EXAMPLE D FS of real sequence IXIkll ↑...., 1012345678910 k ≮Xk k x denotes indeterminate x (magnitude=0) figure 8.2
FIGURE 8.2 EXAMPLE. DFS of real sequence
5. periodic convolution x=团m8m=∑mm-m=团m8 Periods of 3 sequences are all N (1)if:x3[n]=x1[n]8x2[n] then: X3k=X1k]X2k (2)f:x3[m]=x1[n]x2[n then: X3]=X koX2k N
[ ] ~ [ ] ~ [ ] ~ [ ] ~ [ ] ~ [ ] ~ [ ] ~ 5. : 2 1 1 0 3 1 2 1 2 x n x n x n x m x n m x n x n periodic convolution N m = = − = − = [ ] ~ [ ] ~ [ ] ~ (1) : if x3 n = x1 n x2 n : [ ] [ ] 2[ ] ~ 1 ~ 3 ~ then X k = X k X k [ ] ~ [ ] ~ [ ] ~ (2) : if x3 n = x1 n x2 n [ ] [ ] 1 : [ ] 2 ~ 1 ~ 3 ~ X k X k N then X k = Periods of 3 sequences are all N
E2(mI graphic method to calculate periodic tnm convolution N x2[1-m]=x2[-(m-1) x2[2-m=x2[-(m-2) Figure 8.3
Figure 8.3 graphic method to calculate periodic convolution
8.4 fourier representation of finite-duration sequences Definition of the discrete fourier transform x小=∑难n-mN]=xmod]=x(m)] The last two expressions are only suitable to no aliasing. 二-0 x刀]=x[n]RMn EXAMPLE Figure 8.8 xn=z x/n-r12I N=12
8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform [ ] [ ] [ mod ] [(( )) ] ~ N r x n = x n − rN = x n N = x n =− [ ] [ ] ~ x[n] = x n RN n Figure 8.8 EXAMPLE. The last two expressions are only suitable to no aliasing
Two derivations of definition 1. Periodic extension of the finite-duration sequence with period ni DFS of the periodic sequence DFT is the dominant period of DFs 2. DTFT of the finite-duration sequence; DFT is the N-points spectral sampling
Two derivations of definition: 1. Periodic extension of the finite-duration sequence with period N ; DFS of the periodic sequence ; DFT is the dominant period of DFS. 2. DTFT of the finite-duration sequence; DFT is the N-points spectral sampling