Exponential Sequences A=Ael a=lalem xn]=Aa"Aea"emn =Ala"ef(re) =Ala”cos(wn+p)+Ala”sin(w,n+p) Exponentially weighted sinusoids a>1 Exponentially growing envelope a<1 Exponentially decreasing envelope ☑=1 x[n]=Aemon is refered to Complex Exponential Sequences 22 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ
22 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Exponential Sequences 0 jw = e j A = Ae ( ) ( ) ( ) = + + + = = = + A w n j A w n x n A Ae e A e n n n j n j w n n j w n 0 0 cos sin [ ] 0 0 1 =1 1 Complex Exponential Sequences Exponentially weighted sinusoids Exponentially growing envelope Exponentially decreasing envelope 0 [ ] jw n x n Ae = is refered to
difference between continuous-time and discrete-time complex exponentials or sinusoids xIn]=Aei(mo+2r)n Aere2m=Aelon x[川=Acos[(w+2πr）n+p]=Acos(w,n+p) x(t)=AeQ+2r/≠Ae wo frequency of the complex sinusoid or complex exponential phase 23 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ
23 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ. difference between continuous-time and discrete-time complex exponentials or sinusoids j(w )n j w n j n j w n x n Ae Ae e Ae 0 2 0 2 0 [ ] = = = + ◆ : frequency of the complex sinusoid or complex exponential ◆ : phase w0 x n A w r n A w n [ ] cos 2 cos = + + = + ( 0 0 ) ( ) ( 2 ) ( ) j t j t x t Ae Ae + =
Periodic Sequences A periodic sequence with integer period N xIn]=xIn+N]for all n Acos(won+)=Acos(won+woN+) woN =2k,where k is integer N=2nk/wo,where k is integer 24 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ
24 10/18/2022 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
EX.2.1 Examples of Periodic Sequences x[n]=cos(πn/4) Suppose it is periodic sequence with period N x[n]=x[n+N] cos(n/4)=cosla(n+N)/4] πn/4+2πk=πn/4+Nπ/4，k:integer W=2πk/(π/4)=8k k=1,>W=8=2π/W 25 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ
25 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 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.2.1 Examples of Periodic Sequences 2π 8 号-xL-=co6xn8 Suppose it is periodic sequence with period N x [n]=x[n+N] cos(3z n/8)=cos[3x(n+N)/8] 3πn/8+2πk=3πn/8+3Nπ/8，k:integer N=2πk/w,=2πk/(3π/8) k=3,>W=16 W=2π3/w≠2π/w(for continuous signal) 26 10/18/2022 Zhongguo Liu_Biomedical Engineering_Shandong Univ
26 10/18/2022 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. 2.1 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)