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Z-transforms
Computation of the Z-transform for discrete-time signals:
• Enables analysis of the signal in the frequency domain.
• Z - Transform takes the form of a polynomial.
• Enables interpretation of the signal in terms of the roots of the
polynomial.
• z −1 corresponds to a delay of one unit in the signal.
The Z - Transform of a discrete time signal x[n] is defined as
X(z) =
+∞
X
x[n].z −n
(1)
n=−∞
where z = r.ejω
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The discrete-time Fourier Transform (DTFT) is obtained by
evaluating Z-Transform at z = ejω .
or
The DTFT is obtained by evaluating the Z-transform on the unit
circle in the z-plane.
The Z-transform converges if the sum in equation 1 converges
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Region of Convergence(RoC)
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Region of Convergence for a discrete time signal x[n] is defined as a
continuous region in z plane where the Z-Transform converges.
In order to determine RoC, it is convenient to represent the
Z-Transform as:a
P (z)
X(z) =
Q(z)
• The roots of the equation P (z) = 0 correspond to the ’zeros’ of
X(z)
• The roots of the equation Q(z) = 0 correspond to the ’poles’ of
X(z)
• The RoC of the Z-transform depends on the convergence of the
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a Here
we assume that the Z-transform is rational
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polynomials P (z) and Q(z),
• Right-handed Z-Transform
– Let x[n] be causal signal given by
x[n] = an u[n]
– The Z - Transform of x[n] is given by
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X(z) =
+∞
X
$
x[n]z −n
n=−∞
=
+∞
X
an u[n]z −n
n=−∞
=
+∞
X
an z −n
n=0
=
+∞
X
(az −1 )n
n=0
=
=
1
1 − az −1
z
z−a
– The ROC is defined by |az −1 | < 1 or |z| > |a|.
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– The RoC for x[n] is the entire region outside the circle
z = aejω as shown in Figure 1.
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RoC |z| > |a|
a
z−plane
Figure 1: RoC(green region) for a causal signal
• Left-handed Z-Transform
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– Let x[n] be an anti-causal signal given by
y[n] = −bn u[−n − 1]
– The Z - Transform of y[n] is given by
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Y (z) =
+∞
X
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y[n]z −n
n=−∞
=
+∞
X
−bn u[−n − 1]z −n
n=−∞
=
−1
X
−bn z −n
n=−∞
=
+∞
X
−(b−1 z)n + 1
n=0
=
=
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1
1−
z
b
+1
z
z−b
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– Y (z) converges when |b−1 z < 1 or |z| < |b|.
– The RoC for y[n] is the entire region inside the circle
z = bejω as shown in Figure 2
RoC |z| < |a|
a
z−plane
Figure 2: RoC(green region) for an anti-causal signal
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• Two-sided Z-Transform
– Let y[n] be a two sided signal given by
y[n] = an u[n] − bn u[−n − 1]
where, b > a
– The Z - Transform of y[n] is given by
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Y (z) =
+∞
X
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y[n]z −n
n=−∞
=
+∞
X
(an u[n] − bn u[−n − 1])z −n
n=−∞
=
+∞
X
an z −n −
=
(az−1)n −
n=0
=
=
&
+∞
X
(b−1 z)n
n=1
1
1
.
1 − az −1 1 −
z
bn z −n
n=−∞
n=0
+∞
X
−1
X
z
b
+1
z
.
z−a z−b
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– Y (z) converges for |b−1 z| < 1 and |az −1 | < 1 or |z| < |b| and
|z| > |a| . Hence, for the signal
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– The ROC for y[n] is the intersection of the circle z = bejω
and the circle z = aejω as shown in Figure 3
RoC |a| < |z| < |b|
a
b
z−plane
Figure 3: RoC(pink region) for a two sided Z Transform
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• Transfer function H(z)
– Consider the system shown in Figure 4.
y[n] = x[n]*y[n]
x[n]
h[n]
Y(z) = X(z)H(z)
X(z)
H(z)
Figure 4: signal - system representation
– x[n] is the input and y[n] is the output
– h[n] is the impulse response of the system. Mathematically,
this signal-system interaction can be represented as follows
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y[n] = x[n] ∗ h[n]
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– In frequency domain this relation can be written as
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Y (z) = X(z).H(z)
or
H(z) =
Y (z)
X(z)
H(z) is called ’Transfer function’ of the given system.
In the time domain if x[n] = δ[n] then y[n] = h[n],
h[n] is called the ’impulse response’ of the system.
Hence, we can say that
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h[n] ←→ H(z)
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Some Examples: Z-transforms
• Delta function
Z(δ[n]) = 1
Z(δ[n − n0 ]) = z −n0
• Unit Step function
x[n]
= 1, n ≥ 0
= 0, otherwise
1
, |z| > 1
X(z) =
−1
1−z
x[n]
The Z-transform has a real pole at the z = 1.
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• Finite length sequence
x[n]
= 1, 0 ≤ n ≥ N
x[n]
= 0, otherwise
1 − z −N
X(z) =
1 − z −1
N
N −1 z − 1
, |z| > 1
= z
z−1
The roots of the numerator polynomial are given by:
z = 0, N zeros at the origin
and the nth roots of unity:
z=e
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j2πk
N
, k = 0, 1, 2, · · · , N − 1
(2)
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• Causal sequences
1 n
1 n
x[n] = ( ) u[n] − ( ) u[n − 1]
3
2
1
1
1
−1
−
z
,
|z|
>
X(z) =
3
1 − 13 z −1
1 − 12 z −1
The Discrete time Fourier transform can be obtained by setting
z = ejω Figure 5 shows the Discrete Fourier transform for the
rectangular function.
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1
1
−N
2
+N
2
−4π
Ν+1
−2π 2π
Ν+1 Ν+1
4π
Ν+1
Figure 5: Discrete Fourier transform for the rectangular function
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Some Problems
Find the Z-transform (assume causal sequences):
2
3
a a
1. 1, 1!
, 2! , a3! , · · ·
2.
a3
a5
a7
0, a, 0, − 3! , 0, 5! , 0, − 7! , · · ·
3.
a2
a4
a6
0, a, 0, − 2! , 0, 4! , 0, − 6! , · · ·
Hint: Observe that the series is similar to that of the exponential
series.
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