8.1 Representation of Periodic Sequence: the discrete fourier series 2丌 2丌 2丌 k+mN)n 2mn Due to the periodicity of the complex exponential we only need n exponentials for discrete time Fourier series 小]=∑[k]e0xA nk=0
7 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series (2 ) 1 0 1 / [ ] N k j N kn x n X k e N − = = ◆No need 1 (2 / ) [ ] j N kn k x n X k e N = ( ) ( ) 2 2 2 2 j k n j kn j kn N mN N N j mn e e e e + = =
Discrete fourier series pair the Fourier series representation of a periodic sequence. 刘m=∑X[k]e (2T/N)kn N k=0 To obtain the Fourier series coefficients we multiply both sides by e-/(2 N)rn for0≤n≤N-1 and then sum both the sides we obtain .2丌 (-)n x(ne ∑ ∑X(k)eN n=0 n=0 N 2丌 ∑ 27(k-r)n r(n)en ∑X(k)∑eN n=0 k=0 n=0 8
8 Discrete Fourier Series Pair ( ) 1 0 1 2 / [ ] N k j N kn x n X k N e − = = ◆The Fourier series representation of a periodic sequence: 1 1 1 0 0 0 2 ( ) 2 1 ( ) ( ) N N N n n k j n k N N r j r n x n X k N e e − − − = = = − − = 1 1 0 0 1 0 2 ( ) 2 1 ( ) ( ) N N n k N n j r j k r n N n N x n X k N e 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 −
Discrete fourier series pair 2兀 n ∑x n) N ∑)1c解k n=0 k=0 n=0 N 2丌 (k-r)n k-r=mN, m an integer ∑ 0. otherwise Problem 8.51 HW (n) X(r) n=0 2丌 (k)=∑(m)eN n=0 对小=∑[ke k=0
9 Discrete Fourier Series Pair 1 0 2 ( ) 1, - , 0, 1 N n j k r n N k r mN m an integer N otherwi es e − = − = = 1 0 2 ( ) ( ) N n j N r n x n X e r − = − = 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 − = = Problem 8.51, HW 1 1 0 0 1 0 2 ( ) 2 1 ( ) ( ) N N n k N n j r j k r n N n N x n X k N e e − − = = − = − − =
8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆ a periodic sequence with period N n] =n+rN]for any integer A The Fourier series coefficients of xn is N-1 2n kn X(k) x(neN Analysis equation 对=7∑ kn synthesis k=0 equation
11 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆a periodic sequence xn with period N, ~ xn= xn+ rN for any integer r ~ ~ ◆The Fourier series coefficients of is xn ~ 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
8.1 Representation of Periodic Sequence: the Discrete Fourier Series [小=∑ (2 兀/Nkn xne n tThe sequence X [k is periodic with period N []=M]]=XN+ [k+N=∑ 2I/Nk+nin n=0 ∑ 1(2xN)。2z xInle n=0
12 8.1 Representation of Periodic Sequence: the Discrete Fourier Series ◆The sequence is X k periodic with period N 1 ~ 1 ~ , ~ 0 ~ X = X N X = X N + ( )( ) 1 0 2 N n j N k N n X k N x n e − = − + + = ( ) 1 0 2 2 N n j N kn j n x n X k e e − = − − = = ( ) 1 0 2 N n j N kn X k x n e − = − =