8.1 Representation of Periodic Sequence: the discrete Fourier series 2丌 (k+mn)n 2丌m ,k=0,1,2,…,N-1 Due to periodicity, we only need N complex exponentials for Fourier series representation of x[n] 2兀k 刘n=∑X[eN k Ankle N k=0 The Fourier series representation of periodic sequence x[n]
8 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆Due to periodicity,we only need N complex exponentials 1 0 2 1 [ ] j N N k kn x n X k e N − = = 2 1 [ ] N k j kn x n X k e N = ( ) 2 j k N N m n e + ( ) 2 2 j kn N j mn e e = 2 j kn N e = , 0,1,2, , 1 k N = −
Discrete Fourier series pair 2丌 ◆ The fourier series:n=1∑[k]eN k=0 To obtain the Fourier series coefficients we multiply both sides(2m/ N)rn for0≤n≤N-1 and then sum both the sides we obtain ∑ x(n)en"=∑1∑x(k(krm 2丌 0 =0 k=0 2丌 ∑x(21e (k-r)n n=0
9 Discrete Fourier Series Pair ◆The Fourier series: 1 0 2 ( ) N n j n N r x n e − = − 0 1 1 0 2 ( ) ( ) 1 N n N k j k r n N N X k e − − = = − = ◆To obtain the Fourier series coefficients we multiply both sides by for 0nN-1 and then sum both the sides , we obtain j n (2 / ) N r e − 1 1 0 0 2 ( ) ( ) 1 n N k N j n N k r N X k e − − = = − = 1 0 2 1 [ ] j N N k kn x n e X k N − = =
Discrete Fourier series pair e点m(,kr=mN, n an intege 0. otherwise n=0 orthogonality of the complex exponentials Problem x(ne =∑(k)1 8.51 (k-r) 0 石N三X(+mN) (4)=∑(n)eNk=x() Periodic DFS N coefficients[n]=>X[k]e2r/n) k=0 The discrete Fourier series 10
10 Discrete Fourier Series Pair 1 0 2 1 ( ) N n j k r n N N e − = − = X( )r Problem 8.51 1 0 2 ( ) N n j n N r x n e − = − 0 1 1 0 2 ( ) ( ) 1 N n N k j k r n N N X k e − − = = − = 1, - , 0, k r mN m an integer otherwise = = (2 ) 1 / 0 1 [ ] j N N k kn x n e X N k − = = 1 0 2 ( ) N n j n N k x n e − = − X ( ) k = = X( ) r + mN The Discrete Fourier Series coefficients Periodic orthogonality of the complex exponentials. DFS r→k
8.1 Representation of Periodic equence: the Discrete Fourier Series a periodic sequence x[] with period N, [n]=x[n+rN] for any integer The Discrete Fourier series: =1 2丌 kn ∑X[k]e Synthesis equation k=0 ◆ Coefficients: 2丌 X()= Analysis r(n)eN equation n=0 12
12 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆a periodic sequence x n with period N, x n x n rN for any integer r = + ◆The Discrete Fourier Series: 1 0 2 ( ) ( ) , N n j kn X k x n N e − = − = 1 0 2 1 [ ] , N k j kn N x n X k N e − = = Synthesis equation Analysis equation ◆Coefficients:
8. 1 Representation of periodic Sequence: the Discrete Fourier Series X[小]=∑川e j(2T/N) kn CThe sequence X k] is periodic with period N ￥0=X[M,X[=X[N+1 对k+N=∑小]e n=0 ∑划小e|e2=] n=0
13 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆The sequence is X k periodic with period N X X X X 0 , 1 1 = = + N N ( ) 1 0 N 2 n j k n N N X k x n N e − = − + + = 1 0 2 N 2 n j kn N j n x n X k e e − = − − = = ( ) 1 0 2 N n j N kn X k x n e − = − =