Basic Sequences C Exponential sequences xn=aa A and a are real: xn] is real A is positive and o<a<l, n] is positive and decrease with increasing 0-1<a<o, x[n] alternate in sign, but decrease in magnitude with increasing n a>1: [n] grows in magnitude as n increases Real exponential 12 0 Niv
12 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic Sequences uExponential sequences n x[n] A uA and α are real: x[n] is real uA is positive and 0<α<1, x[n] is positive and decrease with increasing n u-1<α<0, x[n] alternate in sign, but decrease in magnitude with increasing n u 1: x[n] grows in magnitude as n increases
EX21(第二版) Combining Basic sequences If we want an exponential sequences that is zero for n <o, then /≈ JAa" n≥0 Cumbersome 0n<0 xn]=Adun simpler 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
13 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. EX. 2.1(第二版) Combining Basic sequences 0 0 0 [ ] n A n x n n uIf 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小]=Acos(n+y)0raln Sinusoidal 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
14 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Basic sequences uSinusoidal sequence x n A w n for all n 0 [ ] cos
Exponential Sequences a =de x[n]=Aa=Aeae on=Alla won+o) Ad" cos(won+)+jAa"sin(won+o) Exponentially weighted sinusoids a>1 Exponentially growing envelope a<1 Exponentially decreasing envelope Won ri n=Ae is refered to Complex exponential sequences 15 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
15 1/30/2021 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 jw 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 xn=aeil o+2 Jwonj 2n won x[n]=Acos(wo +2Tr)n+o=Acos(won+) x(t)=ae/( 2+2x)t ≠Ae Wo: frequency of the complex sinusoid or complex exponential O: phase 16 1/30/2021 Zhongguo Liu_Biomedical Engineering_ Shandong Univ
16 1/30/2021 Zhongguo Liu_Biomedical Engineering_Shandong Univ. difference between continuous-time and discrete-time complex exponentials or sinusoids jw n jw n j n jw n x n Ae Ae e Ae 0 2 0 2 0 [ ] u : frequency of the complex sinusoid or complex exponential u : phase w0 x[n] Acosw0 2rn Acosw0n 2 ( ) j t j t x t Ae Ae