1. 3 THE Z-TRANSFORM 3.0 INTRODUCTION
We have seen that the Fourier transform plays a key role in representing and analyzing discrete-time signals and system .In this chapter ,we develop the z-transform representation of a sequence and study how the properties of a sequence are related to the propertied of its z-transform .

2. 3.1 z-TRANSFORM The z-transform of a sequence is defined as
This equation is, in general ,an infinite sum or infinite power series ,with z being a complex variable. And refer to the z-transform operator ,defined as

3. With this interpretation, the z-transform operator is seen to transform the sequence into the function where z is a continuous complex variable. The correspondence between a sequence and its z-transform is indicates by the notation

4. The z-transform ,as we have defined it in Eq. (3
The z-transform ,as we have defined it in Eq.(3.2) is often referred to as the two –sided or bilateral-transform ,in contrast to the one-sided or unilateral-transform, which is defined as
Clearly ,the bilateral and unilateral transform are equivalent only if
for

5. There is a close relationship between the Fourier transform and
the z-transform. In particular, if we replace the complex variable z in Eq.(32). with the complex, variable ,then the z-transform reduces to the Fourier transform.
More generally ,we can express the complex variable z in polar form as

6. With z expressed in this form ,Eq.(3.2)becomes
(3.6)
Equation (3.6)can be interpreted as the Fourier transform of the product of the original sequence and the exponential sequence

7. Since the z-transform is a function of a complex variable, it is convenient to describe and interpret it using the complex z-transform. In the z-plane, the contour corresponding to is a circle of unit radius ,as illustrated in Figure 3.1. Note that is the angle between the vector to a point z on the unit circle and the real axis of the complex z-plane.

8. If we evaluate at points on the unit circle in the z-plane beginning at
through to we obtain the Fourier transform for
Figure3.1

9. REGION OF CONVERGENCE（ROC收敛域）
As we stated in sec.2.7, if the sequence is absolutely summable ,the Fourier transform converges to a continuous function of ω . Applying this criterion to Eq.(3.6) leads to the condition
(3.7)
for convergence of the z-transform .

10. It should be clear from Eq. (3
It should be clear from Eq.(3.7)that ,because of the multiplication of the sequence by the real exponential it is possible for the z-transform to converge even if the Fourier transform does not. For example ,the sequence is not absolutely summable and therefore ,the Fourier transform does not converge absolutely. However , is absolutely summable if

11. Example 3.1 Right-sided Exponential Sequence （右边指数序列）
Consider the signal Because it is nonzero only for , this is an example of a right-sided sequence . From Eq.(3.2),
For convergence of ,we require that

12. Thus the region of convergence is the range of values of z for which or equivalently, inside the region of convergence, the infinite series converges to
For is the unit step sequence with z-transform

13. Example 3.2 Left-Sided Exponential Sequence （左边指数序列）
Now let since the sequence is nonzero only for ,this is a left –sided sequence . Then

14. If or, equivalently, the sum in Eq.(3.12) converges ,and
The pole-zero plot and region of convergence for this example are shown in Figure 3.4. Note that for ,the sequence grows exponentially as and thus ,the Fourier transform does not exist.

15. Figure Figure3.4
Z平面
Z平面
单位圆
单位圆

16. Example3.3 Sum of two Exponential Sequences
Consider a signal that is the sum of two real exponentials;
The z-transform is then

18. Example 3.4 Sum of Two Exponentials (Again)
Again ,let be given by Eq.(3.14), then using the general result of Example 3.1 with and ,the z-transforms of the two individual terms are easily seen to be

19. and ,consequently,
As we had determined in Example 3.3. the pole-zero plot and ROC for the z-transform of each of the individual terms and for the combined signal are shown in Figure3.5

20. Example 3.5 Two-sided Exponential Sequence （双边指数序列）
Consider the sequence
Note that this sequence grows exponentially as
Using the general result of Example 3.1 with ,we obtain
and using the result of Example 3.2 with yields

21. Thus , by the linearity of the z-transform,
In this case , the ROC is the annular region

22. Note that the rational function in this example is identical to rational function in Example 3.3, but the ROC is different in the three cases. The pole-zero plot and the ROC for this example is shown in Figure 3.6

23. Example 3.6 Finite-Length Sequence （有限长序列）
Consider the signal
Then
where we have used the general formula in Eq.(2.56) to sum the finite series. The ROC is determined by the set of values of z for which

24. The N roots of the numerator polynomial are at
(note that these values satisfy the equation and when ,these complex values are the Nth roots of unity ) the zero at k=0 cancels the pole at z=a.

