*This preview shows
pages
1–3. Sign up
to
view the full content.*

11.04.1
Chapter 11.04
Discrete Fourier Transform
Introduction
Recalled the exponential form of Fourier series (see Equations 18 and 20 from Chapter
11.02),
∑
∞
−∞
=
=
k
t
ikw
k
e
C
t
f
0
~
)
(
(18, Ch. 11.02)
×
=
∫
−
T
t
ikw
k
dt
e
t
f
T
C
0
0
)
(
1
~
(20, Ch. 11.02)
While the above integral can be used to compute
k
C
~
, it is more preferable to have a
discretized formula version to compute
k
C
~
. Furthermore, the Discrete Fourier Transform (or
DFT) [1–5] will also facilitate the development of much more efficient algorithms for Fast
Fourier Transform (or FFT), to be discussed in Chapters 11.05 and 11.06.
Derivations of DFT Formulas
If time “
t
” is discretized at
,
,.......
,
3
,
2
,
3
2
1
t
n
t
t
t
t
t
t
t
n
∆
=
∆
=
∆
=
∆
=
Then Equation (18, of Chapter 11.02) becomes
∑
−
=
=
1
0
0
~
)
(
N
k
t
ikw
k
n
n
e
C
t
f
(1)
To simplify the notation, define
n
t
n
=
(2)
Then, Equations (1) can be written as
∑
−
=
=
1
0
0
~
)
(
N
k
n
ikw
k
e
C
n
f
(3)
In the above formula, “
n
” is an integer counter. However,
)
(
n
f
and
n
t
do NOT have to be
integer numbers.
Multiplying both sides of Equation (3) by
n
ilw
e
0
−
, and performing the summation on “
n
”, one
obtains ( note:
l
= integer number)

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*