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