8.2 Properties of the Discrete Fourier series 8.2.2 Shift of a sequence DES DES then x{n-m]<>W如X[,W柳=c-127ANm DES adW"小平k-小,形”=e2m
24 8.2 Properties of the Discrete Fourier Series ◆8.2.2 Shift of a sequence DFS x n X k − DF N S km x n X k m W ， DFS N n l W l x n X k − − ， j N km (2 ) N km W e− = j N (2 ) N nl nl W e − = ◆if then and
8.2 Properties of the Discrete Fourier series ◆8.23 Duality DES DES i[n>X[k X[n> Nx[-kI X 2 N-1 0 2 N-1 0 X k N [k] k k 012…N1 012 N-1 25
25 8.2 Properties of the Discrete Fourier Series ◆8.2.3 Duality DFS x n X k DFS X n Nx k− 0 1 2 …… N-1 X n n 1 1 X k 0 1 2 …… N-1 k x n 0 1 2 …… N-1 n 1 0 1 2 …… N-1 Nx k − k N
8.2 Properties of the Discrete Fourier series ◆8.23 Duality DES DES x[<)X[ x[n]<>N[- Proof ] N∑F[k1m如 N[小]=∑X[]M k=0 k=0 interchang the roles of n and k [小=∑可→N[小=∑ AnN n=0 n 8.2.4 Symmetry Properties 26
26 8.2 Properties of the Discrete Fourier Series ◆8.2.3 Duality DFS x n X k DFS X n Nx k− ◆8.2.4 Symmetry Properties 1 0 1 N kn N k x n X k W N − − = = 1 0 N kn N n X k x n W − = = Proof： 1 0 N N kn k Nx X W n k − = − = 1 0 N N kn n Nx X W k n − = − = interchang the roles of n and k
8.2.4 Symmetry TABLE 8.1 SUMMARY OF PROPERTIES OF THE DFS Periodic sequence(Period N) DFS Coefficients(Period N) X[一k 10.x[-n X KI 11.Re对 Problem8.53, HW Elk】=(X]+'[-k 12.过m([nl] 文。Dk1=3(文伙1-X[一k 13.en]=号([n]+x*[-n]) ReX [kl] 14. on]=Gi[n]miNd jIm(X [I Properties 15-17 apply only when x[n I is real XIk]=X[k] Re(X联k]=Re{X[一k] 15. Symmetry properties for x[n] real 工m(X]}=-mX[-k]] Xk]=|X[一kl Xk]=-∠X[-k 16.Ren]=5(ln]+x-n]) Rex [kll 17.G0l=3(xn-x-n jIm(X[kl) 27
27 8.2.4 Symmetry Problem 8.53, HW
8.25 Periodic Convolution For two periodic sequence, both with period N DES DES 》小元团々团野=8国 then x3=∑m]3[n-m]=∑[m]年[n-m Proof: m=0 5[]=1∑x5[k]形 ∑X[]x2[]W k=0 N k=0 ∑m∑[=∑[n=m m=0 k=0 28
28 8.2.5 Periodic Convolution X k X k X k 3 1 2 = 3 1 2 1 0 N m x n x m x n m − = = − 1 3 3 0 1 N kn N k x n X k W N − − = = 1 2 1 0 1 0 1 N km N kn N k N m x m W X k W N = − − − = = then Proof: ◆For two periodic sequence, both with period N: 1 1 , DFS x n X k 2 2 . DFS x n X k If 2 1 1 0 N m x m x n m − = = − ( ) 1 2 1 1 0 0 1 N k N k N n m m X k W N x m − − = − − = = 1 1 2 0 N m x m x n m − = = − 1 1 2 0 1 N kn N k X k X k W N − − = =