Periodic Sequences a periodic sequence with integer period N xn]=x[n+m for all n Acos(won+9)=A cos(won+wo N+o woN=2T k, where k is integer N=2T k/wo, where k is integer 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
17 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Periodic Sequences ◆A periodic sequence with integer period N x[n] = x[n + N] for all n A (w n +) = A (w n + w N +) 0 0 0 cos cos 2 , integer 0 w N = k where k is 0 N k w where k is = 2 / , integer
X 2.2 Examples of Periodic Sequences x[n]=cos( n/4 Suppose it is periodic sequence with period n x[m]=x[n+N」 cos(zn/4)=cos(n+N)/4 T n/4+2r k=rn/4+N/4, k: integer N=2兀k/(x/4)=8k k=1,N=8=2兀/ 18 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
18 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.2 Examples of Periodic Sequences ◆Suppose it is periodic sequence with period N [ ] [ ] x1 n = x1 n + N [ ] cos( / 4) x1 n = n cos( n/ 4) = cos(n+ N)/ 4 n / 4 + 2 k = n / 4 + N / 4, k :integer 0 k N w = → = = 1, 8 2 / N k k = = 2 / ( / 4) 8
EX 2.2 Examples of Periodic Sequences 2丌3兀 x[m2=cos(37n/8) Suppose it is periodic sequence with period N xIn=xin+N cos(3 n /8=cos 3(n+N)/8 37n/8+2丌k=37n/8+3N丌/8,k: Integer N=2兀k/w=2k/(3丌/8)k=3,→N=16 N=2 3/w* 2/ wo (for continuous signal 19 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
19 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ◆Suppose it is periodic sequence with period N [ ] [ ] x1 n = x1 n + N cos(3 n/8) = cos3(n+ N)/8 3 n /8+ 2 k = 3 n /8+ 3N /8, k :integer k = 3,→ N =16 EX. 2.2 Examples of Periodic Sequences [ ] cos(3 /8) x1 n = n 8 3 8 2 → 0 N k w k = = 2 / 2 / (3 / 8) 0 0 N w w for = 2 3/ 2 / ( continuous signal)
EX 2.2 Non-Periodic Sequences x,[n]=cos(n) Suppose it is periodic sequence with period N x2]=x2[m+N cos(n)=cos(n+N) for n+2 k=n+N, k: integer there is no integer N 20 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
20 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.2 Non-Periodic Sequences ◆Suppose it is periodic sequence with period N [ ] [ ] x2 n = x2 n + N x [n] cos( n) 2 = cos( n) = cos(n+ N) 2 , : integer, integer for n k n N k there is no N + = +
High and Low Frequencies in Discrete-time signal xn]= acos(won) (a)W=0or2π (b)W=π8or15/8 (c)=πor7m/4 d=π Zhongguo Liu_ Biomedical Engineering_shandong Univ
21 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. High and Low Frequencies in Discrete-time signal 0 x n A w n [ ] cos( ) = (a) w0 = 0 or 2 (b) w0 = /8 or 15/8 (c) w0 = /4 or 7/4 (d) w0 =