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
Duality in Discrete YIk]= Nx[] and y[n]= X [n N points xn n 2-101 N-1NN+1N+2 N_ points k -N-N+1 2-101 2 N-1NN+1N+2 N points k -N-N+1 01 2 N-1NN+1N+2∴ N points f{] -N-N+1 01 2 N-1NN+1N+2 19
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 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 xn ~ Duality in Discrete Fourier Series k n
xample 8.3 The Discrete Fourier Series of a Periodic Rectangular Puise Train Periodic sequence with period N=10 012345678910 [小=∑Wb=∑ (2x/10)km 8_-4k10sn(xk/2) 1-W0=e 1-W 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 − = − ( ) 4 4 10 0 0 kn 2 10 n n j kn X k W e = = − = = ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k k e 1
magnitude ILk 1012345678910 15 20 k ase 14x10)sin(zk/2) sin( k/10) ≮Ⅺ小 k x denotes indeterminate 4 (magnitude =0) 21
21 ( ) ( ) ( ) 4 10 sin 2 sin 10 − = j k k X k k e magnitude phase
X(ejo. IXIkI magnitude 2 4丌 k ase Xk=e i(4zkn1o) sin( 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