Introduction to the z-transform

Chapter 9  z-transforms and applications


    The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis of discrete-time systems.  It is used extensively today in the areas of applied mathematics, digital signal processing, control theory, population science, economics.  These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations.  The role played by the z-transform in the solution of difference equations corresponds to that played by the Laplace transforms in the solution of differential equations.


9.1  The  z-transform

    The function notation for sequences is used in the study and application of z-transforms.  Consider a function  [Graphics:Images/ZTransformIntroMod_gr_1.gif]  defined for  [Graphics:Images/ZTransformIntroMod_gr_2.gif]  that is sampled at times  [Graphics:Images/ZTransformIntroMod_gr_3.gif],  where  [Graphics:Images/ZTransformIntroMod_gr_4.gif]  is the sampling period (or rate).   We can write the sample as a sequence using the notation  [Graphics:Images/ZTransformIntroMod_gr_5.gif].  Without loss of generality we will set  [Graphics:Images/ZTransformIntroMod_gr_6.gif]  and consider real sequences such as,  [Graphics:Images/ZTransformIntroMod_gr_7.gif].  The definition of the z-transform involves an infinite series of the reciprocals  [Graphics:Images/ZTransformIntroMod_gr_8.gif].  


Definition 9.1 (z-transform)  Given the sequence  [Graphics:Images/ZTransformIntroMod_gr_9.gif]  the z-transform is defined as follows

(9-1)            [Graphics:Images/ZTransformIntroMod_gr_10.gif],  

which is a series involving powers of  [Graphics:Images/ZTransformIntroMod_gr_11.gif].  


Remark 9.1.  The z-transform is defined at points [Graphics:Images/ZTransformIntroMod_gr_12.gif] where the Laurent series (9-1) converges.  The z-transform region of convergence (ROC) for the Laurent series is chosen to be

            [Graphics:Images/ZTransformIntroMod_gr_13.gif],    where    [Graphics:Images/ZTransformIntroMod_gr_14.gif].

Remark 9.2.  The sequence notation  [Graphics:Images/ZTransformIntroMod_gr_15.gif]  is used in mathematics to study difference equations and the function notation  [Graphics:Images/ZTransformIntroMod_gr_16.gif]  is used by engineers for signal processing.  It's a good idea to know both notations.  

Remark 9.3.  In the applications, the sequence  [Graphics:Images/ZTransformIntroMod_gr_17.gif]  will be used for inputs and the sequence  [Graphics:Images/ZTransformIntroMod_gr_18.gif]  will be used for outputs.  We will also use the notations

            [Graphics:Images/ZTransformIntroMod_gr_19.gif],  and  



Theorem 9.1 (Inverse z-transform)   Let  [Graphics:Images/ZTransformIntroMod_gr_21.gif]  be the z-transform of the sequence  [Graphics:Images/ZTransformIntroMod_gr_22.gif]  defined in the region  [Graphics:Images/ZTransformIntroMod_gr_23.gif].  Then [Graphics:Images/ZTransformIntroMod_gr_24.gif] is given by the formula  

(9-2)            [Graphics:Images/ZTransformIntroMod_gr_25.gif],  
where [Graphics:Images/ZTransformIntroMod_gr_26.gif] is any positively oriented simple closed curve that lies in the region  [Graphics:Images/ZTransformIntroMod_gr_27.gif]  and winds around the origin.  



9.1.1  Admissible form of a z-transform

    Formulas for  [Graphics:Images/ZTransformIntroMod_gr_38.gif] do not arise in a vacuum.  In an introductory course they are expressed as linear combinations of z-transforms corresponding to elementary functions such as  
            [Graphics:Images/ZTransformIntroMod_gr_39.gif][Graphics:Images/ZTransformIntroMod_gr_40.gif] [Graphics:Images/ZTransformIntroMod_gr_41.gif] [Graphics:Images/ZTransformIntroMod_gr_42.gif] [Graphics:Images/ZTransformIntroMod_gr_43.gif] [Graphics:Images/ZTransformIntroMod_gr_44.gif] [Graphics:Images/ZTransformIntroMod_gr_45.gif] [Graphics:Images/ZTransformIntroMod_gr_46.gif] [Graphics:Images/ZTransformIntroMod_gr_47.gif] [Graphics:Images/ZTransformIntroMod_gr_48.gif][Graphics:Images/ZTransformIntroMod_gr_49.gif] [Graphics:Images/ZTransformIntroMod_gr_50.gif] [Graphics:Images/ZTransformIntroMod_gr_51.gif] [Graphics:Images/ZTransformIntroMod_gr_52.gif][Graphics:Images/ZTransformIntroMod_gr_53.gif].   

