if the region of convergence for
X
(
z
)
includes the
unit circle.
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ECE 513, Fall 2016
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Frequency Representation
Let us multiply
X
(
ω
)
by
e
j
ω
m
and integrate it over one period to
obtain
π
−
π
X
(
ω
)
e
j
ω
m
d
ω
=
π
−
π
∞
n
=
−∞
x
(
n
)
e
−
j
ω
n
e
j
ω
m
d
ω
(52)
If
X
(
ω
)
exist (implies that the region of convergence for
X
(
z
)
includes the unit circle), then we can change the order of
summation and integration.
π
−
π
X
(
ω
)
e
j
ω
m
d
ω
=
∞
n
=
−∞
x
(
n
)
π
−
π
e
j
ω
(
m
−
n
)
d
ω
(53)
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FUNDAMENTAL DSP CONCEPTS
ECE 513, Fall 2016
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Frequency Representation
We observe that
π
−
π
e
j
ω
(
m
−
n
)
d
ω
=
2
π
m
=
n
0
m
=
n
(54)
It follows that
π
−
π
X
(
ω
)
e
j
ω
m
d
ω
=
2
π
x
(
n
)
(55)
or
x
(
m
) =
1
2
π
π
−
π
X
(
ω
)
e
j
ω
m
d
ω
(56)
Thus, the Fourier Transform synthesis, transform pair for
discrete–time sequence is given by
x
(
m
) =
1
2
π
π
−
π
X
(
ω
)
e
j
ω
m
d
ω
(57)
X
(
ω
) =
∞
n
=
−∞
x
(
n
)
e
−
j
ω
n
(58)
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ECE 513, Fall 2016
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Frequency Representation
We can obtain the frequency representation of a sequence by
evaluating its ZTransform on the unit circle as shown in the
previous section.
We will present an example to illustrate the frequency
representation of a discrete–time sequence.
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ECE 513, Fall 2016
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Frequency Representation Example
Example (1.4)
Consider the discrete–time sequence
x
(
n
) =
2
.
0
(
0
.
75
)
n
u
(
n
)
(59)
The corresponding Z–Transform is given by
X
(
z
) =
2
.
0
∞
n
=
0
(
0
.
75
)
n
z
−
n
=
2
.
0
1
−
0
.
75
z
−
1
;

z

>
0
.
75
(60)
Since the region of convergence includes the unit circle,
X
(
ω
)
is
defined and
X
(
ω
) =
2
.
0
1
−
0
.
75
e
−
j
ω
(61)
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ECE 513, Fall 2016
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Frequency Representation Example
Example (1.4)
We can simplify
X
(
ω
)
X
(
ω
) =
2
.
0
(
1
−
0
.
75
e
j
ω
)
(
1
−
0
.
75
e
−
j
ω
)(
1
−
0
.
75
e
j
ω
)
(62)
X
(
ω
) =
2
.
0
−
1
.
5
cos
(
ω
)
−
1
.
5
jsin
(
ω
)
1
−
1
.
5
cos
(
ω
) +
0
.
5625
(63)
Fig. 9 gives a plot of the magnitude of
X
(
ω
)
and Fig. 10 gives a
plot of the phase of
X
(
ω
)
.
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ECE 513, Fall 2016
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Frequency Representation Example
Example (1.4)
0
0.5
1
1.5
2
2.5
3
3.5
1
2
3
4
5
6
7
8
Frequency Plot
Frequency(radians)
Magnitude
Figure ¹:
Magnitude plot of
X
(
ω
)
!
for Example 19.
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