Notice that exin]ex+ninj. Therefore,the Fourier series of a discrete-time periodic signal n]only requires N complex exponentials,so it has the form N-1 1 N-1 =0 k=0 Here,1/N is included in the definition for convenience in future. To obtainXk]fromn], N-1 zn]w, n=0 which can be verified by direct substitution.Thus,n]is periodic in n with period N,and Xk]is periodic in k with period N. Consider two periodic sequences n and yn both with period N,such that n→X[内 and m←一Y[. OSB Figure 8.3 illustrates the periodic convolution of two periodic sequences.Note that as the sequences 2n-m]shifts to the right or left,values that leave the interval between the dotted lines at one end reappear at the other end because of the periodicity. The periodic convolution of periodic sequences corresponds to multiplication of the correspond- ing periodic sequences of Fourier series coefficients. N-1 mn-m一[[ m=0 Other properties of the DFS are discussed in OSB section 5.2. The Discrete Fourier Transform Consider a finite sequence x[n]with length N,which is zero except at n =0,1,...,N-1. Then,we can think about extending this sequence into a periodic sequence of period N. in]=xIn+rN]=x[n mod N]=x[((n))N] 4� � Notice that ek[n] = ek+N [n]. Therefore, the Fourier series of a discrete-time periodic signal x˜[n] only requires N complex exponentials, so it has the form N−1 X˜[k] N−1 ˜ W−nk X[k] N . 1 1 ej 2π N nk x˜[n] = = N N k=0 k=0 Here, 1/N is included in the definition for convenience in future. To obtain X[k] from ˜ ˜ x[n], N−1 Wnk, ˜X[k] = ˜[n] � x N n=0 which can be verified by direct substitution. Thus, x˜[n] is periodic in n with period N, and X˜[k] is periodic in k with period N. Consider two periodic sequences x˜[n] and y˜[n] both with period N, such that x ˜ ˜[n] ←→ X[k] and y˜[n] Y˜ ←→ [k]. OSB Figure 8.3 illustrates the periodic convolution of two periodic sequences. Note that as the sequences x˜2[n − m] shifts to the right or left, values that leave the interval between the dotted lines at one end reappear at the other end because of the periodicity. The periodic convolution of periodic sequences corresponds to multiplication of the correspond￾ing periodic sequences of Fourier series coefficients. N−1 ˜ � x ˜ [m]y˜[n − m] X[k]Y˜ ←→ [k] m=0 Other properties of the DFS are discussed in OSB section 5.2. The Discrete Fourier Transform Consider a finite sequence x[n] with length N, which is zero except at n = 0, 1, . . . , N − 1. Then, we can think about extending this sequence into a periodic sequence of period N. x˜[n] = x[n + rN] = x[n mod N] = x[((n))N ] 4