ENEE 241 02
*
READING ASSIGNMENT 13
Thu 03/26 Lecture
Topic:
examples of the DFT and its inverse; DFT of a realvalued signal
Textbook References:
sections 3.2, 3.3
Key Points:
•
The
N
point vector
s
is a complex sinusoid of frequency
ω
= 2
kπ/N
if and only its DFT (or
spectrum)
S
contains all zeros except for
S
[
k
]
6
= 0.
•
Linearity property of the DFT:
s
=
α
x
+
β
y
DFT
←→
S
=
α
X
+
β
Y
•
The DFT
S
of a
N
point realvalued signal
s
exhibits circular conjugate symmetry:
S
[0] =
S
*
[0]
and
S
[
N

k
] =
S
*
[
k
]
,
k
= 1 :
N

1
(
DFT 2
)
•
Every realvalued signal
s
can be expressed as
s
[
n
] =
1
N
S
[0] +
2
N
·
X
0
<k<N/
2

S
[
k
]

cos
2
πkn
N
+
∠
S
[
k
]
¶
+
1
N
S
[
N/
2](

1)
n
The second term (corresponding to frequency
ω
=
π
) is present only when
N
is even.
Theory and Examples:
1
. In the previous assignment, we considered DFT vectors (or spectra) with a single nonzero
entry. The corresponding time domainsignals are Fourier sinusoids. Thus:
•
If
S
=
£
0
0
0
0
1
0
0
0
/
T
then the timedomain signal
s
is a sinusoid of frequency
ω
= 4(2
π/
8) =
π
. Specifically,
s
=
1
8
v
(4)
=
1
8
£
1

1
1

1
1

1
1

1
/
T
since
s
[
n
] =
1
8
e
jπn
=
(

1)
n
8
,
n
= 0 : 7
•
If
S
=
£
0
0
0
1
0
0
0
0
/
T
then
s
=
v
(3)
/
8. This is a sinusoid of frequency
ω
= 3(2
π/
8) = 3
π/
4, and is given by
s
[
n
] =
1
8
e
j
(3
π/
4)
n
,
n
= 0 : 7
In other words,
s
=
1
8
h
1

√
2
2
+
j
√
2
2

j
√
2
2
+
j
√
2
2

1
√
2
2

j
√
2
2
j

√
2
2

j
√
2
2
i
T
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
•
We now introduce a second nonzero entry in
S
, i.e., a second sinusoidal component in
s
.
Letting
S
=
£
0
0
0
1
0
1
0
0
/
T
we have
s
=
v
(3)
+
v
(5)
8
which is the scaled sum of two columns of
V
. The corresponding frequencies are
ω
=
3
π/
4 (as before); and
ω
= 5
π/
4, which is the same frequency as
ω
=

3
π/
4 for complex
sinusoids. Since
e
jωn
+
e
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 staff
 Frequency, DFT, Complex number, Hilbert space, complex conjugate

Click to edit the document details