Chapter 3 The z-Transform ◆3.0 Introduction ◆3.1z- Transform 3.2 Properties of the Region of Convergence for the z-transform 3.3 The inverse z-Transform 3. 4 z-Transform Properties 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
2 2021/1/31 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
3.0 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 ft, converges for a broader class of signals 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
3 2021/1/31 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 FT, 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 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
4 2021/1/31 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 Z-Transform: two-sided bilateral z-transform X(2)=∑x]z=z{xm n=-00 xn Cone-sided, unilateral z-transform X(2)=∑x= n=0 CIfz=e z-transform is fourier transform X(e")=∑ enel n=-00 5 2021/1/51 zhongguo ulu_biomedIcal Engineering_shandong UniV
5 2021/1/31 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 , z-transform is Fourier transform. jw z = e
Relationship between z-transform and Fourier transform Express the complex variable Z in polar form as 2=7 x(ve)}=∑ykm 1=-00 .The Fourier transform of the product of x[n and the exponential sequence r Ifr=l, X(z)>x(en 2021/1/31 Zhongguo Liu_Biomedical Engineering_shandong Univ
6 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ◆Express the complex variable z in polar form as Relationship between z-transform and Fourier transform jw z = re ( ) ( ) =− − − = n j w n jwn X re x n r e ◆The Fourier transform of the product of and the exponential sequence xn n r − 1, ( ) ( ) jw If r X Z X e = ⎯⎯→