2. THE Z-TRANSFORM
PROAKIS, John G.; MANOLAKIS, Dimitris G. Digital Signal Processing Principles, Algorithms and
Applications. U.S.A. Prentice-Hall International, 1996
2.1. The Direct z-Transform
The z-transform
signal x(n)
series
is
of a discrete-time
defined as the power
where z is a complex variable. The relation is sometimes called the
direct z-transform because it transforms the time-domain signal x(n)
into its comptex-plane representation X(z).
The inverse procedure [i.e., obtaining x(n) from X(z) is called the inverse
z-transform.
For convenience, the z-transform of
signal x(n) is denoted by
a
whereas the relationship between x(n) and X(z) is
indicated by
The region of convergence (ROC) of X(z) is the set of all values of z for
which X(z) attains a finite value.
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Example 2. 1.
Determine the z-transforms of the following finire-duration signals:
Solution
From definition we have :
(a) X1(z)=1+2z-1+5z-2+7z-3+z-5 ,
ROC: entire z-plane except z = 0;
(b) X2(z)=z2+2z+5+7z-1+z-3 ,
ROC: entire z-plane except z=0 and z=∞
(c) X3(z)=z-2+2z-3+5z-4+7z-5+z-7,
ROC: entire z-plane except z = 0;
(d) X4(z)=2z-2+4z+5+7z-1+z-3 , ROC: entire z-plane except z=0 and z=∞
e) X5(z) =1 ROC : entire z-plane .
From a mathematical point of view the z-transform is simply an
alternative representation of a signal.
In many cases we can express the sum of the finite or infinite series
for the Z-transform in a closed-form expression. In such cases the
z-transform offers a compact alternative representation of the signal.
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In many cases we can express the sum of the finite or infinite series
for the Z-transform in a closed-form expression. In such cases the
z-transform offers a compact alternative representation of the signal.
Example 2. 2.
Determine the z-transform of the signal:
Solution The signal x(n) consists of
x(n)=(1/2)nu(n)
an infinite number of nonzero values:
The Z-transform of x(n) is the infinlte power series:
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This is an infinite geometric series. We recall that :
Consequently. for
or equivalently, for:
X(z) converges to :
We see that in this case. the Z-transform provides a compact alternative
representation of the signal.
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Let us express the complex variable z in polar form as :
Then X(z) can be expressed as :
But:
In the ROC of X(z)

Hence |X(z)| is finite if the sequence x(n)r-n is absolutely summable.
The problem of finding the ROC for X(z) is equivalent to determining the
range of values of r for which the sequence x(n)r-n is absolutely summable.
To elaborate, let us express:
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,
If the first sum in (3.1.6) converges. there must exist
values of r small enough such that the product
sequence x(-n)rn is absolutely summable.
Therefore, the ROC for the first sum consists of
all points in a circle of some radius r1 were unde
r1<∞
If the second sum converges, there must exist
values of r large enough such that the product
sequence x(n)/rn is absolutely summable.
Hence the ROC for the second sum consists of all
points outside a circle of radius r > r2,
Since the convergence of X(z) requires that
both sums be finite, it follows that the ROC of
X(z) is generally specified as the annular
region in the z-plane, r2 < r < r1, which is the
common region where both sums are finite.
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Example 2.3.
Determine the Z-transform of the signai:
 n ,
x(n)   n u (n)  
 0,
Solution: From the definition we have :

X (z) 

If
z
 1 or equivalently
n

 n z n   (z 1 )
n0
1
n0
n0
n 0
this power series converges to
Thus we have the Z-transform pair:
The ROC is the exterior of a circle
having radius |a|
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Example 2. 4.
Determine the Z-transform of the signal:
Solution: From the definition we have
were l=-n. Using the formula:
were
provided that
or equivalently
A 1
gives
Thus
ROC
The ROC is now the interior of a circle having radius |a|
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We see that the causal signal and the anticausal signal have identical
closed-form expressions for the Z-transform, that is,
This implies that a closed-form expression for the z-transform does not uniquely
specify the signal in the time domain. The ambiguity can be resolved only if
in addition to the closed-form expression, the ROC is specified.
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2.1.2 The Inverse z-Transform
An inversion formula for obtaining x(n) from X(z) can be derived by using the
Cauchy irzregral theorem, which is an important theorem in the theory of
complex variables.
Suppose that we mulllply both sldes of by zn-1
and inlegrate both sides over a closed conlour
within thc ROC of X(z).
we can interchange the order of
integration and summation on the
right-hand side
Now we can invoke the Cauchy
theorem, which states that
integral
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2.2 PROPERTIES OF THE Z-TRANSFORMI
Linearity.
If
and
then
Example 2. 5.
Determine the Z-transform and
the ROC of the signal:
Solution: If we define the signals
and
then x(n) can be written as
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According to this property, its Z-transform is:
We know that
Thus the overall transform X(z) is:
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Time shifting.
If:
then
Example 2. 6.
If:
The ROC is the same as that of X(z)
except for z = 0 if k>0 and z = ∞ if k<0.
The proof of this property follows
immediately from the definition
of the Z-transform
By applying the time-shifting property, determine the
z-transform of the signals x2(n) şi x3(n)
Solution:
It can easily be seen that :
From property:
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Scaling in the z-domain.
If
then
for any constant a, real or complex. From the definition :
Since the ROC of X(z) is
ROC of X(a-1z) is
or
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Time reversal.
If
then
Proof
where the chanye of variable l = -n. ROC of X(z-1) is:
An intuitive proof is the following. When we fold a signal. the coefficient of z-n
becomes the coefficient of zn. Thus, folding a signal is equivalent to replacing z by z-1 n
the z-transform formula. In other words, reflection in the time domain corresponds to
inversion in the z-domain.
Example 2. 7.
Determine the z-transform of the signal:
It is known that
From property:
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Differentiation in the z-domain.
If
then
Proof
Example 2. 8.
By differentiating both sides of (2.1), we have
Determine the z-transform of the signal:
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Convolution of two sequences. If :
then
The ROC of X(z) is, at least, the intersection of that for X1(z) and X2(z).
Proof. The convolution of x1(n) and x2(n) is defined as :
The Z-transform of x(n) is:
Upon interchanging the order of the summations and applying the time-shifting
property we obtain:
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Exemplul 2. 9.
Compute the convolution of the
signals:
Solution: From definition we have :
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The convolution property is one of the most powerful
properties of the Z-transform because it converts the
convolution of two signals (time domain) to multiplication
of their transforms. Computation of the convolution of
two signals,using the z-transform, requires the following
steps:
1. Compute the z-transforms of the signals to
be convolved.
2. Multiply the two z-transforms..
3. Find the inverse z-transform of X(z).
This procedure is, in many cases, computationally easier
than the direct evaluation of the convolution summation. .
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Correlation of two sequences. If
then
Proof. We recall that:
Using the convolution and time-reversal properties, we easily obtain:
Multiplication of two sequences. If
then
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3.3 RATIONAL Z-TRANSFORMS
Poles and Zeros.
The zeros of a z-transform X(z) are the values of z for
which X(z) = 0. The poles of a z-transform are the
values of z for which X(z) =∞. If X( z) s a rational
function, then
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If a0 ≠ 0 and bo ≠ 0, we can avoid the negative powers of
z by factoring out the terms boz-M and a0z-N as follows:
Since N(z) and D(z) are polynomials in L,
expressed in factored form as:
they can be
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