Chapter 10 The z-transform Property 6 If xn] is two sided, 3r cRoc The RoC of X(z is n<z<n2 Example107x团]=b,b>0 b"uIn+b-nul-n 1 b < 1-bz 1-6 Z b ①b≥1X() does not exist ②0<b<1X(z)=, b<z|< 1-b 16
16 Chapter 10 The Z-Transform Property 6: If is two sided, xn r0 ROC 1 2 0 The ROC of X(z) is r z r r Example 10.7 xn= b b 0 n , = + − −1 − x n b u n b u n n n 1 1 1 1 − − − − b z b z 1 1 1 1 − −bz z b ② 0 b 1 ( ) 1 1 1 1 1 1 1 − − − − − + − = bz b z X z ① b 1 X(z) does not exist. b b z 1
Chapter 10 The z-transform Example x()=71x27 1/3)z-2) 3 多 2 2 3 ①z>2 ②;<z<2 < xn is right sided xn] is two sided x[n] is left sided 17
17 Chapter 10 The Z-Transform ( ) ( ) 1 1 1 1 2 1 3 1 − − − − = z z X z Example ( 1/ 3)( 2) 2 − − = z z z ① z 2 xn is right sided 0 2 3 1 0 2 3 1 0 2 3 1 2 3 1 ② z xn is two sided xn is left sided 3 1 ③ z
Chapter 10 The z-transform Basic z-Transform pairs: δ]<2)1|z20 a"l<2→>,1z>l z Z n一 1 -z<a 1z -n-] kz <1
18 Chapter 10 The Z-Transform Basic Z-Transform pairs: n⎯→1 z 0 Z 1 1 1 − − ⎯→ az a u n n Z z a 1 1 1 1 − − − − − ⎯→ az a u n n Z z a 1 1 1 − − ⎯→ z u n Z z 1 1 1 1 1 − − − − − ⎯→ z u n Z z 1
Chapter 10 The z-transform X()=T小n ROC→ x[n]<z> X(z);ROC
19 Chapter 10 The Z-Transform ( ) n X z x n r − =F ( ) ( ) ROC z =1 j z e j X e X z = = ( ) Z x n X z ROC ⎯→ ;
Chapter 10 The z-transform 510.3 The Inverse Z-Transform xng X(zada 1. Partial-Fraction Expansion 3-(5/6)z Example 10.9 X(z)= -(/4)z2)1-(0/32) Determine xn] for all possible roC. 20
20 Chapter 10 The Z-Transform §10.3 The Inverse Z-Transform X(z)z dz j x n n 1 2 1 − = 1. Partial-Fraction Expansion Example 10.9 ( ) ( ) ( ( ) )( ( ) ) 1 1 1 1 1/ 4 1 1/ 3 3 5 / 6 − − − − − − = z z z X z Determine for all possible ROC. xn