Exponential Sequences a=ale n wOn [n]=Ad=Aeae n j(wont) alla" cos(won+o)+j ala sin(won Exponentially weighted sinusoids Exponentially growing envelope a<1 Exponentially decreasing envelope 1 x[n]=ae Twon is refered to Complex exponential sequences 22 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
22 2/6/2021 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 j(1+2) j2 Twon ee x[n]=Acos(wo+2rr)n+o=Acos(won+p) x(t)=ae (2+2x) ≠Ae Wo: frequency of the complex sinusoid or complex exponential o: phase 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
23 2/6/2021 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 xn]=x[n+m for all n Acos(wn+o)=Acos(won+woN+o woN=2T k, where k is integer n=2T k /w, where k is integer 24 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
24 2/6/2021 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 2.1 Examples of Periodic Sequences x[n]=cos( n/4 Suppose it is periodic sequence with period n x[m]=x[n+N」 cos(In/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=27/W 25 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
25 2/6/2021 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 2丌3兀 88 →xm=coS3n/8) Suppose it is periodic sequence with period n x n]=xIn+N cos(3n/8=cos B/(n+n/8 3xn/8+2兀k=3丌n/8+3N兀/8,k: integer N=2k/w=2zk/(3/8)k=3,→)N=16 N=2丌3/W≠2n/W( for continuous signal) 26 2/6/2021 Zhongguo Liu_Biomedical Engineering_shandong Univ
26 2/6/2021 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. 2.1 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)