6.4Example of practical implementation
Assuming 8 sweeps are to be taken for each Vibrated Point, and harmonic lines 2, 3 and 5 are to be removed :
For harmonic lines 2:
As a result, = 180° removes harmonic line2.
For harmonic line 3 :
= 90° removes harmonic line 3.
For harmonic line 5 :
= 45° removes harmonic line 5.
Combinations of 8 successive sweeps in pairs, resulting in the removal of harmonics through the stacking process.
6.5Applying the principle
The harmonics can be removed by adding two signals with opposite phase, but also by adding N signals shifted by 360°/N.
Example: 6 sweeps shifted by 60°
|
Phase of harmonic before correlation |
|
Phase of harmonic after correlation |
Vector |
||||||||||||
Harmonic |
Sweep |
|
Sweep |
sum |
||||||||||||
N |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
2 |
3 |
4 |
5 |
6 |
|
||
Fundamental |
0 |
60 |
120 |
180 |
240 |
300 |
|
0 |
0 |
0 |
0 |
0 |
0 |
6 |
||
2 |
0 |
120 |
240 |
0 |
120 |
240 |
|
0 |
60 |
120 |
-180 |
-120 |
-60 |
0 |
||
3 |
0 |
180 |
0 |
180 |
0 |
180 |
|
0 |
120 |
-120 |
0 |
-240 |
-120 |
0 |
||
4 |
0 |
240 |
120 |
0 |
240 |
120 |
|
0 |
180 |
0 |
-180 |
0 |
-180 |
0 |
||
5 |
0 |
300 |
240 |
180 |
120 |
60 |
|
0 |
240 |
120 |
0 |
-120 |
-240 |
0 |
||
6 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
-60 |
-120 |
-180 |
-240 |
-300 |
0 |
||
7 |
0 |
60 |
120 |
180 |
240 |
300 |
|
0 |
0 |
0 |
0 |
0 |
0 |
6 |
||
8 |
0 |
120 |
240 |
0 |
120 |
240 |
|
0 |
60 |
120 |
-180 |
-120 |
-60 |
0 |
||
9 |
0 |
180 |
0 |
180 |
0 |
180 |
|
0 |
120 |
-120 |
0 |
-240 |
-120 |
0 |
||
10 |
0 |
240 |
120 |
0 |
240 |
120 |
|
0 |
180 |
0 |
-180 |
0 |
-180 |
0 |
||
11 |
0 |
300 |
240 |
180 |
120 |
60 |
|
0 |
240 |
120 |
0 |
-120 |
-240 |
0 |
||
12 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
-60 |
-120 |
-180 |
-240 |
-300 |
0 |
||
13 |
0 |
60 |
120 |
180 |
240 |
300 |
|
0 |
0 |
0 |
0 |
0 |
0 |
6 |
||
We can see that after correlation the primary (first harmonic) is in phase.
The terms in the second harmonic cancel since there are three sets of wavelets each 180° out of phase.
The terms in the 3rd harmonic cancel since the first, second and third sweeps are 120 degrees out of phase and have vector sum of zero.
Fourth harmonic: three sets 180° out of phase.
Fifth harmonic: same as third.
Sixth harmonic: six wavelets each out of phase by 60 degrees.
You can see that all terms up to and including the sixth harmonic are zero, while the seventh harmonic does not cancel.