25. Figure 3. 7 Pole-zero for Example 3
Figure 3.7 Pole-zero for Example 3.6 with N=16 and a real such that 0<a<1. the region of convergence in this example consists of all values of z except z=0

26. Table 3.1 SOME COMMON z-TRANSFORM PAIRS( see book on the page of 104)

27. 3.2 PROPERTIES OF THE REGION OF CONVERGENCE FOR THE Z-TRANSFORM

28. PROPERTY 1: The ROC is a ring or disk in the z-plane centered at the origin: i.e.,
PROPERTY 2; The Fourier transform of converges absolutes if and only if the ROC of the z-transform of includes the unit circle
PROPERTY 3; The ROC cannot contain any poles.
PROPERTY 4; If is a finite-duration sequence ,i.e., a sequence that is zero except in a finite interval then the ROC is the entire z-plane, except possibly or

29. PROPERTY 5 ; If is a right-sided sequence ,i. e
PROPERTY 5 ; If is a right-sided sequence ,i.e., a sequence that is zero for ,the ROC extends outward from the outermost (i.e., largest magnitude) finite pole in to ( and possibly including)
PROPERTY 6 ;If is a left-sided sequence ,i.e., a sequence that is zero for ,the ROC extends inward from the innermost (smallest magnitude) for pole in to ( and possibly including)
PROPERTY 7; A two-sided sequence is an infinite-duration sequence that is neither right sided nor left sided . If is a two-sided sequence ,the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and, consistent with property ,not containing any poles.
PROPERTY; the ROC must be a connected region.

30. As discussed in Section 3
As discussed in Section 3.1 ,property 1 results from the fact that convergence of Eq.(3.2) for a given is depended only on ,and property 2 is a consequence of the fact that Eq.(3.2) reduces to the Fourier transform when Property 3 follows from the recognition that is infinite at a pole and therefore , by definition . does not converge.
Properties 4 through 7 can all be developed more or less directly from the interpretation of the z-transform as the Fourier transform of the original sequence, modified by an exponential weighting.
Property 8 is somewhat more difficult to develop formally , but at least intuitively, it is strongly suggested by our discussion of properties 4 through 7.

31. 3.3 THE INVERSE z-TRANSFORM
One of the important roles of the z-transform is in the analysis of discrete-time linear systems . Often ,this analysis involves finding the z-transform of sequence and, after some manipulation of the algebraic expressions, finding the inverse z-transform.
In Sections we consider some of these procedures , specifically the inspection method, partial fraction expansion , and power series expansion.

32. 3.3.1 Inspection Method（观察法）
Evaluate the z-transform for sequence of the form
where a can be either real or complex.
If we need to find the inverse z-transform of
and we recall the z-transform pair of Eq.(3.35), we shall recognize “by inspection” the associated sequence as

33. If the ROC associated with in Eq(3.36) had been
we can recall transform pair 6 in Table 3.1 to find by inspection that

34. 3.3.2 Partial Fraction Expansion 部分分式展开法
To see how to obtain a partial fraction expansion ,let us assume that is expressed as a ratio of polynomials in ; i.e.,
an equivalent expression is
(3.37)
(3.38)

35. Equation (3.38 )explicitly shows that for such functions there will be M zeros and N poles at nonzero locations in the z-plane. In addition ,
there will be either M-N poles at z=0 if M>N
or N-M zeros at z=0 if N>M
Note that could be expressed in the form

36. where the ‘s are the nonzero zeros of and the ‘s are nonzero poles of
where the ‘s are the nonzero zeros of and the ‘s are nonzero poles of If M<N and the poles are all first order , then can be expressed as
Obviously , the common denominator of the fractions in Eq.(3.40)is the same as the denominator in Eq.(3.39). Multiplying both sides of Eq.(3.40). by and evaluating for shows that the coefficients, can be found from
(3.40)
(3.41)

37. Example 3.8 Second-Order z-Transform
Consider a sequence with z-transform
The pole-zero plot for is shown in Figure from the region of convergence and property 5 ,Section 3.2 ,we see that is a right-sided sequence .
3.42

38. Since the poles are both first order , can be expressed in the form of Eq.(3.40);i.e.,
From Eq.(3.41)

39. Therefore
Since is right sided , the ROC for each term extends outward from the outermost pole . From Table 3.1 and the linearity of the z-transform , it then follows that

40. Figure 3.12 pole-zero and ROC for Example 3.8

43. Example 3.9 Inverse by partial Fractions
Consider a sequence with z-transform
The pole-zero plot for is shown in Figure3.13 .From the region of convergence and property 5. Section 3.2 it is clear that is a right-sided sequence . Since M=N=2 and the poles are all first order can be represented as
(3.47)

