453.701 Linear Systems, S.M. Tan, The University of Auckland
91
Chapter 9
The Discrete Fourier transform
9.1
De
°
nition
When computing spectra on a computer it is not possible to carry out the integrals involved in
the continuous time Fourier transform.
Instead a related transform called the
discrete Fourier
transform
is used. We shall examine the properties of this transform and its relationship to the
continuous time Fourier transform.
The discrete Fourier transform (also known as the
°
nite Fourier transform) relates two
°
nite se
quences of length
N
. Given a sequence with components
x
[
k
]
for
k
= 0
;
1
; :::; N
¡
1
, the discrete
Fourier transform of this sequence is a sequence
X
[
r
]
for
r
= 0
;
1
; :::; N
¡
1
de
°
ned by
X
[
r
] =
1
N
N
−
1
X
k
=0
x
[
k
] exp
°
¡
j
2
…rk
N
¶
(9.1)
The corresponding inverse transform is
x
[
k
] =
N
−
1
X
r
=0
X
[
r
] exp
°
j
2
…rk
N
¶
(9.2)
Notice that we shall adopt the de
°
nition in which there is a factor of
1
=N
in front of the forward
transform. Other conventions place this factor in front of the inverse transform (this is used by
MatLab) or a factor of
1
=
p
N
in front of both transforms.
Exercise:
Show that these relationships are indeed inverses of each other.
It is useful
°
rst to
establish the relationship
N
−
1
X
r
=0
exp
°
j
2
…rk
N
¶
=
‰
N
if
k
is a multiple of
N
0
otherwise
(9.3)
9.2
Discrete Fourier transform of a sampled complex exponential
A common application of the discrete Fourier transform is to
°
nd sinusoidal components within a
signal. The continuous Fourier transform of
exp(
j
2
…”
0
t
)
is simply a delta function
–
(
”
¡
”
0
)
at the
frequency
”
0
. We now calculate the discrete Fourier transform of a sampled complex exponential
x
[
k
] =
A
exp (
j
2
…”
0
k¿
)
;
for
k
= 0
;
1
; : : : ; N
¡
1
(9.4)
The sampling interval is
¿
and we denote the duration of the entire sampled signal by
T
=
N¿
.
Substituting (9.4) into the de
°
nition of the discrete Fourier transform (9.1) yields
X
[
r
] =
1
N
N
−
1
X
k
=0
A
exp
°
j
2
…k
N
¶
[
”
0
T
¡
r
]
(9.5)
If
”
0
T
is an integer, i.e., if there are an integer number of cycles in the frame of duration
T
, we see
that
X
[
r
] =
‰
A
if
r
¡
”
0
T
is a multiple of
N
0
otherwise
(9.6)
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453.701 Linear Systems, S.M. Tan, The University of Auckland
92
In this case, there is only a single nonzero term in the discrete Fourier transform at an index which
depends on the frequency
”
0
. The value of this nonzero term is
A
, the complex amplitude of the
component.
If
”
0
T
is not an integer however, we
°
nd that
X
[
r
] =
A
N
exp (
¡
j…
(
N
¡
1)(
r
¡
”
0
T
)
=N
)
sin[
…
(
r
¡
”
0
T
)]
sin[
…
(
r
¡
”
0
T
)
=N
]
(9.7)
This is nonzero for all values of
r
. Thus even though there is only a single frequency component in
the signal, it can a
ff
ect all the terms in the discrete Fourier transform.
A plot of the last factor in the equation (9.7) is shown in Figure 9.1. Since
r
only takes on integer
values, the terms in the output sequence are samples from this function taken at unit spacing.
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