Chapter 3 The z-Transform ◆3.0 Introduction ◆3.1 z-Transforn 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 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
2 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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. 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
3 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 and it is useful to have a generalization of the Fourier transform In analytical problems the z-Transform notation is more convenient than the Fourier transform notation 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
4 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.0 Introduction ◆Motivation of z-transform: ◆The Fourier transform does not converge for all sequences and it is useful to have a generalization of the Fourier transform. ◆In analytical problems the z-Transform notation is more convenient than the Fourier transform notation
3.1 z-Transform xin n=-0 ◆Ifz=e/, Fourier transform→ z-transform Z-Transform: two-sided bilateral z-transform X(z)=∑x[]z=2{xn 对m<>X(z) ◆ one-sided, unilateral X(2)=∑x z-transform n=0 5 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
5 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 z = e Fourier transform z-transform
Relationship between z-transform and Fourier transform Express the z in polar form as Z=re X(2)=∑x]zn n- n xInIn e Jwn The Fourier transform of the product of x n and r(the exponential sequence 矿r=1,X(Z)=X(e") 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
6 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ◆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)=∑x[]z Region of Convergence (ROC) n=-00 ∮m Z-plane unit circle Z= ≤<元 gRe 2021/2/6 Zhongguo Liu_ Biomedical Engineering_shandong Univ
7 2021/2/6 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 St-nT fs-1/T: sampling rate; n=-00 Q2s=2T/T: sampling rate x(t)=x()s() ()∑8(t-n7)=∑x(m)5(t-m7 n=-00 x[n]=x (t)len=x nr) 2T-T0T 2T -4-3-2-101234n 8
8 periodic sampling [ ] ( ) | ( ) c t nT c x n x t x nT = = = T：sampling period; fs=1/T: sampling rate; 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
Review Relation between Laplace Transform and Z-transform Continuous Laplace transform X Time domain X(s)= x(teat Complex frequency domain e已 S=σ+jg2 ijS2 S-pla ane C2=2丌 O Region Of Convergence(roc)
9 Continuous Time domain： x(t) Complex frequency domain： − − X s = x t e dt s t ( ) ( ) Laplace transform s = + j = 2f s- plane j 0 Relation between Laplace Transform and Z-transform Review Region Of Convergence (ROC) = st j e e e − − −
Laplace transform and fourier transform X(s)= x(t)e sdt, st Since s=σ+g2 0 s-plane S frequency domain X(iQ)= x(te dt Fourier transform Fourier Transform is the laplace transform when s have the value only in imaginary axis, s=iQ
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()∑(t-n)=∑x(nT)6(t-n) n=-00 n=-00 the laplace transform LIx,(t)=x,(t)e"dt S=+ jQ2 ∑x:(n)」(-mT) if s=jQ2, =Q2T =∑x(n)em=X(e)=X(e n=-00 Jon z-transform 令zae ,(0+19)7 of discrete x[n]z=X(z) e T time signal n=-0o
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 令 z e e @ [ ] ( ) 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 =