Basic Sequences Exponential sequences xn]=aa A and a are real: in] is real A is positive and o<a<1, x[nj is positive and decrease with increasing n -1<a<0, x[n] alternate in sign, but decrease in magnitude with increasing n oa>1: xIn] grows in magnitude as n increases Real exponential 0 n jni
12 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic Sequences ◆Exponential sequences n x[n] = A ◆A and α are real: x[n] is real ◆A is positive and 0<α<1, x[n] is positive and decrease with increasing n ◆-1<α<0, x[n] alternate in sign, but decrease in magnitude with increasing n ◆ 1 : x[n] grows in magnitude as n increases
EX 2.1 Combining Basic sequences If we want an exponential sequences that is zero for n <0 then Aa" n>0 In Cumbersome 0n<0 xn=aaun simpler 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
13 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1 Combining Basic sequences = 0 0 0 [ ] n A n x n n ◆If we want an exponential sequences that is zero for n <0, then x[n] A u[n] n = Cumbersome simpler
Basic sequences ◆ Sinusoidal sequence x[n]=A cos(wn+o) for all n Sinusoidal 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
14 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences ◆Sinusoidal sequence x[n] = Acos(w0 n +) for all n
Exponential Sequences a=ale x[n]=Aa"=Aea"/"=aa n,j(10n+d) Ala"cos(won+o)+jl la"sin(won+D) Exponentially weighted sinusoids a>1 Exponentially growing envelope a<1 Exponentially decreasing envelope n xin= Ae is refered to Complex exponential sequences 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
15 2/2/2021 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
Frequency difference between continuous-time and discrete- trme complex exponentials or sinusoids j(wo+2T) n won ran Ae/wo x[n]=Acos(wo+2tr)n+p=Acos(won+) 1o frequency of the complex sinusoid or complex exponential o: phase 16 2/2/2021 Zhongguo Liu_ Biomedical Engineering_shandong Univ
16 2/2/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Frequency 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 ) ( )