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How can it be stated the precision of any measure?
How it is stated the precision of the ADDC measure
How it is stated the precision of the ADDC measure
How the ADDC measure copes with small up to tiny output inaccuracies
How the ADDC measure copes with small up to tiny output inaccuracies
How the ADDC measure copes with small up to tiny output inaccuracies
How the ADDC measure copes with small up to tiny output inaccuracies
How the ADDC measure copes with not small up to huge output inaccuracies
How the ADDC measure copes with not small up to huge output inaccuracies
How the ADDC measure copes with not small up to huge output inaccuracies
How the ADDC measure copes with not small up to huge output inaccuracies

How can it be stated the precision of any measure?
 
 

No matter the theory behind any measure, the only definitive proof of its correctness and precision should be a wide range of real tests.

This said, come to think of it for a moment: how can you assert the exactness of any results got from a real system if you, don’t perfectly knowing its behaviour, are obliged to make some assumptions?

Indeed every measure is aimed at showing how a system behaves, but you cannot state how much that measure is wrong if you don’t have complete knowledge of the system being measured (and if you had such knowledge perhaps you wouldn’t need to measure it). Should this not be enough, to make things worse they may be embedded in the measure results the side effects of the “a priori” assumptions made in order to apply it.

Every measure is aimed at getting new pieces of information but often, if you can’t exactly cope with something, some assumptions are made upon the system (and what it is expected from it) or from the measure.

As an example, have you ever heard something similar to “given a LTI system…”

To avoid this I tried to not take for granted a measure be right given any “a priori” assumptions, as it would be definitively better to know as exactly as possible (I mean mathematically) how much precise it is in absolute terms (otherwise how could it be known when it can be used and when it cannot)?

How it is stated the precision of the ADDC measure
To definitively clear this hurdle you could intentionally inject a spurious and well known signal mucking up the good one and then try to recognize the former making no assumption.
This is the way in which the robustness of the ADDC measure results are verified: you compare what the ADDC measure guesses about the spurious signal against what you already know of it, making no assumptions on anything, but you are the only one who knows about the bad signal, the measure instead knows nothing about it and will do its best only based on what sees.
So doing you can state how much the measure is wrong comparing its results against the real and well known properties of the injected error signal.

This simulations allow to test the ADDC measure in many working conditions, verifying its robustness and its limits. As soon as these tests give results good enough (for your purposes), it is likely the measure will be also able to well state the behaviour of a real system, as the simulations act as a precision database to compare with when measuring a real system. Given even a rough idea of what happens at its output with a first measure, you can then tune it and know the measure limits comparing the real case with the most similar simulation.

To achieve such results it is crucial the precision threshold the ADDC measure accepts as one of its input parameters. This parameter answer to the two apparently simple questions: how much have I to push the measure precision to get precise enough results for my purposes and given that system and that operative conditions? And is there a solution to this issue or am I just asking too much?

Coping with small up to tiny output inaccuracies
In general the results of these tests depend upon the kind of spurious signal you use in your simulations. In the ADDC measure theory a wide range of test cases have been made to develop its precision scenario and in the next table they are shown a few of them, which are referred to a random signal mucking up the output.
In this table the column index is the percentage ratio between the signal output amplitude and the spurious random signal amplitude injected on it.
The row index instead is the precision threshold as a percentage of the output signal amplitude. At the intersection between each row and each column we have the percentage inexactness of the ADDC measure while trying to guess that spurious signal (column index) given that threshold precision (row index). This percentage number describe how much the measure is wrong when guessing that bad signal in those operative conditions, that is the relative percentage error the measure makes in those conditions.
 
Random spurious signal
Columns: random spurious signal level % with respect to the output amplitude
Cell: error % in measuring that spurious signal (column) given that precision threshold (row)
Output level % precision threshold
 0.4 %
 4· 10-2 %
 4· 10-3%
 4· 10-4 %
 4·  10-5 %
 4· 10-6 %
4·  10-7 %
4· 10-3 %
5.47141e-008
3.51723e-008
1.44664e-007
2.40183e-007
2.18817e-006
0.0278063
4.83405
4· 10-4 %
2.33638e-008
5.89073e-009
2.15535e-007
5.31951e-005
1.1772
19.7178
206.085
4· 10-5 %
7.5374e-009
6.20804e-009
1.18779e-008
1.22897e-007
1.63156e-005
0.660232
13.5215
4· 10-6 %
3.87863e-009
9.52046e-009
3.27087e-009
1.11998e-008
2.42773e-007
0.000396092
1.11282
4· 10-7 %
3.01083e-009
4.84306e-009
6.07827e-009
2.53359e-009
5.363e-009
6.01084e-008
0.00018034
4· 10-8 %
2.45579e-009
3.33451e-009
1.03925e-008
3.08852e-008
2.86897e-010
1.57789e-008
1.28236e-007
4· 10-9 %
4.80063e-010
1.45063e-009
2.39229e-009
2.05512e-009
1.20647e-009
4.91333e-009
4.10708e-008
 
It can be seen that the measure precision is really high unless the number expressing the precision threshold is greater than or comparable with the spurious signal amplitude, and this sounds natural.
For instance, the cell in position (6,5) (precision threshold = 4· 10-8 % of the output signal level, spurious signal level = 4·10-5 % of the output signal level) states that the ADDC measure exhibits a relative percentage error equal to 2.86897· 10-10 % in recognizing such spurious signal.
 
Coping with not small up to huge output inaccuracies
 
Below it is shown the ADDC measure inexactness when a not small up to huge sinusoidal spurious signal is injected on the output.
The horizontal axe is the spurious signal amplitude with respect to the output (for example 0.15 means the spurious signal amplitude is 0.15 times the output level).
The vertical axe is the percentage error the measure makes while recognizing that spurious signal.
 
  
 
 

 

It can be seen that the measure inexactness is roughly 2.3% when the spurious signal amplitude reaches the same level of the output, and quickly goes down as soon as the output decreases (for instance it becomes less than 0.07% when the error level value is 0.2 times the output level).

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