C2 accuracy analisys

As said in the C2 results, we will study the accuracy of the C2 error reporting of the drives, under normal conditions of error correction (see the definition in the C2 results page).
Thus we need to filter the unsignificant data, and to process the remaining ones in order to compute the C2 undetection rate versus the error rate. We will use Microsoft Excel for this purpose.
You'll find how to analyse the data in Excel into the appendix 1

There are two possible kinds of graphs. I first used one-parameter graphs (the horizontal axis being defined by the record number of the Excel file) before finding the possibility of creating two-parameters graphs (a column plotted against another). However, the averaging process that we will use in order to reduce the statistical noise is easier in a one parameter study. So we will process the data with one parameter, and switch to two parameters only at the end.

Memorex DVD Maxx 1648 firmware GWH2 DVD ROM

Errors per second
Undetected errors per second x100
Undetected errors per second/errors per second x3,000,000
Errors, undetected errors, and undetection ratio vs time in seconds

The high yellow peaks screw the scaling. What are they ?
Let's search an example : line 3376, 8 errors, 1 undetected.
This comes from the fact that the numbers are too small to provide a statistic sample. Let's sort the data on the number of errors.

Errors per second
Undetected errors per second x100
Undetected errors per second/errors per second x3,000,000
Errors, undetected errors, and undetection ratio recorded each second, sorted on errors Seconds with zero errors have been discarded.

All the peaks in the undetection ratio come from recordings on too small statistic samples. We should eliminate them, but what if the undetection rate doesn't fall down with the error rate ? We must first analyse them as a whole. Let's cut the data into two parts. Here are the data with an error rate superior to 55 per second.

Errors per second
Undetected errors per second x100
Undetected errors per second/errors per second x3,000,000
Errors, undetected errors, and undetection ratio recorded each second, sorted on errors, for error rate >=55

Here, we can see that the undetection ratio variations decrease as the errors increase. That's not surprising, granted that the number of undetected errors at the beginning of the graph is often 0, 1, or 2. Let's check the other part of the data, under 55 errors per second.
Any second showing 0 errors show 0 undetetced errors.
For the rest, between 1 and 54, we have a total of 1164 errors, and 15 undetected ones. That would give a yellow plot around 38000 on the y-axis, rather to the left, since 14 undetections are 1400 on the y-Axis for the pink curve. This fits perfectly.

Thus the peaks in the undetection ratio are definitely statisical, and don't come from a defect of the drive.

Since the yellow curve varies very much, we must change the method of analyse. The problem comes from the fact that most recordings have a too small number of undetected errors. We must sum the recordings, the more recording we will sum, the more accurate will be the analysis, but the fewer the X-axis resolution will be. There are 1464 recording with an error rate superior to 54. Let's sum the cells by groups of 14, so that we keep an horizontal resolution of 100 on the graph, but increase the stability of the curves.

Errors per second averaged on 14 seconds.
Undetected errors per second x100 averaged on 14 seconds
Undetected errors per 14 seconds/errors per 14 seconds x3,000,000 for each second
Errors, undetected errors, and undetection ratio recorded each second, averaged on 14 seconds, sorted on errors, for error rate >=55 per second.

This graph is 330 pixels wide, so the horizontal accuracy is about 3 pixels.

We've seen that its variatons on the left part of the graph are statistical. In order to get an accurate ratio for lower error rates, we must collect about 5,000x14 = 70,000 errors (because the yellow curve becomes stable when the blue one is around 5,000). But an artifact in the data comes into play. Look at this line

We're in the low error rates : there are 7 errors only (40322+4-40319). But 40,322 are reported ! How is it possible ?
The time gives the answer. We are 79.6 minutes into the wav. That is in plain digital silence. It is the end of the CD, that is very damaged, thus featuring many errors (about 40,000 per second), but the interpolation of these errors generates only null samples, since the drive just holds the last good value, that is always zero. Therefore the analyse program doesn't detect them as errors, since all samples are perfectly correct... by chance !
Let's discard these data from the analysis, since we want to analyse data read for low error rates only.
Note that the effect of digital silence in the analysis is very little, since we are comparing the ratio between undetected errors and the total number of errors. The problem of interpolation in the digital silence makes us assume that there are very few errors, but the undetected errors taken into account are necessarily part of them. A lack of C2 into an interpolated error is considered as a normal successful reading, and the presence of C2 as C2 overdetection, that we don't analyse, since it doesn't affect the secureness of the extracton.
Thus the only effect is translating the good undetected errors ratio from its actual error rate to very low error rates. but we especially discarded this kind of report for the low error rate measurment, and it doesn't affect the ratio computed for high error rates.
Let's manually select clusters of records featuring at least 70,000 errors each, until we cover all the range where the graph is unaccurate.

