Dorf, R.C., Wan, Z. “The z-Transfrom”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
8
The z-Transform
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.
8.2 Properties of the z-Transform
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{()} () ()xn Xz xnz
n
n
==
-
=-¥
¥
?
Richard C. Dorf
University of California, Davis
Zhen Wan
University of California, Davis
? 2000 by CRC Press LLC
Translation
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)} = z
k
F(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.
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? 2000 by CRC Press LLC
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 a
n
This operation corresponds to a rescaling of the z-plane. For a > 0,
where F(z) is defined for R
1
< ?z? < R
2
.
Time Reversal
where F(z) is defined for R
1
< ?z? < R
2
.
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
n
{ ()} fn F
z
a
aR z aR=
?
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?
÷
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+
-
+= - - ---{( )} () () () ( )fnk zFz zf zf zfk
kkk
01 1
1
. . .
? 2000 by CRC Press LLC
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,
8.4z-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.
fFz() ()0=
T¥
lim
z
ffn Fz() () ( )()¥= =
T¥ T¥
lim lim1–z
nz
–1
Fz cz
n
n
n
()=
-
=-¥
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? 2000 by CRC Press LLC
A complex-valued function G(z) has a pole of order k at z = z
0
if it can be expressed as
where G
1
(z
0
) is finite.
The residue of a complex function G(z) at a pole of order k at z = z
0
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] < s
0
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.)
Gz
Gz
zz
k
()
()
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=
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0
Res
0 0
[()]
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k
k
k
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-
-
=
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-
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1
1
1
1
0
fn
Fzz Fz C m
Fzz Fz Cm
n
n
()
() () ,
() () ),
sum of residues of at poles of inside
–(sum of residues of at poles of outside
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? 2000 by CRC Press LLC
and *z* > r
0
for the Laplace and z-transforms, respectively. The parameters s
0
and r
0
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 8.2z-Transforms for Sampled Data
f(t), t = nT,
n = 0, 1, 2, . . . F(s), Re[s] > s
0
F(z), * z * > r
0
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.
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7.
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++ - +
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w
w
w
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w
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w
w
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sa
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et
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2
2
8.
9.
? 2000 by CRC Press LLC
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.
z-transform:A transform from the s-domain to the z-domain by z = e
sT
.
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
? 2000 by CRC Press LLC