The electrical engineering handbook CH008

The electrical engineering handbook

Dorf, R.C., Wan, Z “The z-Transfrom”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

8.1 Introduction 8.2 Properties of the z-Transform Linearity • Translation • Convolution • Multiplication by

a n • Time Reversal 8.3 Unilateral z-Transform Time Advance • Initial Signal Value • Final Value 8.4 z-Transform Inversion

Method 1 • Method 2 • Inverse Transform Formula (Method 2) 8.5 Sampled Data

8.1 Introduction

Discrete-time signals can be represented as sequences of numbers Thus, if x is a discrete-time signal, its values can, in general, be indexed by n as follows:

x = {…, x (–2), x (–1), x (0), x (1), x (2), …, x ( n ), … }

In order to work within a transform domain for discrete-time signals, we define the z-transform as follows The z-transform of the sequence x in the previous equation is

in which the variable z can be interpreted as being either a time-position marker or a complex-valued variable, and the script Z is the z-transform operator If the former interpretation is employed, the number multiplying the marker z –n is identified as being the nth element of the x sequence, i.e., x(n) It will be generally beneficial

to take z to be a complex-valued variable

The z-transforms of some useful sequences are listed in Table 8.1

Linearity

Both the direct and inverse z-transform obey the property of linearity Thus, if Z{f(n)} and Z{g(n)} are denoted

by F(z) and G(z), respectively, then

Z { af ( n ) + bg ( n )} = aF ( z ) + bG ( z )

where a and b are constant multipliers

Z{ ( )} ( ) ( ) x n X z x n z n

n

-=-¥

¥

å

Richard C Dorf

University of California, Davis

Zhen Wan

University of California, Davis

An important property when transforming terms of a difference equation is the z-transform of a sequence shifted in time For a constant shift, we have

Z { f ( n + k )} = zkF ( z )

Table 8.1 Partial-Fraction Equivalents Listing Causal and Anticausal z-Transform Pairs

z-Domain: F(z) Sequence Domain: f(n)

Source: J A Cadzow and H.F Van Landingham, Signals, Systems and Transforms, Englewood Cliffs,

N.J.: Prentice-Hall, 1985, p 191 With permission.

1a for ,

1b for ,

2a for

1

1 0 1

1 1 1 1

1

1

2

z a

n

n

- > - = { }

- < - - =

- - -ì

í

ï îï

ü ý

ï þï

- >

-, ( ) -, -, -,

, ( ) , ,

( )

, (

* * * *

* * * *

* * * * n a u n a a

z a

z a

n

n

- - = { }

- < - - - =

ì í

ï îï

ü ý

ï þï

- >

-1 1 0 1 2 3

1

1 3 2 1

1

2

2

3

) ( ) , , ,

( )

, ( ) ( ) , ,

( )

,

,

2b for ,

3a for

* * * *

* * * **

* * * *

1

2 1 2 1 0 0 1 3 6

1 1

2 1 2

6 3 1

3

3

( )( ) ( ) , , , ,

( ) , ( )( ) ( ) , ,

n

n

- - - = { }

- <

- - - - = ì - -

-í

ï îï

ü ý

ï þï

-,

3b for ,

( )

,

( )! ( ) ( )

( ) , ( )! ( ) ( )

, ,

4a for

4b for

5a for

1 1

1 1

1 1

1

0

1 1

1 1

z a

m

n m

k m

m

n m

k m

m

- > - -

<

-

-¹ ³

-=

-=

-Õ Õ

* * * *

* * * *

0 0 0 1 0 0

0 0 0 1 0 0

d d

( ) , , , ,

, , ( ) , , ,

- = { }

< ¥ ³ + ={ }

+

, , , ,

5b for * * , , , , , ,

for positive or negative integer k The region of convergence of z k F(z) is the same as for F(z) for positive k;

only the point z = 0 need be eliminated from the convergence region of F(z) for negative k

Convolution

In the z-domain, the time-domain convolution operation becomes a simple product of the corresponding

transforms, that is,

Z { f ( n ) * g ( n )} = F ( z ) G ( z )

Multiplication by an

This operation corresponds to a rescaling of the z-plane For a > 0,

where F(z) is defined for R1 < ½z½ < R2

Time Reversal

where F(z) is defined for R1 < ½z½ < R2

8.3 Unilateral z-Transform

The unilateral z-transform is defined as

where it is called single-sided since n ³ 0, just as if the sequence x(n) was in fact single-sided If there is no

ambiguity in the sequel, the subscript plus is omitted and we use the expression z-transform to mean either

the double- or the single-sided transform It is usually clear from the context which is meant By restricting

signals to be single-sided, the following useful properties can be proved

Time Advance

For a single-sided signal f(n),

Z+{ f (n + 1)} = zF(z) – zf (0)

More generally,

This result can be used to solve linear constant-coefficient difference equations Occasionally, it is desirable to

calculate the initial or final value of a single-sided sequence without a complete inversion The following two

properties present these results

Z a { ( )} nf n F z

= æ èç

ö ø÷ for 1 < * * < 2

Z ± { ( ) f ( )} for n = F z- 1 R- < z < R

-2 1

1 1

* *

-=

¥

{ ( )} ( ) ( ) x n X z x n z z Rn

n 0

for * *

Z+{ ( )} ( ) ( ) ( ) ( ) f n + k z F z z f z = k - k 0 - k-1f z f k 1 - - - 1

Initial Signal Value

If f (n) = 0 for n < 0,

where F(z) = Z{ f (n)} for *z* > R.

