Digital Signal Processing Reference
In-Depth Information
FOURIER- analysis of periodic signals
Periodic signals always fulfill the following formalism:
xt
(
+=
T
)
xt
( ) für alle
t
∈
\
The cycle duration of the signal
T
0
is the smallest possible positive value for
T
. The most
important periodic signal in this context is without doubt the sine oscillation, shown here
as
xt
()
=
cos(
ω
t
+
ϕ
)
0
According to Chapter 2 any periodic signal can be regarded as the sum of amplitude
weighted sine oscillations whose frequencies are an integral multiple of the fundamental
frequency
f
0
.
()
()
()
()
u
t
=
u
t
+
u
t
+
u
t
+
...
periodisch
1
2
3
ˆ
ˆ
ˆ
(
)
(
)
(
)
=
U
sin
ωϕ
t
+
+
U
sin 2
ωϕ
t
+
+
U
sin 3
ωϕ
t
+
+
...
1
0
1
2
0
2
3
0
3
n
→∞
ˆ
¦
(
)
=
Un
sin
ωϕ
t
+
n
0
n
n
=
1
In mathematical notation this usually is formulated as follows:
∞
¦
(
)
xt
()
=+
C
C
cos
k
ω
t
−
ϕ
0
k
0
k
k
=
1
This representation is called the
harmonic form
of the FOURIER- series. Seen from the
perspective of physics,
C
0
is the direct current component (offset) of the signal as with,
for example, the periodic ramp or the sawtooth wave.
As seen in Illustration 98, every phase- shifted sine oscillation can be written as the sum
of a (weighted) sine and a cosine oscillation:
a
∞
(
)
¦
(
)
(
)
0
x t
()
=+
a
cos
k
ω
t
+
b
sin
k
ω
t
k
0
k
0
2
k
=
1
This notation is called the
trigonometric form
of the FOURIER- series.
NB: Any information of a periodic signal have to lie within a period
T
0
, because
the signal then repeats itself. Furthermore, the momentary temporally process of
x(t)
has to be linked to the accordant frequency
kf
0
.
Thus, it is hardly surprising that both coefficients
a
k
and
b
k
are calculated by the arithme-
tic average of the product of
x(t)
and the sine oscillation of the correspondending
frequency
kf
0
:
2
2
³
³
a
=
x t
()cos(
k
ω
t dt
) bzw.
b
=
x t
()sin(
k
ω
t dt
)
k
0
k
0
T
T
0
T
0
T
0
0
The exact mathematical derivation has yet to be described in the following.
