IX(ejo)l. IXIk 4丌 20 a ≮X(e),≮X|k T 10 20 (b) FIgure 8.5
FIGURE 8.5
8.3 properties of the discrete fourier series DES DES DES [n]X[k],x[]X1[k,x2[m]x2[k DES 1.linearity: ax,[n]+bx2In axi[k]+bx2k N=4, 12 points DFS two periodic sequences with different period N=6, 12 points DFS both period=12 compositive sequence N=12, 12 points dFs
[ ] ~ [ ] ~ [ ], ~ [ ] ~ [ ], ~ [ ] ~ 1 1 2 2 X k DFS X k x n DFS X k x n DFS x n 8.3 properties of the discrete fourier series [ ] ~ [ ] ~ [ ] ~ [ ] ~ 1. : 1 2 1 2 aX k bX k DFS linearity ax n bx n + + N=4，12 points DFS N=6，12 points DFS compositive sequence N=12，12 points DFS two periodic sequences with different period both period=12
des km 2. shift of a sequence: xn-m w Xk l DES w xi Xk-a DES 3.duality: XIn Nx[k]
[ ] ~ [ ] ~ 2. : X k N k m W DFS shift of a sequence x n m − [ ] ~ [ ] ~ X k l DFS x n N nl W − − [ ] ~ [ ] ~ 3. : Nx k DFS duality X n −
4.sy mmetry properties DFS DFS x*[n]X*[一k],x*[一1 ⅹ*[k] DFS 1 Re{X[m}=(x山+X*[m)(X[]+*[-k])=Xk] <)2 DFS 1 jImin=-(x[n]-x *nD) Xk]—X*[k])=X。[k )2 DFS 1 x=(x]+x*[一n)(X队]+X*k])=Re{[k]} DES x[]=(x[n]-X*[-m) 付2(X-Xk)=jm(k}
X*[k] DFS~ x *[ n] ~ X*[ k], DFS~ x *[n] ~ 4.symmetry properties: − − X [k] ~ X*[ k]) ~ X[k] ~ ( 2 DFS 1 x *[n]) ~ x[n] ~( 2 1 x[n]} ~ Re{ = + + − = e X [k] ~ X*[ k]) ~ X[k] ~ ( 2 DFS 1 x *[n]) ~ x[n] ~( 2 1 x[n]} ~ jIm{ = − − − = o X[ ]} ~ X*[k]) Re{ ~ X[k] ~ ( 2 DFS 1 x *[ n]) ~ x[n] ~( 2 1 x [n] ~ e + = k = + − [ ]} ~ *[ ]) Im{ ~ [ ] ~ ( 2 1 *[ ]) ~ [ ] ~( 2 1 [ ] ~ X k X k j X k DFS xo n x n x n − = = − −
For a real sequence: [n]=xIn Xk]=X*[k Rexik=refxl-k Im(Xk=-ImXl-k X[]=X[-k] 4X[k]=-4X[-k]
[ ] *[ ] ~ ~ x n = x n [ ] ~ [ ] ~ [ ]| ~ [ ]| | ~ | [ ]} ~ [ ]} Im{ ~ Im{ [ ]} ~ [ ]} Re{ ~ Re{ X k X k X k X k X k X k X k X k = − − = − = − − = − *[ ] ~ [ ] ~ X k = X −k For a real sequence: