Purdue University: ECE438  Digital Signal Processing with Applications
1
ECE438  Laboratory 6:
Discrete Fourier Transform and Fast Fourier
Transform Algorithms
(Week 1)
October 6, 2010
1
Introduction
This is the first week of a two week laboratory that covers the Discrete Fourier Transform
(DFT) and Fast Fourier Transform (FFT) methods. The first week will introduce the DFT
and associated sampling and windowing effects, while the
second week
will continue the
discussion of the DFT and introduce the FFT.
In previous laboratories, we have used the DiscreteTime Fourier Transform (DTFT)
extensively for analyzing signals and linear timeinvariant systems.
(DTFT)
X
(
e
jω
)
=
∞
summationdisplay
n
=
∞
x
(
n
)
e

jωn
(1)
(inverse DTFT)
x
(
n
)
=
1
2
π
integraldisplay
π

π
X
(
e
jω
)
e
jωn
dω.
(2)
While the DTFT is very useful analytically, it usually cannot be exactly evaluated on a com
puter because equation (1) requires an infinite sum and equation (2) requires the evaluation
of an integral.
The discrete Fourier transform (DFT) is a sampled version of the DTFT, hence it is
better suited for numerical evaluation on computers.
(DFT)
X
N
(
k
)
=
N

1
summationdisplay
n
=0
x
(
n
)
e

j
2
πkn/N
(3)
(inverse DFT)
x
(
n
)
=
1
N
N

1
summationdisplay
k
=0
X
N
(
k
)
e
j
2
πkn/N
(4)
Here
X
N
(
k
) is an
N
point DFT of
x
(
n
). Note that
X
N
(
k
) is a function of a discrete integer
k
, where
k
ranges from 0 to
N
−
1.
Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman,
School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494
0340; [email protected]
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Purdue University: ECE438  Digital Signal Processing with Applications
2
In the following sections, we will study the derivation of the DFT from the DTFT, and
several DFT implementations. The fastest and most important implementation is known as
the fast Fourier transform (FFT). The FFT algorithm is one of the cornerstones of signal
processing.
2
Deriving the DFT from the DTFT
2.1
Truncating the Timedomain Signal
The DTFT usually cannot be computed exactly because the sum in equation (1) is infinite.
However, the DTFT may be approximately computed by truncating the sum to a finite
window. Let
w
(
n
) be a rectangular window of length
N
:
w
(
n
) =
braceleftBigg
1
:
0
≤
n
≤
N
−
1
0
:
else
.
(5)
Then we may define a truncated signal to be
x
tr
(
n
) =
w
(
n
)
x
(
n
)
.
(6)
The DTFT of
x
tr
(
n
) is given by:
X
tr
(
e
jω
)
=
∞
summationdisplay
n
=
∞
x
tr
(
n
)
e

jωn
(7)
=
N

1
summationdisplay
n
=0
x
(
n
)
e

jωn
.
(8)
We would like to compute
X
(
e
jω
), but as with filter design, the truncation window
distorts the desired frequency characteristics;
X
(
e
jω
) and
X
tr
(
e
jω
) are generally not equal.
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 Spring '08
 Staff
 Digital Signal Processing, Algorithms, Signal Processing, DFT

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