Chapter 2 the -Transform 2.1 definition 2.2 properties of ROC 2.3 the inverse -transform 2.4 -transform properties
2.1 definition One motivation for introducing -transform is that the Fourier transform doesn t converge for all sequences (such as u[n], nu[n]), and it is useful to have a generaliation of the Fourier transform that encompasses a broader class of signals. A second advantages is that in analytical problems the -form notation is often more convenient than the Fourier transform notation. The -transform of a sequence [n] is defined as n j X ( ) [ n], re is a comple variable n
The region of convergence The region of convergence (ROC: 收敛域 ) are the set of values of for which the -transform converges. The condition for convergence of the -transform is: n [ ] n n n [ n] n So the convergence depends only on. Thus if some values of, Say, = 1, is in the ROC, then all values of on the circle defined By = 1 will also be in the ROC. As one consequence of this, the region of convergence will consist of a ring in the -plane centered about the origin. This is epressed as R R
Some common -transform pairs Sequence Transform ROC [n] 1 all u[n] -u[-n - 1] 1-1 1-1 -1 1-1 1 n 1 a u[n] -1 1 - a n 1 -a u[-n - 1] -1 1 - a a a
Zeros and poles When X() is a rational function inside the ROC X ( ) P( ) / Q( ) the values of for which P()=0 are called the eros( 零点 ) of X(), and the values of for which Q()=0 are called the poles( 极点 ) of X(). An eample is shown in Figure 3.2, where a o denotes the eros and a denotes the poles. The ROC is bounded with the pole. 零点和极点个数相等, 不要漏掉 =0 和 = 处的零点或极点
2.2 Properties of the ROC 1. The ROC is always bounded by a circle since the convergence condition is on the magnitude ; 2. The ROC for right-sided sequences (n<n0, [n]=0) is always outside of a circle of radius R-. (if n0>=0, [n] is a causal sequence) 3. The ROC for left-sided sequences (n>n0, [n]=0) is always inside of a circle of radius R+. (if n0<=0, [n] is a anti-causal sequence) 4. The ROC for two-sided sequences is always an open ring R-< <R+ if it eists.
5. The ROC for finite-duration sequences(n<n 1 and n>n 2, [n]=0) is the entire -plane. If n 1 <0, then = is not in the ROC. If n 2 >0, then =0 is not in the ROC; 6. The ROC cannot include a pole since X() converges uniformly in there; 7. There is at least one pole on the boundary of a ROC of a rational X(); 8. The ROC is one contiguous region, the ROC does not come in pieces.
X 2 0 1 2 ( )... [ 2] [ 1] [0] [1] [2]... right-side : [ ] two-side : or no ROC left-side: 0 [ ] causal finite-duration : 0 [ ] [ ] FT eists, then ROC includes 1 Anti-causal
2.3 the inverse -transform 1.inspection method 2.partial fraction epansion N ( ) A m X ( ) 1 D( ) 1 p A p X 1 m (1 m ) ( ) p m N M N m1 m k0 3.power series epansion (1) X() 是有限项的整式 (2) X() 是分式, 采用长除法右边序列 : -1 升幂排列左边序列 : -1 降幂排列 k ck M N
[ n] X ( ) ROC R [ n] X ( ) ROC R 1 1 [ n] X ( ) ROC R 2 2 2.4 -transform properties In this section, the transform pairs are denoted as 1. Linearity a [ n] b [ n] ax ( ) bx ( ) ROC R R 1 2 1 2 2. Time shifting 0 [ 0] n n n X ( ) ROC R 3. Multiplication by an eponential sequence n [ n] X ( / ) ROC R 0 0 0 1 2 1 2
4. Differentiation of X() dx ( ) n[ n] ROC R d EXAMPLE 1 u[ n] 1 1 1 1 d 1 1 1 nu[ n] 1 2 d 1 1
5. Conjugation of a comple sequence * * * [ n] X ( ) ROC R * * X( ) X ( ) Re [ n] 2 ROC R * * X ( ) X ( ) Im [ n] 2 j ROC R 实数序列的 Z 变换的零 / 极点共轭对称 6. Time reversal [ n] X (1/ ) ROC 1/ R * * * [ n] X (1/ ) ROC 1/ R 实偶序列的零 / 极点共轭对称且倒数 ( 反演 )4 个一组
7.[n] is causal [0] lim X ( ) lim [ n] lim( 1) X ( ) n 1 if : [ n] 0, n 0 then : [0] lim X ( ) 0 Prove: 1 n1 1 1 right lim X ( ) X ( ) lim( Z[ [ n 1] Z[ [ n]]) n lim [ ( [ n 1] [ n]) ] ( [0] [ 1]) ( [1] [0]) ( [ n 1] [ n])...( [ ] [ 1]) left
8. Convolution of sequences [ n] [ n] X ( ) X ( ) ROC R R 1 2 1 2 1 2 Prove: Z( [ n] [ n]) ( [ n] [ n]) 1 2 1 2 n n 1 2 1 2 n k k n ( [ k] [ n k]) [ k] [ n k] k k [ k] [ n'] 1 2 n ( n' k ) n k n' 1 2 1 2 n [ k] [ n'] X ( ) X ( ) n
第 2 章总结 2.1 正变换 n 2 0 1 2 ( ) [ ]... [ 2] [ 1] [0] [1] [2]... X n n 2.2 ROC 的特点右边 : 园内左边 : 圆外有限长 : 整个双边 : 圆环因果 : 包含无穷大稳定 : 包含单位圆 2.3 反变换观察法部分分式法 X ( ) m 幂级数法 2.4 性质 A N M N m 1 1 1 pm k0 k ck M N n n X ROC R n X ROC R n0 [ ] ( ) n [ ] ( / ) 0 0 0 0 [ n] [ n] X ( ) X ( ) ROC R R 1 2 1 2 1 2 [0] lim X( ), lim [ n] lim( 1) X( ) n 1 实序列的 Z 变换的零 / 极点共轭对称 ; 实偶序列的 Z 变换的零 / 极点共轭 倒数 4 个一组
第 2 章作业 2-1 2-2 2-8 2-10( 不用写步骤 ) 2-16