Chapter 3 The z-Transform ◆3.0 Introduction ◆3.1z- Transfor 3.2 Properties of the region of Convergence for the z-transform 3.3 The inverse z-Transform 93.4 Z-Transform Properties 3.5 z-Transform and lti Systems 3.6 the Unilateral z-Transform
2 Chapter 3 The z-Transform ◆3.0 Introduction ◆3.1 z-Transform ◆3.2 Properties of the Region of Convergence for the z-transform ◆3.3 The inverse z-Transform ◆3.4 z-Transform Properties ◆3.5 z-Transform and LTI Systems ◆3.6 the Unilateral z-Transform
3.0 Introduction Fourier transform plays a key role in analyzing and representing discrete-time signals and systems but does not converge for all signals Continuous systems: Laplace transform is a generalization of the Fourier transform Discrete systems Z-transform, generalization of dtFT converges for a broader class of sIgnals
3 3.0 Introduction ◆Fourier transform plays a key role in analyzing and representing discrete-time signals and systems, but does not converge for all signals. ◆Continuous systems: Laplace transform is a generalization of the Fourier transform. ◆Discrete systems : z-transform, generalization of DTFT, converges for a broader class of signals
3.0 Introduction Motivation of z-transform The Fourier transform does not converge for all sequences PIt is useful to have a generalization of the fourier transform→→z- transform In analytical problems the z-Transform notation is more convenient than the Fourier transform notation
4 3.0 Introduction ◆Motivation of z-transform: ◆The Fourier transform does not converge for all sequences; ◆It is useful to have a generalization of the Fourier transform→z-transform. ◆In analytical problems the z-Transform notation is more convenient than the Fourier transform notation
3. 1Z-Transform X(e)=∑xem→∑小]zn 1=-00 Z=r ◆Ife川→>z, Fourier transform→z- transform Z-Transform two-sided bilateral z-transform X(z)=∑x]z=z{xn} x[n ◆one- sided unilateral x(z)=∑m” z-transform
5 3.1 z-Transform ( ) jw n jwn X e x n e =− − = ( ) [ ] n n X z x n x n z =− − = = Z ( ) = − = n 0 n X z x n z ◆one-sided, unilateral z-transform ◆z-Transform: two-sided, bilateral z-transform x n X z [ ]⎯→ ( ) Z ◆If , jw e z → Fourier transform z-transform n n x n z =− → − jw Z = re
Relationship between z-transform and Fourier transform Express the z in polar form as Z=re X(z)=∑x]z re e vwn The Fourier transform of the product of x n and r(the exponential sequence r=1,X(2)=→X(e-)
6 ◆Express the z in polar form as Relationship between z-transform and Fourier transform jw Z = re ( ) ( ) − =− − = = jw n jwn n X re x n r e ◆The Fourier transform of the product of and (the exponential sequence ). x n n r − If r X Z =1, ( ) ( ) n n X z x n z =− − = ( ) jw X e
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 =