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The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC 212 Chapter 8
Discrete Fourier Transform
2
0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT)
±
Recall what we have learnt in Chapter 02,
3
0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT)
±
In discretetime signal processing, the DTFT is not directly applicable
to the digital analysis because:
– We cannot use a computer / digital processing unit / DSP chip to evaluate
the DTFT, since
X
(
e
j
ω
)
is a
continuous function
of
ω
±
For practical applications, we use a closely related transform known as
Discrete Fourier transform (DFT).
–U
s
e
d
t
o
evaluate the frequency response of a finite sequence
– Can be used to evaluate the samples of the continuous spectrum (ie. we
can regard the DFT as a discrete version of the DTFT)
– Can be evaluated using efficient algorithms (next lecture)
±
DFT is defined based on the Discrete Fourier Series (DFS)
4
0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
±
DFS gives a
discrete
frequency domain representation for periodic signals
±
For periodic signals, the DTFT and
z
transform do not uniformly converge
– For a periodic sequence
, the sequence
is not absolutely summable
for any
±
DFS is based on fact that any
periodic sequence
with period
N
can be
represented as a
finite
weighted sum
of
N
harmonicallyrelated complex
exponentials of the form
±
Definition:
]
[
~
n
x
r
n
r
n
x
−
]
[
~
]
[
~
n
x
]
[
~
of
(IDFS)
DFS
Inverse
]
[
~
1
]
[
~
]
[
~
of
DFS
]
[
~
]
[
~
1
0
/
2
1
0
/
2
k
X
e
k
X
N
n
x
n
x
e
n
x
k
X
N
k
N
kn
j
N
n
N
kn
j
⇒
=
⇒
=
∑
∑
−
=
−
=
−
π
integers
are
and
,
]
[
]
[
)
/
2
(
l
k
n
e
e
n
e
lN
k
kn
N
j
k
+
=
=
Discrete Fourier Series (DFS)
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0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
]
[
~
of
(IDFS)
DFS
Inverse
]
[
~
1
]
[
~
]
[
~
of
DFS
]
[
~
]
[
~
1
0
/
2
1
0
/
2
k
X
e
k
X
N
n
x
n
x
e
n
x
k
X
N
k
N
kn
j
N
n
N
kn
j
⇒
=
⇒
=
∑
∑
−
=
−
=
−
π
±
Point to note:
The periodic sequence
is represented by
only
N
coefficients
,
–
can be defined arbitrarily for
k
outside this range
– Convention is to define
to be
periodic
with period
N,
since it leads to
simple duality relations
between time and frequency domains for DFS
]
[
~
n
x
]
[
~
k
X
1
0
]
[
~
−
≤
≤
N
k
k
X
for
]
[
~
k
X
Discrete Fourier Series (DFS)
6
0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
∑
∞
−∞
=
⎩
⎨
⎧
=
=
−
=
r
rN
n
rN
n
n
x
otherwise
,
0
,
1
]
[
]
[
~
δ
Discrete Fourier Series (DFS)
±
Example 8.1:
DFS of a periodic impulse train
– Consider the periodic impulse train
– Since
for
, the DFS coefficients are found as
– In this case,
is the same for all
k
– Substituting
,
into the IDFS formula leads to the
equivalent timedomain representation
1
]
[
]
[
~
1
0
/
2
=
=
∑
−
=
−
N
n
N
kn
j
e
n
k
X
]
[
]
[
~
n
n
x
=
1
0
−
≤
≤
N
n
]
[
~
k
X
1
0
−
≤
≤
N
k
1
]
[
~
=
k
X
∑
−
=
=
1
0
/
2
1
]
[
~
N
n
N
kn
j
e
N
n
x
7
0910 Fall ELEC 212
Discrete Fourier Transform (DFT)
Properties of the DFS
±
Linearity
±
Shift of a Sequence
(Time Domain)
– Any shift greater than the period (ie.
m
≥
N
) cannot be distinguished in the
timedomain from a shorter shift
m
1
such that
m
=
m
1
+
rN
, where
m
1
and
r
are
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 Spring '11
 Prof.MatthewMcKay
 Digital Signal Processing, Signal Processing

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