4. Exponential sequence xn=a 实指数序列 a=05 40 =15 05 20 10 40 a=-15 05 a=05 20 20 0.5 40 10 10
11 4. Exponentialsequence n x[n] = a
xn]=a=(re) 复指数序列 06 实部 虚部 0.4 05 02 -05 -02 5 10 幅度 相位 05 05 0.5 5
12 n j n x[n] a (re ) = =
5. Sinusoidal sequence x[n]=Acos(n+Φ) 0: regency. radians/ sample (a)x(n/cos(0. 1 I n)=cos(0.9m)(b)x(nCos( 2r n)=cos(1. 8m) -1 (ax[n]=cos( n) (a][n]=cos(2.1 I n)=cos(O 1 I n x[n]=Acos(on+①)=Acos(2m-(om+①)=Acos(2x-)n-①) x[n]=Acos(om+①)=Acos(+2mk)n+①)
13 5. Sinusoidal sequence x[n] = Acos(n + ) = cos(1.8n) : frequency,radians /sample = cos(0.9n) [ ] cos( ) cos(( 2 ) ) [ ] cos( ) cos(2 ( )) cos((2 ) ) = + = + + = + = − + = − − x n A n A k n x n A n A n n A n
For convenience, sinusoidal signals are usually expressed by exponential sequences Asin(om+①) A f(omn+①) e/(om+Φ) Acos(n+①)=(e(m)+em The relationship between o and Q2 x[n]=Asin(nO+①) x(D)l=nr=Asi(9t+①)l=nr=Asn(n7+Φ) ∴O=9T=9/f f=9/(2丌) t f: Hz(period/ sec ond) Q: radians/sec ond @: radians/sample
14 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 cos 2 sin + − + + − + + = + + = − j n j n j n j n e e A A n e e j A A n ( )| sin( )| sin( ) [ ] sin( ) = = + = + = + x t = A t = A nT x n A n c t n T t n T s = T = / f radians sample radians ond f Hz period ond unit f : / : /sec : ( /sec ) : /(2 ) = For convenience, sinusoidal signals are usually expressed by exponential sequences. The relationship between ω and Ω：
2.1.4 Period of sequence if xn=xn+N]-oo<n< oo, then period is N x()=Asn(2+①)=Asin(t+Φ+2) =Asin(92(t+2r/92)+Φ) →T=2丌/g2 xm=Asin(mO+由ASin(n+Φ+21n) Asin((n+2l/@o+dp =xn+2x/0 integer N, period =N 2r/o=rational number P/@, period =P Hiena -number neried=eg-
15 2.1.4 Period of sequence if x[n] = x[n + N],− n ,then period is N x(t) = Asin(t + ) = + + A n l sin(( 2 / ) ) x[n] = Asin(n + = + + ) A n l sin( 2 ) T = 2 / = Asin((t + 2 /）+ ) = Asin(t + + 2 ) = + x n l [ 2 / ] = = = = irrational number period rational number / period integer period 2 / ， ， ， P Q P N N