In Table 9.1, we will see that the z-transform of each function in  [Graphics:Images/ZTransformIntroMod_gr_54.gif]  is a rational function of the complex variable  [Graphics:Images/ZTransformIntroMod_gr_55.gif].  It can be shown that a linear combination of rational functions is a rational function.  Therefore, for the examples and applications considered in this book we can restrict the z-transforms to be rational functions.  This restriction is emphasized this in the following definition.


Definition 9.2 (Admissible z-transform)  Given the z-transform  [Graphics:Images/ZTransformIntroMod_gr_56.gif]  we say that [Graphics:Images/ZTransformIntroMod_gr_57.gif] is an admissible z-transform, provided that it is a rational function, that is  

(9-3)            [Graphics:Images/ZTransformIntroMod_gr_58.gif],

where [Graphics:Images/ZTransformIntroMod_gr_59.gif], are polynomials of degree [Graphics:Images/ZTransformIntroMod_gr_60.gif], respectively.  

    From our knowledge of rational functions, we see that an admissible z-transform is defined everywhere in the complex plane except at a finite number of isolated singularities that are poles and occur at the points where  [Graphics:Images/ZTransformIntroMod_gr_61.gif].  The Laurent series expansion in  (9-1) can be obtained by a partial fraction manipulation and followed by geometric series expansions in powers of [Graphics:Images/ZTransformIntroMod_gr_62.gif].  However, the signal feature of formula (9-3) is the calculation of the inverse z-transform via residues.  


Theorem 9.2 (Cauchy's Residue Theorem)  Let  D  be a simply connected domain, and let  C  be a simple closed positively oriented contour that lies in  D.  If  f(z)  is analytic inside  C  and on  C,  except at the points  [Graphics:Images/ZTransformIntroMod_gr_63.gif]  that lie inside  C,  then  




Corollary 9.1 (Inverse z-transform)   Let  [Graphics:Images/ZTransformIntroMod_gr_65.gif]  be the z-transform of the sequence [Graphics:Images/ZTransformIntroMod_gr_66.gif].  Then [Graphics:Images/ZTransformIntroMod_gr_67.gif] is given by the formula  


where  [Graphics:Images/ZTransformIntroMod_gr_69.gif]  are the poles of  [Graphics:Images/ZTransformIntroMod_gr_70.gif].  


Corollary 9.2 (Inverse z-transform)   Let  [Graphics:Images/ZTransformIntroMod_gr_71.gif]  be the z-transform of the sequence.  If   [Graphics:Images/ZTransformIntroMod_gr_72.gif]  has simple poles at the points  [Graphics:Images/ZTransformIntroMod_gr_73.gif]  then  [Graphics:Images/ZTransformIntroMod_gr_74.gif] is given by the formula  



Example 9.1.  Find the z-transform of the unit pulse or impulse sequence   [Graphics:Images/ZTransformIntroMod_gr_76.gif].  

Solution 9.1.  This follows trivially from Equation (9-1)      




Explore Solution 9.1.


Example 9.2.  The z-transform of the unit-step sequence   [Graphics:Images/ZTransformIntroMod_gr_84.gif]  is   [Graphics:Images/ZTransformIntroMod_gr_85.gif].  

Solution 9.2.  From Equation (9-1)      




Explore Solution 9.2.


Example 9.3.  The z-transform of the sequence   [Graphics:Images/ZTransformIntroMod_gr_97.gif]  [Graphics:Images/ZTransformIntroMod_gr_98.gif]   is   [Graphics:Images/ZTransformIntroMod_gr_99.gif].  

