Fourier Transforms, DFTs, and FFTs
Author: John M. Cimbala, Penn State University
Latest revision: 21 February 2008
Introduction
•
In
spectral analysis
,
our goal is to determine the
frequency content
of a signal
.
•
For analog signals, we use
Fourier series
, as discussed in a previous learning module.
•
For digital signals, we use
discrete Fourier transforms
, as discussed in
this
learning module.
Fourier transform (FT)
•
The
Fourier transform
(
FT
) is
a generalization of the Fourier series
.
•
Instead of the
sines
and
cosines
in a Fourier series, the Fourier transform uses
exponentials
and
complex
numbers
; instead of the
summations
in a Fourier series, the Fourier transform uses
integrals
.
•
For a signal or function
f
(
t
), the
Fourier transform
is defined as
() (
)
it
fte d
t
ω
∞
−
−∞
=
∫
F
, and the
inverse
Fourier transform
is defined as
()
1
2
tF
e
d
f
π
∞
−∞
=
∫
, where
i
is the
unity imaginary number
, defined
as the square root of
−
1,
1
i
=−
, and
is the range of angular frequencies associated with the signal – the
frequency content of the signal.
•
When working with time and the signal
f
(
t
), we are working in the
time domain
, and the variables are
real
.
•
When working with angular frequency and the Fourier transform
F
(
), we are working in the
frequency
domain
, and
F
(
) is
complex
.
•
The FT is an
analog
tool – it is used for analyzing the frequency content of
continuous
signals.
Discrete Fourier transform (DFT)
•
We define the
discrete Fourier transform
(
DFT
) – a Fourier transform for a
discrete
(digital) signal.
•
The DFT is a
digital
tool – it is used for analyzing the frequency content of
discrete
signals.
•
The discrete Fourier transform is defined as
(
)
(
)
1
2
0
N
ik
f
n
t
n
Fkf
fnte
−
−
ΔΔ
=
Δ=
Δ
∑
for
k
= 0, 1, 2, .
..,
N
−
1.
•
Note that
summation
has replaced
integration
since
discrete
rather than
continuous
data are being examined.
•
In the
time domain
, the relevant variables are:
o
N
= total number of discrete data points taken.
o
T
= total sampling time.
o
Δ
t
= time between data points,
/
tTN
Δ =
.
o
f
s
= sampling frequency,
1/
/
s
f
tNT
=Δ
=
.
•
Note that
integers
n
and
k
in the above definition of the
DFT have values from 0 to
N
−
1, not from 1 to
N
.
•
For example, consider data sampled as in the plot to the
right. Data are sampled discretely at a sampling
frequency of 0.5 Hz. Starting at time
t
= 0,
N
= 4 data
points are taken. Here,
T
= 8 s,
Δ
t
=
T
/
N
= (8 s)/4 = 2 s, and
f
s
= 1/
Δ
t
= 0.5 Hz. The discrete data used to
calculate the DFT are
those at
t
= 0, 2, 4, and
6 seconds. The data
point at
n
=
N
(at
t
=
T
=
8 s) is
not used
.
f
(
t
)
t
(s)
t
= 0
n
= 0
4
2
Δ
t
Data point
not
used
6
3
8
4
2
1
Data points used
-3
-2
-1
0
1
2
3
0
0.02
0.04 0.06
0.08
0.1
0.12
0.14 0.16
0.18
0.2
t
(s)
f
(
t
)
Actual
period of
the signal
T
•
Consider a periodic
signal that repeats itself
every 0.1 s, as plotted.