Digital Signal Processing Reference
In-Depth Information
We first assume the FOURIER- series still exists. Only then we examine the exact
influence of
T
0
towards infinity.
T
0
∞
2
π
1
2
¦
jkt
ω
−
kt
ω
³
x t
()
=
c e
with
ω
=
=
2
π
f
and
c
=
x t e
()
dt
0
0
k
0
0
k
T
T
k
=−∞
T
0
0
−
0
2
The index
T
0
only suggests that formalism applies to all cycle durations, also for
T
0
∞
2
1
1
-
jkt
ω
-
kt
ω
³
³
T
→∞
. Thus
c
=
x t e
( )
dt
=
x t e
( )
dt
0
0
0
k
T
T
0
T
0
-
∞
0
-
2
Here, the frequency response
X(Ȧ)
that was included in the solution of the integral is of
particular interest. We define the FOURIER- transformation
FT
as
∞
()
()
³
−
jt
ω
X
f
=
x t
e
dt
with
ωπ
=
2
f
−∞
The complex amplitudes or FOURIER- coefficients can then be written as
1
c
=
X k
(
ω
)
0
T
0
When inserted in the FOURIER- series (see above) this results in:
∞
1
¦
-
jkt
ω
xt
()
=
X k
(
ω
)
e
0
0
T
k
=−∞
0
2
π
1
∞
¦
jk
ω
t
Because of
ω
==
2
π
f
it follows
x
=
X k
(
ω
)
e
ω
0
0
0
T
0
0
T
2
π
0
k
=−∞
0
∞
¦
jk
Δ
ω
t
In terms of limits: ( )
xt
=
lim
x t
( )
=
lim
X k
(
Δ
ω
)
e
Δ
ω
T
T
→∞
0
ω
→
0
0
0
k
=−∞
The geometrical interpretation for this formalism results in the area beneath the curve of
X(
)e
j
ω
t
as the sum of the step- like small rectangles.
ω
jk
Δ
ω
t
(
) (
)
k
Δ⋅
ω
X k
(
Δ
ω
e
)
Consideration of the limit value results in:
∞
∞
1
()
()
jt
ω
jt
ω
³
³
x t
()
=
X
ω
e
d
ω
=
X
f
e
df
2
π
−∞
−∞
This is the inverse FOURIER- transformation
IFT
, by means of which the chronological
curve can be calculated from the information of the frequency area. The structural com-
parison of the formalism of
FT
and
IFT
shows the perfect symmetry of both transforma-
tions, to some extent the source information for chapter 5.
