Digital Signal Processing Reference
In-Depth Information
Sine axis
Imaginary part
4,00
-f
π
/6
2,00
1
2
−
i
ϕ
e
Results at any angle
ϕ
to zero !!!
2
c
os
ϕ
i
sin
ϕ
Cosine axis
Real part
0,00
1
2
e
ϕ
π
/6
-2,00
8,00
4,00
0,00
-4,00
-8,00
Phase shift
π
/6
+f
0.25
0.50
0.75
1.0
s
-4,00
-4,00
-2,00
0,00
2,00
4,00
V
(
)
(
)
it
ωϕ
+
−
it
ω
+
ϕ
e
0
+
e
0
1
(
)
1
(
)
it
ω
+
ϕ
−
it
ω
+
ϕ
(
)
0
0
c
o
s
ωϕ
t
+
=
=
e
+
e
0
2
2
2
(
)
(
)
it
ω
+
ϕ
−
it
ω
+
ϕ
0
0
(
)
(
)
e
−
e
1
1
it
ω
+
ϕ
−
it
ω
+
ϕ
(
)
i
s
in
ωϕ
t
+
=
=
e
0
−
e
0
0
2
2
2
4
2
0
-2
-4
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
4,00
2,00
0,00
-2,00
-4,00
4,00
2,00
0,00
-2,00
-4,00
x- components always point in
the same direction and add .
Therefore we speak of the “
real
part
”.
y- components always add
up to zero .
Therefore
w
e
speak of the “
imaginary
part
”.
-125
-100
-75
-50
-25
0
25
50
75
100
125
Hz
Sine frequency 12 Hz; phase shift 30
0
or
π
/6
Illustration 306:
Complex rotating vectors as the cause sine and cosine oscillation
Here we have recourse to graphic material from Chapter 5 to demonstrate the mathematics of rotating fre-
quency vectors using the data measured.
At a frequency of 12 Hz the frequency vectors travel 12 times a turn of 2ʌ per second. The graphs describe
the momentary conditions at a phase shift of ʌ/6 or 30ࢮ. In the measurements below a sine oscillation of
12Hz with an amplitude of 4V was used. This amplitude was not included in the formulaic display in the
interests of clarity.