Solution 9.3.  From Definition 9.1




Explore Solution 9.3.


Example 9.4.  The z-transform of the exponential sequence   [Graphics:Images/ZTransformIntroMod_gr_116.gif]   is   [Graphics:Images/ZTransformIntroMod_gr_117.gif].  

Solution 9.4.  From Definition 9.1  




Explore Solution 9.4.


9.1.2  Properties of the z-transform

    Given that   [Graphics:Images/ZTransformIntroMod_gr_134.gif]   and   [Graphics:Images/ZTransformIntroMod_gr_135.gif].  We have the following properties:
(i)    Linearity.            [Graphics:Images/ZTransformIntroMod_gr_136.gif].  

(ii)   Delay Shift.        [Graphics:Images/ZTransformIntroMod_gr_137.gif].  

(iii)  Advance Shift.        [Graphics:Images/ZTransformIntroMod_gr_138.gif],   or


(iv)   Multiplication by [Graphics:Images/ZTransformIntroMod_gr_140.gif].    [Graphics:Images/ZTransformIntroMod_gr_141.gif].  


Example 9.5 (a).   The z-transform of the sequence   [Graphics:Images/ZTransformIntroMod_gr_142.gif]   is   [Graphics:Images/ZTransformIntroMod_gr_143.gif].  

Example 9.5 (b).  The z-transform of the sequence      is   .  

Solution 9.5 (a).


Solution 9.5 (b).  This is left as an exercise for the reader.


Remark 9.4.  When using the residue theorem to compute inverse z-transforms, the complex form is preferred, i. e.




Explore Solution 9.5 (a).



9.1.3  Table of z-transforms  

    We list the following table of z-transforms.  

It can also be used to find the inverse z-transform.       




Theorem 9.3 (Residues at Poles)

(i)      If [Graphics:Images/ZTransformIntroMod_gr_205.gif] has a simple pole at  [Graphics:Images/ZTransformIntroMod_gr_206.gif],  then the residue is   


(ii)      If [Graphics:Images/ZTransformIntroMod_gr_208.gif] has a pole of order [Graphics:Images/ZTransformIntroMod_gr_209.gif] at  [Graphics:Images/ZTransformIntroMod_gr_210.gif],  then the residue is   


(iii)      If [Graphics:Images/ZTransformIntroMod_gr_212.gif] has a pole of order [Graphics:Images/ZTransformIntroMod_gr_213.gif] at  [Graphics:Images/ZTransformIntroMod_gr_214.gif],  then the residue is   




Subroutines for finding the inverse z-transform


Example 9.6.  Find the inverse z-transform   [Graphics:Images/ZTransformIntroMod_gr_218.gif].   Use (a) series, (b) table of z-transforms, (c) residues.  

Solution 9.6.

Explore Solution 9.6.


    The following two theorems about z-transforms are useful in finding the solution to a difference equation.  


Theorem 9.4 (Shifted Sequences & Initial Conditions)  Define the sequence [Graphics:Images/ZTransformIntroMod_gr_238.gif] and let  [Graphics:Images/ZTransformIntroMod_gr_239.gif]  be its z-transform.  Then

    (i)          [Graphics:Images/ZTransformIntroMod_gr_240.gif]  

    (ii)         [Graphics:Images/ZTransformIntroMod_gr_241.gif]   

    (iii)        [Graphics:Images/ZTransformIntroMod_gr_242.gif]   


Theorem 9.5 (Convolution)  Let  [Graphics:Images/ZTransformIntroMod_gr_243.gif]  and  [Graphics:Images/ZTransformIntroMod_gr_244.gif]  be sequences with z-transforms  [Graphics:Images/ZTransformIntroMod_gr_245.gif], respectively.  Then  

where the operation  [Graphics:Images/ZTransformIntroMod_gr_247.gif]  is defined as the convolution sum  [Graphics:Images/ZTransformIntroMod_gr_248.gif].  




9.1.4  Properties of the z-transform

    The following properties of z-transforms listed in Table 9.2 are well known in the field of digital signal analysis.  

