5k j55(2x/10) 2丌 X([小]=∑W如=1_M= 701-W61-a-k(2xA0)W 5k(x/10),15k(x/10)。j5k(x/10 (4rk/1o)sin (z k/2) k(x/10),k(x/10)k(x/10) sin(k/10) 21
21 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: 10 2 10 j W e − = ( ) ( ) ( ) ( ) ( ) ( ) 10 10 10 10 10 10 j j j 5 5 5 j j j k k k k k k e e e e e e − − − − − = − ( ) ( ) 2 10 2 10 5 1 1 jk j k e e − − − = − x n %
magnitud of the DFS IX kI magnitude 1012345678910 15 20 k e v(ark/10) sin(T k/2 ≮X[k sin(I k/10) phase x denotes indeterminate phase (magnitude =0) 22
22 magnitud of the DFS magnitude phase x denotes indeterminate phase (magnitude =0). ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k X k k e
n]一个周期的Dm:X(e")=2em=em sin(5012) EX,8.6 X(e/e) n=0 sin(o/2 magnitude sampling 4丌 k i(4zkn1o) sin(T k/2) phase ≮X(e),≮X[k sin( k/10) 20 k 23
magnitud of the DFS 23 ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k X k k e magnitude phase ( ) 4 n 0 jw jwn X e e = − = ( ) ( ) 2 sin 5 / 2 sin / 2 j e − = sampling x n % 一个周期的DTFT: Ex. 8.6
8.2 Properties of the Discrete Fourier series ◆8.2.1 Linearity For two periodic sequence, both with period n DES 元四<>X,<》k DES ax,[n+bx2n]t>ax[k]+bx2[] 24
24 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 + +