Final Value

If f (n) = 0 for n < 0 and Z{ f (n)} = F(z) is a rational function with all its denominator roots (poles) strictly inside the unit circle except possibly for a first-order pole at z = 1,

8.4 z-Transform Inversion

We operationally denote the inverse transform of F(z) in the form

f(n) = Z–1{F(z)}

There are three useful methods for inverting a transformed signal They are:

1 Expansion into a series of terms in the variables z and z–1

2 Complex integration by the method of residues

3 Partial-fraction expansion and table look-up

We discuss two of these methods in turn

Method 1

For the expansion of F(z) into a series, the theory of functions of a complex variable provides a practical basis

for developing our inverse transform techniques As we have seen, the general region of convergence for a

transform function F(z) is of the form a < *z* < b, i.e., an annulus centered at the origin of the z-plane This

first method is to obtain a series expression of the form

which is valid in the annulus of convergence When F(z) has been expanded as in the previous equation, that

is, when the coefficients c n , n = 0, ±1, ±2, … have been found, the corresponding sequence is specified by

f (n) = c n by uniqueness of the transform

Method 2

We evaluate the inverse transform of F(z) by the method of residues The method involves the calculation of

residues of a function both inside and outside of a simple closed path that lies inside the region of convergence

A number of key concepts are necessary in order to describe the required procedure

f ( ) ( ) 0 = F z

Þ¥

lim z

f ( ) ( ) ( ) ( ) ¥ = = f n F z

Þ¥ Þ¥

lim lim 1 – z

–1

F z c zn n

n

-=-¥

¥

å

A complex-valued function G(z) has a pole of order k at z = z0 if it can be expressed as

where G1(z0) is finite

The residue of a complex function G(z) at a pole of order k at z = z0 is defined by

Inverse Transform Formula (Method 2)

If F(z) is convergent in the annulus 0 < a < *z* < b as shown in Fig 8.1 and C is the closed path shown (the path C must lie entirely within the annulus of convergence), then

where m is the least power of z in the numerator of F(z)z n–1 , e.g., m might equal n – 1 Figure 8.1 illustrates

the previous equation

8.5 Sampled Data

Data obtained for a signal only at discrete intervals (sampling period) is called sampled data One advantage

of working with sampled data is the ability to represent sequences as combinations of sampled time signals

Table 8.2 provides some key z-transform pairs So that the table can serve a multiple purpose, there are three

items per line: the first is an indicated sampled continuous-time signal, the second is the Laplace transform of

the continuous-time signal, and the third is the z-transform of the uniformly sampled continous-time signal.

To illustrate the interrelation of these entries, consider Fig 8.2 For simplicity, only single-sided signals have

been used in Table 8.2 Consequently, the convergence regions are understood in this context to be Re[s] < s0

FIGURE 8.1 Typical convergence region for a transformed discrete-time signal (Source: J A Cadzow and H F Van

Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 191 With permission.)

z z k

=

-1 0 0

Res

0 0

[ ( )]

G z

k

d

z z

k k

k

z z

=

=

=

-1 1

1

n n

( ) ( ) ),

sum of residues of at poles of inside –(sum of residues of at poles of outside

-³

<

ì

í

î

1 1

0 0

and *z * > r0 for the Laplace and z-transforms, respectively The parameters s0 and r0 depend on the actual

transformed functions; in factor z, the inverse sequence would begin at n = 0 Thus, we use a modified partial-fraction expansion whose terms have this extra z-factor.

FIGURE 8.2 Signal and transform relationships for Table 8.2.

Table 8.2 z-Transforms for Sampled Data

f (t), t = nT,

n = 0, 1, 2, F(s), Re[s] > s 0 F(z), * z * > r0

Source: J A Cadzow and H F Landingham, Signals, Systems and Transforms,

Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 191 With permission.

1 1 (unit step)

2 (unit ramp)

3

4

5

6 sin

7

1

1 1

1

2 1

1 1

1

2 1

2

3

2 3

2 2 2

2 2

s

z z

t

s

Tz z

t

s

T z z z

e

s a

z

z e

te

s a

Tze

z e

t

s

s

z z

at

aT

aT

-+

+

-+

-+ - +

+

-( ) ( ) ( )

( ) ( )

sin cos

cos (

w w

w

w w

w

w

- +

+ + - + +

+ +

+

-cos ) cos

sin

( )

sin cos

cos

( )

( cos ) cos

w w

w w

w

w w

w

w

w w

T

s a

ze T

s a

aT aT

at

aT

aT aT

2

2 1

2

2 8

9

Defining Terms

Sampled data: Data obtained for a variable only at discrete intervals Data is obtained once every sampling period

Sampling period: The period for which the sampled variable is held constant

Related Topics

17.2 Video Signal Processing • 100.6 Digital Control Systems

References

J A Cadzow and H F Van Landingham, Signals, Systems and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985

R C Dorf, Modern Control Systems, 7th ed Reading, Mass.: Addison-Wesley, 1995.

R E Ziemer, Signals and Systems, 2nd ed., New York: MacMillan, 1989.

Further Information

IEEE Transactions on Education

IEEE Transactions on Automatic Control

IEEE Transactions on Signal Processing

Contact IEEE, Piscataway, N.J 08855-1313