The reader will be asked to prove some of these properties in the exercises.      




Example 9.7.  Given   [Graphics:Images/ZTransformIntroMod_gr_317.gif].   Use convolution to show that the z-transform is  [Graphics:Images/ZTransformIntroMod_gr_318.gif].  

Solution 9.7.

    Let both  [Graphics:Images/ZTransformIntroMod_gr_319.gif]  be the unit step sequence, and both  [Graphics:Images/ZTransformIntroMod_gr_320.gif]  and  [Graphics:Images/ZTransformIntroMod_gr_321.gif].  Then  

so that  [Graphics:Images/ZTransformIntroMod_gr_323.gif]  is given by the convolution  




9.1.5  Application to signal processing

    Digital signal processing often involves the design of finite impulse response (FIR) filters.  A simple 3-point FIR filter can be described as
(9-4)            [Graphics:Images/ZTransformIntroMod_gr_325.gif].  

Here, we choose real coefficients [Graphics:Images/ZTransformIntroMod_gr_326.gif] so that the homogeneous difference equation  

(9-5)            [Graphics:Images/ZTransformIntroMod_gr_327.gif]  

has solutions  [Graphics:Images/ZTransformIntroMod_gr_328.gif].  That is, if the linear combination  [Graphics:Images/ZTransformIntroMod_gr_329.gif]  is input on the right side of the FIR filter equation, the output [Graphics:Images/ZTransformIntroMod_gr_330.gif] on the left side of the equation will be zero.  

    Applying the time delay property to the z-transforms of each term in (9-4), we obtain  [Graphics:Images/ZTransformIntroMod_gr_331.gif].  Factoring, we get

(9-6)            [Graphics:Images/ZTransformIntroMod_gr_332.gif],  where

(9-7)            [Graphics:Images/ZTransformIntroMod_gr_333.gif]

represents the filter transfer function.  Now, in order for the filter to suppress the inputs [Graphics:Images/ZTransformIntroMod_gr_334.gif], we must have


and an easy calculation reveals that  

            [Graphics:Images/ZTransformIntroMod_gr_336.gif],   and


A complete discussion of this process is given in Section 9.3 of this chapter.


Example 9.8.  (FIR filter design)  Use residues to find the inverse z-transform   [Graphics:Images/ZTransformIntroMod_gr_338.gif]   of   [Graphics:Images/ZTransformIntroMod_gr_339.gif].  
Then, write down the FIR filter equation that suppresses



Solution 9.8.

Explore Solution 9.8.



9.1.6  First Order Difference Equations

The solution of difference equations is analogous to the solution of differential equations.  Consider the first order homogeneous equation  


[Graphics:Images/ZTransformIntroMod_gr_385.gif] is a constant.  The following method is often used.

Trial solution method.

    Use the trial solution   
[Graphics:Images/ZTransformIntroMod_gr_386.gif],   and substitute it into the above equation and get   [Graphics:Images/ZTransformIntroMod_gr_387.gif].  Then divide through by [Graphics:Images/ZTransformIntroMod_gr_388.gif]

and simplify to obtain  [Graphics:Images/ZTransformIntroMod_gr_389.gif].  The general solution to the difference equation is


    Familiar models of difference equations are given in the table below.       




9.1.7  Methods for Solving First Order Difference Equations

    Consider the first order linear constant coefficient difference equation (LCCDE)  

            [Graphics:Images/ZTransformIntroMod_gr_430.gif]  with the initial condition   [Graphics:Images/ZTransformIntroMod_gr_431.gif].

Trial solution method.

    First, solve the homogeneous equation   [Graphics:Images/ZTransformIntroMod_gr_432.gif]  and get   [Graphics:Images/ZTransformIntroMod_gr_433.gif].   Then use a trial solution that is appropriate for the sequence [Graphics:Images/ZTransformIntroMod_gr_434.gif] on the right side of the equation and solve to obtain a particular solution  [Graphics:Images/ZTransformIntroMod_gr_435.gif].   Then the general solution is  

The shortcoming of this method is that an extensive list of appropriate trial solutions must be available.  Details can be found in difference equations textbooks.  We will emphasize techniques that use the z-transform.  

z-transform method.