44. The constant B0 can be found by long division
Since the remainder after one step of long division is of degree 1 in the variable it is not necessary to continue to divide . Thus , can be expressed as

45. Now the coefficient A1 and A2 can be found by applying Eq. (3
Now the coefficient A1 and A2 can be found by applying Eq.(3.41) to Eq.(3.46) or equivalently ,Eq.(3.47). Using Eq.(3.47), we obtain
Therefore

46. From Table3.1 ,we see that since the ROC is
Thus ,from the linearity of the z-transform

47. Figure 3.13 Pole-zero plot for the z-transform in Example 3.9

48. 2.2 Inverse z-Transform 求z反变换的方法通常有三种：围线积分法（留数法）、部分分式展开法和长除法。
一、Residue Theorem（留数法）

56. 二、 Power Series幂级数数展开法

57. 3.3.3 Power Series Expansion幂级数数展开法
The defining expression for the z-transform is a Laurent series where the sequence values are the coefficients of Thus ,if the z-transform is given as a power series in the form
We can determine any particular value of the sequence by finding the coefficient of the appropriate power of

58. Example 3.10 Finite-length Sequence
Suppose is given in the form
We can express as

59. Therefore ,by inspection , is seen to be
Equivalently

60. Example 3.11 Inverse transform by Power Series Expansion
Consider the z-transform
Using the power series expansion for log(1+x) ,with
we obtain
Therefore

61. Example 3.12 Power Series Expansion by Long Division
Consider the z-transform
Since the region of convergence is the exterior of a circle ,the sequence is a right-sided one. Furthermore, since approaches a finite constant as z approaches infinity ,the sequence is causal. Thus , we divide, so as to obtain a series in power of

62. Carrying out the long division, we obtainOr
hence,

63. Example 3.13 Power Series Expansion for a Left-Sided Sequence
As another example, we can consider the same ratio of polynomials as in Eq.(3.53), but with a different region of convergence
Because of the region of convergence , the sequence is a left-sided one, and since at z=0 is finite , the sequence is zero for n>0.

64. thus , we divide, so as to obtain a sequence in powers of z as follows:
Therefore

65. 3.4 z-TRANSFORM PROPERTIES (Z变换性质)
Many of the properties of the z-transform are particularly useful in studying discrete-time signals and systems.
In the following discussion , denotes the z-transform of , and the ROC of is indicates by Rx: i.e.,

66. As we have seen ,Rx represents a set of values of z such that
For properties that involve two sequence and associated z-transform, the transform pairs will be denoted as

67. 3.4.1 Linearity线性
The linearity property states that
For example

68. 3.4.2 Time Shifting 序列的移位 According to the time-shifting property,
ROC=Rx( except for the possible
addition or deletion of z=0 or z= )
The quantity is an integer. If is positive, the original sequence is shifted right ,and if is negative, is shifted left.
The derivation of this property follows directly from the z-transform is
With the substitution of variables

69. Example 3.14 Shifted Exponential Sequence
Consider the z-transform
From the ROC, we identify this as corresponding to a right-sided sequence. We can first rewrite in the form

70. This z-transform is of the form of Eq. (3
This z-transform is of the form of Eq.(3.39) with M=N=1, and its expansion in the form of Eq.(3.43) is
From Eq.(3.56), it follows that can be expresses as

71. An expression for can be obtained more directly by applying the time-shifting property . first can be written as
From the time-shifting property
It is easily verified that Eqs.(3.57) and (3.59) are the same for all values of n; i.e.

72. 3.4.3 multiplication by an Exponential Sequence 乘以指数序列
The exponential multiplication property is expressed mathematically as
The notation denotes that the ROC is Rx scaled by ; i.e., if Rx is the set of values of z such that
then is the set of values of z such that

73. Example 3.15 Exponential Multiplication
Staring with the transform pair
we can use the exponential multiplication property to determine the z-transform of
First , is expressed as

74. Then ,using Eq.(3.60) and the exponential multiplication property ,we see that
From the linearity property, it follows that

75. 3.4.4 Differentiation of X(z) X(z)的微分
The differentiation property states that
This property is verified by differentiating the z-transform expression of Eq.(3.2); i.e.,

76. Example 3.16 Inverse of Non-Rational z-Transform
In this use the differentiation property together with the time-shifting property to determine the inverse z-transform considered in Example3.11.with
We first differentiation to obtain a rational expression:

