Extended algorithms test

(a) Maximum angular error of non extended algorithms (b) Mean angular error of non extended algorithms

0 10 20

LoG 30

40

Maximum angular error [°]

50 60

3x3 70 80

Approximation Sensor array size

5x5 7x7 9x9 11x11 13x13 15x15 Extended 2D-DFT

(c) Maximum angular error of extended algorithms

0 5 10

LoG 15

Mean angular error [°]

20 25 30

3x3 35

Approximation Sensor array size

5x5 7x7 9x9 11x11 13x13 15x15 Extended 2D-DFT

(d) Mean angular error of extended algorithms

Figure 5.37: Comparison of maximum and mean angular errors of non extended and AMM point tracking extended algorithms. A significant improve of angular accuracy is observed by an increase of array size from 3 x 3 to 5 x 5 elements.

Interesting fact is that the angular errors of the 2D-DFT algorithms (extended with field realign-ment) observable in Figure 5.37(a) and 5.37(b) reduces when the array size is increased. This is achieved because the 2D-DFT algorithm evaluates the total field information while with LoG and approximation algorithms exclusively only the field curvature is evaluated. Due to this, for large misalignment effects, the 2D-DFT approach including field realignment feature possibly represents a potential advantage. However, the remained angular error of 2D-DFT algorithm is in this setup still not satisfied.

The direct comparison of angular errors values depending on array size are shown in Table 5.16.

Tracking Array size: 3x3 5x5 7x7 9x9 11x11 13x13 15x15

extension Error []: max max max max max max max

not Ext. 2D-DFT 61.52 62.28 59.97 58.39 57.39 56.94 56.18

activated LoG 61.54 65.39 68.06 69.71 71.01 72.45 73.03

Approx. 61.52 65.29 67.84 69.31 70.37 71.54 71.76

Ext. 2D-DFT 61.52 15.19 8.862 6.380 5.116 4.461 3.849

activated LoG 61.54 15.19 8.863 6.380 5.116 4.461 3.849

Approx. 61.52 15.19 8.862 6.380 5.116 4.461 3.849

Table 5.16: Direct comparison of mean and maximum angular errors of non extended and AMM point tracking extended algorithms.

To get a clear knowledge about the angular error reduction tendency, the reduction of relative maximum error related to the initial one in percentage was calculated and is recorded in Table 5.17.

Array size: 5x5 7x7 9x9 11x11 13x13 15x15

Relative Ext. 2D-DFT 75.31 85.59 89.63 91.68 92.75 93.74

error LoG 75.31 85.59 89.63 91.68 92.75 93.74

reduction [%] Approx. 75.31 85.59 89.63 91.68 92.75 93.74 Table 5.17: Relative reduction of maximal angular error of AMM point tracking extended algorithms in dependence of sensor array size.

With the application of tracking algorithm the maximal angular error reduction in the first step when the array size is increased from 3 x 3 to 5 x 5 is around 75%. In the step afterwards, the reduction of 85% is achieved. The maximum reduction in frame of this setup amounts to 93%.

All the methods show completely the same behavior.

Conclusions: All the investigated algorithms exhibit the same behavior. Their robustness against misalignment effect combined with the tracking algorithm is identical.

With the extension of all algorithm candidates by AMM point tracking algorithm and by applica-tion of an extracapplica-tion mask size of 3 x 3 elements the maximum error caused by worst misalign-ment case within the SOA limitations could be maximally reduced by 93.74 %. However, the remained error amounts to 3.8 and is still not acceptable, what leads to the conclusion, that no acceptable angular error compensation can be achieved with this algorithmic extension and therefore an angular error correction procedure is definitely needed.

Robustness against signal-to-noise ratio variation (subblock 4.1)

Of course, this extension of all algorithms by in decrease of effective area for angle information extraction associated with it plays directly against the noise effects of the signals. The smaller

the extraction mask the worse the impact of noise effects become. In order to prove the appli-cability of the tracking extension the following test was carried out. The total array size within the sensor die area was set to 15 x 15. The SNR was varied in defined limit, from 100 dB to 20 dB. The extraction mask size was varied from 15 x 15 to 3 x 3 elements. No misalignment was applied here in order to get a maximum possible angular error of 0. Since the limits of this extension need to be found in this test setup the z-distance was set to maximum of 5 mm, in order to obtain the worst possible noise effects.

In analogy to 5.7.1, absolutely identical evaluation images were created to obtain a visual com-parison of the noise impacts. In Figure 5.38 the maximum and minimum error associated with the noise impact for the approximation algorithm is shown. The '_err 1 boundary is set visually by scaling, as well as G.5.2.

All algorithms exhibit the same performance in this test. Their evaluated figures are included in Appendix G.7.1. From Figure 5.38 it can be taken that SNR value for the required angular accuracy increases rapidly, if the extraction mask size is reduced. The highest tendency for the required increase of the SNR value is observable between 9 x 9 to 7 x 7, 7 x 7 to 5 x 5 and 5 x 5 to 3 x 3 extraction masks. The tendency of required SNR for the adherence to angular error criteria depending on the array size is in both figures is not linear.

In Table 5.18 the determined limits of SNR for the angular accuracy requirement of'err 1 are represented.

Extraction

mask size: 3x3 5x5 7x7 9x9 11x11 13x13 15x15

^{SNR for 1}

angular Mean error 87 72 62 56 51 46 43

error limit

[dB] Max. error >100 85 76 70 65 60 56

Table 5.18: Limits for an angular accuracy'_err 1 for AMM point tracking extended algorithms in dependence of extraction mask size for an array size of 15 x 15. Results of approximation algorithm.

Additional bar plots were created so that the angular error behind the'err 1 limit can also be evaluated and included in Appendix G.7.1 as well.

(a)

(b)

Figure 5.38: Investigated impact of noise on the angular accuracy of AMM point tracking extended al-gorithms. Results of approximation algorithm. (a) Mean error depending on SNR and extraction mask size. (b) Maximum error depending on SNR and extraction mask size. Note: the maximum limit of color was set to 1 in order to see the'_err 1 boundary clearly. Simulation setup: 15 x 15 array, no misalignment, z-distance 5 mm, sphere encoder magnet.

Conclusions: It will not be practically possible at all to rely on the tracking concept with a extraction mask size of 3 x 3 and array size of 15 x 15. There are more than 100 dB SNR signal quality needed. For the extraction mask size of 5 x 5 elements and an array size of 15 x 15 a SNR value around 85 dB is needed, what is still very ambitious and practically less possible.

The combination with 7 x 7 extraction mask for a 15 x 15 array is applicable. With a decrease of amount of sensor pixels on the die the SNR of the signals for the same extraction mask size will definitely will become better because a higher field curvature capture will automatically be obtained. For an expected SNR value of the signal quality an optimal combination between array size and extraction mask size can be found. The search of the optimal combination can be carried out in the context further work.