(i)    Use the time forward property  [Graphics:Images/ZTransformIntroMod_gr_437.gif]  and take the z-transform of each term and get

(ii)    Solve the equation in (i) for  [Graphics:Images/ZTransformIntroMod_gr_439.gif].

(iii)    Use partial fractions to expand  [Graphics:Images/ZTransformIntroMod_gr_440.gif]  in a sum of terms, and look up the inverse z-transform(s) using Table 1, to get


Residue method.

    Perform steps (i) and (ii) of the above z-transform method.  Then find the solution using the formula

(iii)            [Graphics:Images/ZTransformIntroMod_gr_442.gif].  

where  [Graphics:Images/ZTransformIntroMod_gr_443.gif]  are the poles of   [Graphics:Images/ZTransformIntroMod_gr_444.gif].  

Convolution method.

(i)    Solve the homogeneous equation   [Graphics:Images/ZTransformIntroMod_gr_445.gif]   and get   [Graphics:Images/ZTransformIntroMod_gr_446.gif].  

(ii)    Use the transfer function   [Graphics:Images/ZTransformIntroMod_gr_447.gif]  

    and construct the unit-sample response   [Graphics:Images/ZTransformIntroMod_gr_448.gif].

(iii)    Construct the particular solution   [Graphics:Images/ZTransformIntroMod_gr_449.gif],    

    in convolution form   [Graphics:Images/ZTransformIntroMod_gr_450.gif].  

(iv)    The general solution to the nonhomogeneous difference equation is  


(v)     The constant  [Graphics:Images/ZTransformIntroMod_gr_452.gif]  will produce the proper initial condition  [Graphics:Images/ZTransformIntroMod_gr_453.gif].  Therefore,


Remark 9.6.  The particular solution  [Graphics:Images/ZTransformIntroMod_gr_455.gif]  obtained by using convolution has the initial condition  



Example 9.9.  Solve the difference equation  [Graphics:Images/ZTransformIntroMod_gr_457.gif]  with initial condition  [Graphics:Images/ZTransformIntroMod_gr_458.gif].
9 (a).  Use the z-transform and Tables 9.1 - 9.2 to find the solution.
9 (b).  Use residues to find the solution.

Solution 9.9.

Explore Solution 9.9.


Example 9.10.  Solve the difference equation  [Graphics:Images/ZTransformIntroMod_gr_490.gif]  with initial condition  [Graphics:Images/ZTransformIntroMod_gr_491.gif].
9.10 (a).  Use the z-transform and Tables 9.1 - 9.2 to find the solution.
9.10 (b).  Use residues to find the solution.

Solution 9.10.

Explore Solution 9.10.


Example 9.11.  Given the repeated dosage drug level model  [Graphics:Images/ZTransformIntroMod_gr_523.gif] with the initial condition  [Graphics:Images/ZTransformIntroMod_gr_524.gif].
9.11 (a).
  Use the trial solution method.  
9.11 (b).  Use z-transforms to find the solution.
9.11 (c).  Use residues to find the solution.
9.11 (d).  Use convolution to find the solution.

Solution 9.11.

    An illustration of the dosage model using the parameters [Graphics:Images/ZTransformIntroMod_gr_572.gif] and initial condition  [Graphics:Images/ZTransformIntroMod_gr_573.gif] is shown in Figure 1 below.


            Figure 9.1.  The solution to [Graphics:Images/ZTransformIntroMod_gr_575.gif] with [Graphics:Images/ZTransformIntroMod_gr_576.gif].

Explore Solution 9.11 (a).

Explore Solution 9.11 (b).

Explore Solution 9.11 (c).

Explore Solution 9.11 (d).


Exercises for Section 9.1.  The z-Transform


Library Research Experience for Undergraduates

The z-Transform

Nyquist Stability Criterion





The Next Module for Z-Transforms is

Homogeneous Difference Equations



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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
























(c) 2012 John H. Mathews, Russell W. Howell