Complex z plane X(z)=2x[nz- Region of convergence (ROC) ∮n z-plane unit circle Z <形<丌 Re
7 Complex z plane ( ) =− − = n n X z x n z jw unit circle Z = e − w Z =1 Region Of Convergence (ROC)
Review periodic sampling T: sampling period(单位s); s()=∑(-m7)6=1: sampling rate(Hz模拟频率 2s=2π/: sampling rate模拟角频率 x(t)=x2(t)(t) rad/s x:()∑6(-n)=∑x(n)5(-m7) 1=-00 xn]=x(tlenr=x( nt x() x() xI n 2T-T0T 2T 4-3-2-01234n
8 periodic sampling [ ] ( ) | ( ) c t nT c x n x t x nT = = = T: sampling period (单位s); fs=1/T: sampling rate(Hz); Ωs=2π/T: sampling rate ( ) ( ) =− = − n s t t nT x t x t s t s c ( ) = ( ) ( ) Review ( ) ( ) =− = − c n ( ) ( ) x nT t nT =− = − c n x t t nT t n 模拟频率 模拟角频率 (rad/s) x t s ( ) x n[ ]
Review y Relation between Laplace Transform and z-transform Continuous Laplace transform Time domain X(s)= x(te dt Complex frequency domain: S=0+Q2 S-plane 9=2m Region Of Convergence(Roc)
9 Continuous Time domain： x(t) Complex frequency domain： − − X s = x t e dt s t ( ) ( ) Laplace transform s- plane j 0 Relation between Laplace Transform and Z-transform Review Region Of Convergence (ROC) = st j e e e − − − s = + j = 2f
Laplace Transform and Fourier transform X(s)=x(t)e stdt, est=ee/s2 Since s=σ+j j So o=0-S=jQ2 S-plane frequency domain O X((D)=「x()eoat 少 Fourier transform Fourier Transform is the laplace transform when s have the value only in imaginary axis, sjs 10
10 − − X j = x t e dt j t ( ) ( ) Fourier Transform frequency domain ： j 0 − Fourier Transform is the Laplace transform when s have the value only in imaginary axis, s=jΩ Since s = + j So = 0 s j = ( ) ( ) st X s x t e dt − − = ， Laplace Transform and Fourier transform s- plane = st j e e e − − −
For sampling signal x()=x()∑6(t-m7)=∑x(m7)6(t-n7) the laplace transform [x、()=x()e"t S=+j2 ∑x(nT)J。(-nl"t if s=jQ2, a=QT ∑x(n7)e"=X(e)=X(e0) 1= an z-transform of discrete-=2x[n-n=X() 令zQ@e7=e T time signal n=-0o0 e 9 11
11 For sampling signal， ( ) ( ) st c n x nT t nT e dt − − = − ( ) − =− = sTn c n x nT e [ ] ( ) ( ) st s s x t x t e dt − − = L sT ( j T) z e e + 令 @ = [ ] ( ) n n x n z X z − =− = = z-transform of discretetime signal the Laplace transform s c c ( ) ( ) ( ) ( ) ( ) n n x t x t t nT x nT t nT =− =− = − = − ( ) ST = X e if s j , = = T = x n[ ] j re = s j = + − j n e ( ) j X e ; T r e =