As long as there is no time aliasing,we can recover xn]perfectly from n]. xin]in]RnIn], where RN[n]is a rectangular window: Rw[m=1n=0,1,.,N-1 0 otherwise The discrete Fourier transform of a finite sequence x[n is defined as the discrete Fourier series of iin. N- N-1 X[内= ∑mWW=∑z[n]Wkk=0,1,V-1 0 n=0 m=∑X[网Wkn=0,1,w、 k=0 Consider a finite-duration sequence shown in OSB Figure 8.10(a).If we consider xin]as a sequence of length N =5,the corresponding periodic sequence is in]in OSB Figure 8.10(b). Fourier series coefficients X[k]for n]is shown in OSB Figure 8.10(c).To emphasize that the Fourier series coefficients are samples of the Fourier transform,X(ew)is also shown.The 5-point DFT X[k]corresponds to one period of X[k],as shown in OSB Figure 8.10(d). If we consider zn]to be a sequence of length N =10,however,we get completely different DFT values.The corresponding periodic sequence i[n]is shown in OSB Figure 8.11(b).The 10-point DFT X[k]is shown in OSB Figure 8.11(c)and (d). We can interpret the relationship between a finite-length sequence x[n]and a periodic sequence n by displaying x[n]around the circumference of a cylinder with a circumference of exactly N points.As we repeatedly traverse the circumference of the cylinder,the sequence that we see is the periodic sequence n.Then,a linear shift of this sequence corresponds to a rota- tion of the cylinder.Such a shift is called a circular shift,which is illustrated in OSB Figure 8.12. A circular shift in time results in multiplying the DFT of the sequence by a linear phase factor. x(n-m)wl,0≤n≤N-1一e-2mk/NmX[ Consider two finite-duration sequences zin]and z2[n],both of length N,with DFTs X1[k]and X2[k],respectively.Then,X3k]=Xi[k]X2[k]corresponds to the DFT of the N-point circular convolution of zi[n and x2[n],defined as follows: N-1 x3m=∑xmx2l(n-m)wl,0≤n≤N-1. m=0 OSB Figure 8.14 illustrates the circular convolution of two finite-length sequences. 5As long as there is no time aliasing, we can recover x[n] perfectly from x˜[n]. x[n] = x˜[n]RN [n], where RN [n] is a rectangular window: RN [n] = 1 n = 0, 1, . . . , N − 1 0 otherwise The discrete Fourier transform of a finite sequence x[n] is defined as the discrete Fourier series of x˜[n]. N−1 N−1 X Wnk [k] = � x[n] = � x[n]Wnk ˜ N N k = 0, 1, . . . , N − 1 n=0 n=0 N−1 1 x[n] = � X[k]W−nk n = 0, 1, . . . , N − 1 N N k=0 Consider a finite-duration sequence shown in OSB Figure 8.10(a). If we consider x[n] as a sequence of length N = 5, the corresponding periodic sequence is x˜[n] in OSB Figure 8.10(b). Fourier series coefficients X[k] for ˜ ˜ x[n] is shown in OSB Figure 8.10(c). To emphasize that the Fourier series coefficients are samples of the Fourier transform, |X(ejω)| is also shown. The 5-point DFT X[k] corresponds to one period of X˜[k], as shown in OSB Figure 8.10(d). If we consider x[n] to be a sequence of length N = 10, however, we get completely different DFT values. The corresponding periodic sequence x˜[n] is shown in OSB Figure 8.11(b). The 10-point DFT X[k] is shown in OSB Figure 8.11(c) and (d). We can interpret the relationship between a finite-length sequence x[n] and a periodic sequence x˜[n] by displaying x[n] around the circumference of a cylinder with a circumference of exactly N points. As we repeatedly traverse the circumference of the cylinder, the sequence that we see is the periodic sequence x˜[n]. Then, a linear shift of this sequence corresponds to a rota￾tion of the cylinder. Such a shift is called a circular shift, which is illustrated in OSB Figure 8.12. A circular shift in time results in multiplying the DFT of the sequence by a linear phase factor. e x[((n − m))N ], 0 ≤ n ≤ N − 1 ↔ −j(2πk/N)mX[k] Consider two finite-duration sequences x1[n] and x2[n], both of length N, with DFTs X1[k] and X2[k], respectively. Then, X3[k] = X1[k]X2[k] corresponds to the DFT of the N-point circular convolution of x1[n] and x2[n], defined as follows: N−1 x3[n] = � x1[m]x2[((n − m))N ], 0 ≤ n ≤ N − 1. m=0 OSB Figure 8.14 illustrates the circular convolution of two finite-length sequences. 5