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