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EE4150
DIGITAL SIGNAL PROCESSING
HOMEWORK # 4
CHAPTER 5
(Discrete Fourier Transform)
OBJECTIVES
:
1. Let the students understand and use the DFT (Discrete Fourier Transform) as a tool to
analyze and process experimental data in frequency domain.
2. Let the students familiarize with MATLAB.
1
Introduction
The DFT of a discrete sequence
{
x
[
n
]
}
is deﬁned as sampled version of the FT at equally spaced
frequency points
X
[
k
]=
X
±
2
πk
N
²
=
N

1
X
n
=0
x
[
n
]
e

j
2
πkn/N
k
=0
,
1
,...,N

1(
1
)
where
N
is the number of samples used in the transformation. Thus, the DFT provides fre
quency information of the sequence
{
x
[
n
]
}
in the interval 0
≤
ω
≤
2
π
with spacing Δ
ω
=2
π/N
.
However, for sequences with inﬁnite length the DFT provides just an
approximation
of the
frequency content of the time sequence.
Now, if the discrete sequence
{
x
[
n
]
}
is obtained from sampling a continuous time signal
x
a
(
t
), i.e.
x
[
n
x
a
(
nT
)
∞
<n<
∞
(2)
where
T
is the sampling period. Then, there exists a relation between frequency in discrete
time
f
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