duration of sequence is N X[k]=>xm]WMk=0…N-1 xin N ∑XW如,n X[=∑xn x(0 =∑Xk N X[k]=X(e)|2x=X(=)
k N j z e k N j X k X e X z 2 [ ] ( )| ( )| 2 = = = = = = = = − − = − − = X k W n N N x n X k x n W k N N k kn N N n kn N [ ] , 0,1,... 1 [ ] [ ] [ ] , 0,1,.... 1 1 0 1 0 duration of sequence is N = = − = − = 1 0 1 0 [ ] 1 [0] [0] [ ] N k N n X k N x X x n
EXAMPLE x[叫 Figure 8.10 periodic extension with period 5 DESIn 4 (c) XkI X[kRs[k]
Figure 8.10 EXAMPLE. [ ]} ~ DFS { x n [ ] [ ] 5 ~X k R k [(( )) ] n 5 x periodic extension with period 5
explanations: 1. DFT and dfs have the same expression, but DFT are samples of frequency spectrum of the finite-duration sequence, dfs is frequency spectrum of periodic sequence 2. the periods of dFS in time and frequency domain is n, DFTin frequency domain is defined to be finite duration, but has the immanent period n 3. the meaning of dFt not only is samples of frequency spectrum, but also can reconstruct time-domain signal
explanations：1. DFT and DFS have the same expression, but DFT are samples of frequency spectrum of the finite-duration sequence , DFS is frequency spectrum of periodic sequence。 2. the periods of DFS in time and frequency domain is N, DFT in frequency domain is defined to be finite duration, but has the immanent period N。 3. the meaning of DFT not only is samples of frequency spectrum , but also can reconstruct time-domain signal