Digital Signal Processing Reference
In-Depth Information
Sinus
Time domain
FFT
Compl.Spectr
IF FT
Sinusoidal signal 12 Hz; phase shift
0
degree
1, 00
0, 75
0, 50
0, 25
0, 00
-0, 25
-0, 50
-0, 75
-1, 00
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
x
- components at +12 and -12 Hz
0
0
y
- co
m
ponents at +12 and -12
H
z
+0,5
-0 ,5
-125
-100
-75
-50
-25
0
25
50
75
100
125
Hz
Sinusoidal signal 12 Hz; phase shift
30
degree
1, 00
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
-1,00
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
ms
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
0, 25
0, 25
x
- components at +12 and -12 Hz
N
ote
: y-co
m
ponents always have
opposite
direction with the sum
„zero“. Therefore this axis is called
imaginary axis
y
- co
m
ponents at +12 and -12
H
z
+0 ,43
-0 ,43
-125
-100
-75
-50
-25
0
25
50
75
100
125
Hz
Sinusoidal signal 12 Hz; phase shift
230
degree
1, 00
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
-1,00
.
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
Note
: x-components always have
same
direct
i
on with a nonzero su
m
(except 0 +
0). Therefore this axis is called
real axis
ms
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
0, 75
0, 50
0, 25
0, 00
-0,25
-0,50
-0,75
x
- components at +12 and -12 Hz
-0 ,38
- 0 ,38
+0,32
y
- components at +12 and -12 Hz
-0,32
-125
-100
-75
-50
-25
0
25
50
75
100
125
Hz
Illustration 96:
Symmetrical spectra consisting of x and y components
(x-y-representation)
We must now clarify what information appears at the two outputs of this FFT module. Giving the
amplitude and phase, apart from frequency, is part of every sinusoidal signal. One might therefore suspect
that the amplitude and phase of the positive and negative frequency + and - 12 Hz will appear at the two
outputs. The diagrams give a different result.
A reminder: Illustration 24 links the sinusoidal signal with a rotating pointer. If we pursue the idea further
the rotating pointer can be represented like a vector by x and y components which may change over time.
Following this idea in Illustration 97 an x-y-module is selected in order to visualize the two channels.
Result: two pointers rotating in the opposite direction!