77. From the differentiation property
The inverse transform of Eq.(3.63) can be obtained by the combined use of the z-transform pair of Example3.1 , the linearity property and the time-shifting property. Specifically , we can express as
Therefore

78. Example 3.17 Second-Order pole
As another example of the use of the differentiation property, let us determine the z-transform of the sequence
From the z-transform pair of Example 3.1 and the differentiation property, it follows that
therefore

79. 3.4.5 conjugation of a Complex Sequence 共轭序列
The conjugation property is expressed as
This property follows in a straightforward manner from the definition of the z-transform, the details of which are left as an exercise( Problem 3.51)

80. 3.4.6 Time Reversal翻褶序列
The notation ROC=1/Rx implies that Rx is inverted ; i.e., if the set of values of z such that , then the ROC is the set of values of z such that Thus, if is in the ROC for , then is the ROC for the z-transform of If the sequence is real or we do not conjugate a complex sequence, the result becomes

81. Example 3.18 Time-Reversal Exponential Sequence
As an example of the use of the property of time reversal , consider the sequence
Which is a time-reversed version of From the time-reversal property, it follows that

82. 3.4.7 Convolution of Sequences序列的卷积
According to the convolution property
contains
To derive this property formally, we consider
So that

83. If we interchange the order of summation
Changing the index of summation in the second sum from n to m=n-k, we obtain
Thus, for values of z inside the regions of convergence of both
and ,we can write
Where the region of convergence includes the intersection of the regions of convergence of and

84. Example 3.19 Evaluating a Convolution Using the z-Transform
Let and The corresponding are
and
If , the z-transform of the convolution of with
is then

85. The poles and zeros of are plotted in Figure 3
The poles and zeros of are plotted in Figure 3.14 , and the region of convergence is seen to be the overlap region. The sequence can be obtained by determining the inverse z-transform . Expanding in Eq.(3.64) in a partial fraction expansion, we get
Therefore ,

86. Figure 3.14Pole-zero plot for the z-transform of the convolution of the sequence u [n] and an u [n]

87. Input z-Transform X(z) and system function H(z), determine
the ROC for output z-Transform Y(z)

88. 3.4.8 Initial-Value Theorem初值定理
If is zero for ( i.e., if is causal), then

90. 2.4 System Function

91. 一、因果稳定系统
已知线性移不变系统稳定的充要条件：
当|z|=1时，上式变成
这就是系统稳定的充要条件。
因此，若系统函数在单位圆上收敛，则系统是稳定的。这也意味着，如果系统函数H(z)的收敛域包括单位圆，则系统是稳定的。反之，如果系统稳定，则系统函数H(z)的收敛域一定也包括单位圆。

92. 显然，一个因果系统的系统函数的收敛域应该是

93. 二、系统函数和差分方程的关系
线性移不变系统可以用线性常系数差分方程描述：
对上式两边求Z变换，利用线性性质和时不变性质，得 因此
可见系统函数的系数也正是其差分方程的系数。

94. 系统函数还可以进一步分解成：
式中，{dk)和{cr}分别表示H(z)在z平面上的极点和零点。这样，系统函数可以用z平面上的极点、零点和常数A来确定。
例 根据系统函数求差分方程
求该系统的差分方程。

95. 为了求满足该系统输入输出的差分方程，可以将H(z)的分子和分母各因式乘开，而得到如下的形式：
于是，
其差分方程就是

96. 三、系统的频率响应的意义

97. 四、频率响应的几何确定法

103. 四、IIR系统与FIR系统
1.IIR系统：若系统的单位抽样（冲激）响应延伸到无穷长，称之为“无限长单位冲激响应系统”，简写成IIR系统。其对应的差分方程有输出端递归到到输入断，又称递归系统。
2.IIR系统：若系统的单位抽样（冲激）响应是一个有限长序列，称之为“有限长单位冲激响应系统”，简写成FIR系统。其对应的差分方程无输出端递归到到输入断，又称非递归系统。

104. 3.4.9 Summary of Some z-Transform Properties
We have presented and discussed a number of the theorems and properties of z-transforms , many of which are useful in manipulating z-transform . These properties and a number of others are summarized for convenient reference in Table3.2( see the book on page.126)

105. 3.5 SUMMARY
In this chapter , we have defined the z-transform of a sequence and shown that it is a generalization of the Fourier transform. The discussion focused on the properties of the z-transform and techniques for obtaining the z-transform of a sequence and vice versa.
An important part of the chapter was a discussion of some of the many properties of the z-transform that make it useful in analyzing discrete-time signals and systems.