Discretetime Fourier Analysis (DTFT)
Discretetime Fourier analysis (transform) is a powerful tool to
analyze the frequency characteristics of discretetime signals and
systems. If the discretetime signal is periodic, discretetime Fourier
series will be used instead.
Continuoustime Fourier Transform
If a continuoustime signal
x
(
t
) is nonperiodic, the frequency
characteristics can be expressed in terms of Fourier transform
∫
∞
∞


=
dt
e
t
x
f
X
ft
j
π
2
)
(
)
(
The above equation is referred to as analysis equation. The time signal
x
(
t
) can be obtained (synthesized) by the corresponding inverse
transform
∫
∞
∞

=
df
e
f
X
t
x
ft
j
π
2
)
(
)
(
Discretetime Fourier Transform
The frequency analysis of a discretetime signal
x
[
n
] is to express
the signal sequence in terms of complex exponential sequences {
n
j
e
ϖ
ˆ
},
where
ϖ
ˆ
is the frequency variable called as normalized radian
frequency. The analysis is called as discretetime Fourier transform
(DTFT) and defined by
∑
∞
∞
=

=
n
n
j
j
e
n
x
e
X
ϖ
ϖ
ˆ
ˆ
]
[
)
(
,
)
,
(
ˆ
π
π
ϖ

∈
The DTFT is a continuous periodic function of
ϖ
ˆ
with period of 2
π
.
Please note that {
n
j
e
ϖ
ˆ
} are periodic functions with period of 2
π
Example: DTFT of an exponential sequence
Consider the causal sequence
.
1


],
[
]
[
<
=
α
α
n
u
n
x
n
The DTFT of
x
[
n
] is given by
∑
∑
∑
∞
=

∞
∞
=

∞
∞
=

=
=
=
0
ˆ
ˆ
ˆ
ˆ
]
[
]
[
)
(
n
n
j
n
n
n
j
n
n
n
j
j
e
e
n
u
e
n
x
e
X
ϖ
ϖ
ϖ
ϖ
α
α
ϖ
ϖ
α
α
ˆ
0
ˆ
1
1
)
(
j
n
n
j
e
e

∞
=


=
=
∑
Note: The DTFT of most practical discretetime sequences is expressed
in using geometric series.
The time signal
x
[
n
] can be computed from the following inverse
integral
∫

=
π
π
ϖ
ϖ
ϖ
π
ˆ
)
(
2
1
]
[
ˆ
ˆ
d
e
e
X
n
x
n
j
j
Proof:
Substituting the DTFT definition into the inverse transform gives
∫
∫
∑
∑



∞
∞
=
∞
∞
=

=
⋅
π
π
ϖ
π
π
ϖ
ϖ
ϖ
π
ϖ
π
ˆ
2
1
]
[
ˆ
]
[
2
1
)
(
ˆ
ˆ
ˆ
d
e
m
x
d
e
e
m
x
m
n
j
m
n
j
m
m
j
We have
n
m
n
m
d
e
m
n
j
≠
=
=
∫


0
1
)
(
ˆ
2
1
ϖ
π
π
ϖ
π
Using the above result into the inverse transform, we obtain
∫
∑

∞
∞
=

=
⋅
π
π
ϖ
ϖ
ϖ
π
]
[
]
[
2
1
ˆ
ˆ
n
x
d
e
e
m
x
n
j
m
m
j
The inverse transform is called as synthesis equation because the time
signal can be synthesized by using the Fourier coefficients and complex
exponential functions.
Discretetime Fourier transform pair:
∑
∞
∞
=

=
n
n
j
j
e
n
x
e
X
ϖ
ϖ
ˆ
ˆ
]
[
)
(
∫

=
π
π
ϖ
ϖ
ϖ
π
ˆ
)
(
2
1
]
[
ˆ
ˆ
d
e
e
X
n
x
n
j
j
Note: If the DTFT is expressed in terms of a rational function of
n
j
e
ϖ
ˆ
, to
find the inverse transform, we can express the rational function in terms
of partial fraction expansion and use binomial expansion to find the
inverse. For example:
ϖ
β
α
β
ϖ
β
α
α
ϖ
ϖ
ϖ
β
α
β
α
ˆ
ˆ
ˆ
ˆ
ˆ
1
1
)
1
)(
1
(
1
)
(
j
j
j
j
j
e
e
e
e
e
X









=


=
∑
∑
∞
=


∞
=



=
0
ˆ
0
ˆ
n
n
j
n
n
n
j
n
e
e
ϖ
β
α
β
ϖ
β
α
α
β
α
Comparing the expansion with the DTFT,
∑
∞
∞
=

=
n
n
j
j
e
n
x
e
X
ϖ
ϖ
ˆ
ˆ
]
[
)
(
gives
F
]
[
]
[
]
[
n
u
n
x
n
n
β
α
β
α
β
β
α
α



=
Magnitude Spectrum and Phase Spectrum
The magnitude spectrum and phase spectrum of a discretetime
signal
x
[
n
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Frequency, Signal Processing, jω