Signals and SystemsLecture 27The z-Transform·ROC of z-Transform·Inverse z-Transform
1 Signals and Systems Lecture 27 •The z-Transform •ROC of z-Transform •Inverse z-Transform
Chapter10The Z-Transform10CHAPTERTHE Z-TRANSFORM
2 Chapter 10 The Z-Transform CHAPTER 10 THE Z-TRANSFORM
Chapter10The Z-Transform$10.1 The Z-TransformX(2)- x[nk-"n=x[n]k z→x(z)X(z)= F(x[n]r=" unitcircleX(eji0)= X(z)z=lioZR0C2z=1Z-plane2
3 ( ) n n X z x n z − + =− = §10.1 The Z-Transform xn⎯→ X(z) Z ( ) n X z x n r − =F Chapter 10 The Z-Transform ROC z =1 ( ) ( ) j z e j X e X z = = 0 1 Z-plane z =1 unit circle
Chapter 10The Z-TransformS10.2 The Region of Convergence for the Z-TransformPropertyl:.The ROC of X(z) consists of a ring in the z-planecentered about the origin.X(z)= F([n/r-")Z [x[n] /r"<00n=Property 2:TheROCdoes not contain anypoles
4 Chapter 10 The Z-Transform §10.2 The Region of Convergence for the Z-Transform Property 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin. ( ) n X z x n r − =F n n x n r + − =− Property 2: The ROC does not contain any poles
Chapter10The Z-TransformExample 10.5S[n]< [nk-"=1 z≥0n=-8Example 10.6x[n] = a" (u[n] -u[n- N]Nbist order polezN-aNO(d)- -(z-a)B2OOOX(rejo)=aNej2kn22Or=aZeros:z>00k = j2k元 / N5
5 Chapter 10 The Z-Transform Example 10.6 xn a un un N n = − − ( ) z (z a) z a X z N N N − − = −1 ( ) N j k N j re a e 2 = = = ω j k N r a k 2 / Zeros: (N-1)st order pole a z 0 Example 10.5 ⎯→ =1 − + =− n n n n z Z z 0