These results can be manually typed into the undetection ratio column in order to replace the data for low error rates. The new graph has four yellow dots on the left replacing the original data, but with much more accuracy.

Errors per second averaged on 14 seconds.
Undetected errors per second x100 averaged on 14 seconds
Undetected errors/errors x3,000,000 for each second averaged on 70000 errors (error rate <1498) or 14 seconds (>=1498).
Errors, undetected errors, and undetection ratio recorded each second, averaged on 14 seconds or 70000 errors, sorted on errors

Now we can plot directly the C2 accuracy against the error rate.

C2 accuracy in %, averaged on 70000 errors under 1498 errors per seconds, and on 14 seconds beyond, vs error rate in mono samples

Sony DDU1621 DVD ROM firmware S1.6

Since Analyse.exe 1.2 was bugged, we'll use the data returned by version 1.4 (second Sony graph in the C2 results page). As we previously said in that page, we will discard all the high error rate part, because, the CD being too damaged, some very long burst errors appeared which turned the extracted wav file unusable, while the C2 reporting stopped working.

Study of the C2 anomalies

Errors per second
Undetected errors/errors x 300,000 for each second
Errors and undetected errors/errors ratio recorded each second, sorted on ratio, for error rate >0

The point where the pink curve starts to rise is the point where the long burst errors appear, that is where the drive is supposed to loose the groove. From this point, the undetection rate grows progressively, because the data are collected for whole seconds of music. Therefore at the beginning, the burst errors being a few, they contribute little to the total undetection, because they fill only a little part of the 88200 samples collected each second. Then, as their number increase progressively, the number of undetected errors grows with them, rather than with the random error rate, because, when burst errors are a significant part of the extracted data, nearly all the undetections come from them..
The blue dots are the numbers of errors. They are clearly divided into two clusters. The cluster on the right stands for a high error rate, and match high undetection ratios. It is the burst errors. The one at the bottom stands for low error rates, and match low undetection ratios. This is the part we are interested in. We can already see that the error rates increases to the left, thus the undetection ratio decreases as the error rate increase, like with the Memorex. There are blue dots at the very bottom for all low undetection ratios, because at low error rates, the statistical samples are little, and the ratio oscillates randomly, covering all low values.
There are some blue dots on the left of the right cluster. Especially two of them, that stands for a high error rate, but a low undetection ratio. Maybe they are worth studying. According to Excel, one of them is the recordings for 78'12".

Here are the waves :

Extracted wave (up) and its differences from the original between 78'12" and 78'13"
The burst error visible in the wave is about 1700 samples long.
The data are 43,533 errors, 480 undetected. According to the graph, for this error rate, the undetection ratio is usually bigger.

Looking in the wave, we can see some entire seconds of data without any burst error in them. At 78'13", for example. Since there is no burst error, the undetection should be very lower. The data gives 44,430 errors, 217 undetected. Thus in fact, it is a blue dot that is on the complete left of the graph, at about the same level. What about the following second ? 63,614 errors, 15,202 undetections, about 4,000 samples of burst errors.
Here, the undetections are bigger than the burst errors !

Therefore the idea of a bug in the firmware that would be related to the burst errors doesn't stand anymore. Once the error rate is too big, the C2 reporting doesn't work properly anymore.

C2 accuracy

C2 accuracy in %, averaged on 70,000 errors under 753 errors per seconds, and on 14 seconds beyond, over error rate in mono samples

Close up of the above

Teac CD 540E firmware 1.0a CD ROM

C2 accuracy in %, averaged on 70,000 errors under 1770 errors per seconds, and on 14 seconds beyond, over error rate in mono samples

Teac CD 540E firmware 3.0a CD ROM

C2 accuracy in %, averaged on 200,000 errors under 2114 errors per seconds, and on 14 seconds beyond, over error rate in mono samples

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