Chapter 10 The z-transform §10.1TheZ- Transform X(z)=∑xk n=-00 x小]<2→>X(z) X(z=Fkn/- unit circle Xle z-e ROC=z=1 Z-plane 6
6 ( ) 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 10 The z-transform Example 10.1 x/n]=a"un n<2 z>a z Example 10.2 xn]=-a"ul-n-1 n"l-n-<2)1-0nrk c< a 7
7 Chapter 10 The Z-Transform Example 10.1 xn a un n = 1 1 1 − − ⎯→ az a u n n Z z a 0 a Example 10.2 xn= −a u− n −1 n 1 1 1 1 − − − − − ⎯→ az a u n n Z z a 0 a z a
Chapter 10 The z-transform x[n]<)X(a):ROC Example103x]=7(1/3yu]-60/2yu X()211=12)>2 ,X×
8 Chapter 10 The Z-Transform Example 10.3 xn ( ) un ( ) un n n = 7 1/3 −6 1/ 2 ( ) ( ) ( 1/ 3)( 1/ 2) 3/ 2 − − − = z z z z X z 2 1 z 0 2 1 3 1 2 3 ( ) Z x n X z ROC ⎯→ ;
Chapter 10 The z-transform 510.2 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 x(a)=FkInI Property 2: The roc does not contain any poles
9 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
Chapter 10 The z-transform Property 3: If xIn is of finite duration, then the Roc is the entire z-plane, except possibly F0 and/or zoo x{四]=0,n<N1;n>N2 xnr“<0 1=N ①N1<0z=∞女ROC ②N2>0z=0女ROC X(z)=∑ -n nz+ ∑|[lz n=M n=0 positive powers ofz negative powers of乙
10 Chapter 10 The Z-Transform Property 3: If is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and/or z=∞. xn 1 2 x n = 0 , n N ;n N 2 1 N n n N x n r − = ① N1 0 z = ROC ② N2 0 z = 0 ROC ( ) n N n n n N X z x n z x n z − = − − = = + 2 1 0 1 positive powers of z negative powers of z