{"id":4233,"date":"2016-12-15T22:19:39","date_gmt":"2016-12-15T20:19:39","guid":{"rendered":"http:\/\/www.tonmeister.ca\/wordpress\/?p=4233"},"modified":"2018-05-29T07:44:08","modified_gmt":"2018-05-29T05:44:08","slug":"probability-and-death","status":"publish","type":"post","link":"https:\/\/www.tonmeister.ca\/wordpress\/2016\/12\/15\/probability-and-death\/","title":{"rendered":"Probability and Death"},"content":{"rendered":"

Inspired by a conversation with Jamie Angus following my last posting<\/a>, I did some digging into the probability density functions (PDF’s) of a bunch of test tracks that I use for tuning and testing loudspeakers.<\/p>\n

The plots below are the results of this analysis.<\/p>\n

Some explanations, to start…<\/p>\n

A PDF of an audio signal is a measurement of the probability that a given level (or sample value) will happen in a given period of time. In the case of the plots below, I just counted each time every sample value the 16-bit range of possibilities (from -32768 to 32767, if you think in binary – from -1 to +1-2^-15 in steps of 2^-15 if you prefer to think in floating point decimal) occurred in an entire track (usually the full tune). \u00a0That’s plotted below as the gray “curve” in a linear world.<\/p>\n

I converted the linear levels to “instantaneous” (sample-by-sample) dB values (since they’re instantaneous, they’re not in dB FS – but let’s not get into that discussion), but kept the positive and negative polarities of the linear values separate. Those are plotted as the red (negative values, expressed in dB) and black (positive values, expressed in dB). I cheated a little here, since a linear value of “0” isn’t really -96 dB – it’s -infinity dB… but, except for that one value, everything else is plotted correctly.<\/p>\n

When I did these analyses, I noticed lots of sample values in lots of tracks that had no probability of ever occurring. Sometimes, this is just because the track is mastered low in level – so the “upper” sample values are not used. Sometimes, there are “dead values” well inside the range. This likely points to an error in the converter and\/or digital mixing and\/or mastering equipment.<\/p>\n

Finally, I made a plot of the number of “dead” sample values per 128 sample values in 512 blocks (65536\/128 = 512). That’s the red line above the gray one.<\/p>\n

 <\/p>\n

Some other things are noticeable in the plots, but we’ll take those as they come, below…<\/p>\n

Note that I will not reveal the names of the tracks I used, since it’s not my purpose here to make anyone look bad. It’s to look at the differences between different recordings, types, and even equipment… Don’t ask which recordings I used.<\/em><\/p>\n

 <\/p>\n

\"\"<\/p>\n

This plot of an orchestral recording (“orchestral8”) looks fairly normal. As Jamie pointed out in the last posting, the distribution looks to be\u00a0Laplacian. There is a big spike at the “0” mark – due to the silence at the beginning and end of the track. As can be seen, the track peaks around a linear value of about +\/- 0.2 or about -14 dB below full scale. So, the sample values above 0.2 (or below -0.2) are unused. This can be seen in the blue lines (comprised of a blue dot at each sample value) in the top plot, and the red line at 128 (the number of “dead values” per 128 possible values) in the bottom plot.<\/p>\n

 <\/p>\n

\"\"<\/p>\n

The “solo inst3” is very similar in behaviour, even though it’s a completely different recording. This one is of a solo stringed instrument in a fairly reverberant space. Notice that its basic characteristics are very similar to those shown in “orchestra8”.<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

The “voice3” recording is also similar. It’s a little interesting in all three of these recordings to note the transition between levels where there are no dead sample values (around the “0” line) and levels where there are nothing but dead values. In this area (in the “voice3” recording, for example, around +\/- 0.3, there are sample values that are used – but others that are skipped. This is because the track has a reasonably large crest factor (the ratio\u00a0between the peak and the RMS of the track) – in other words, it has noticeable peaks. When the levels peak positively or negatively, there will be some values skipped along the way…<\/p>\n

Although these are very different recordings of different instruments in different spaces on different labels – they all appear to have some characteristics in common.<\/p>\n