I know you jitter-meisters like to talk about nothing but jitter but I think there are many more interesting topics in the art and practice of digital audio. One interesting area of exploration is duplicate samples.
What’s wrong with duplicate samples? Two duplicate samples are the equivalent of halving the sample rate for that instant in time. Each time you halve the sample rate you lower the frequency of the sampling noise by an octave making the noise frequency lower than the LPF cut-off frequency.
How do duplicate samples come about? From too few bits in the sample. Consider a pure 60Hz sine wave recorded at –60dB (16 bits @ 44.1K). That signal can be fully encoded in just 7 bits or 128 unique sample values. Each half-wave contains 367 or 368 samples. There are more sample periods than there are unique sample values therefore a number of samples must be duplicated to fill the time. My simulation shows there will be up to 58 duplicates in a row. Adding a 2-bit dither signal will reduce the average number of duplicates to 16 but at the expense of more noise. Recording with 24-bit samples at 192K also helps but still puts out as many as 14 duplicates in a row.
How often do duplicate samples appear in music CDs? More often than I would have guessed. Here are a few histograms I created after analyzing the occurrence of duplicate samples in a variety of commercial audio CDs. The there is a ‘|’ in the center of the plot which denotes ‘zero’. Moving to the right, the values of the samples increase; moving to the left, they decrease. Each ‘-’ indicates samples with a value proportional to the lateral displacement were present in the CD track. If the line of ‘-’ ends before the ‘:’ is reached, it means the track was recorded at a low level. If the line of ‘-’ goes to the ‘:’ it means the track was probably normalized. If a sample with the absolute maximum/minimum value was found, the last ‘-’ is replaced with a ‘+’.
Asterisks rising above the dashes indicate increasing numbers of duplicates at each sample value. Asterisks descending below the dashes indicate longer strings of duplicate samples. I won’t explain the numbers at this time because understanding the histogram is complicated enough.
Track05 is a purist, AAD recording. Note the low recording level and lack of duplicates at the highest and lowest sample values.
Track03 is typical of a modern studio recording. The track was dithered and normalized to the max. Notice the duplicates at the highest and lowest sample values, which is indicative of clipping.
Track01 is typical rock & roll. Notice the extreme clipping.
The final question: Are duplicate samples audible? Yes, but it first takes some education. From the plots I knew there were lots of duplicates but I didn’t know where they occurred in the music. Of course, duplicates associated with clipping occurred at or near the peaks, but what about all the others? To find out I find out, I rerecorded a number of CDs after converting every duplicate sample to zero. The zero samples highlighted the location of the duplicates and, with a little practice, I was able to recognize duplicate samples in unmodified recordings.
The next step was to eliminate the duplicates by modifying the CD tracks in a benign way. After a few tries I think I succeeded. I fine-tuned the algorithm by ear and I think it sounds very good.
The final step will be to implement the algorithm in hardware. I am working on a DAC that will do that and a few other interesting ideas I’ve been working on.
You see, there is more to life in digital audio then jitter.
What’s wrong with duplicate samples? Two duplicate samples are the equivalent of halving the sample rate for that instant in time. Each time you halve the sample rate you lower the frequency of the sampling noise by an octave making the noise frequency lower than the LPF cut-off frequency.
How do duplicate samples come about? From too few bits in the sample. Consider a pure 60Hz sine wave recorded at –60dB (16 bits @ 44.1K). That signal can be fully encoded in just 7 bits or 128 unique sample values. Each half-wave contains 367 or 368 samples. There are more sample periods than there are unique sample values therefore a number of samples must be duplicated to fill the time. My simulation shows there will be up to 58 duplicates in a row. Adding a 2-bit dither signal will reduce the average number of duplicates to 16 but at the expense of more noise. Recording with 24-bit samples at 192K also helps but still puts out as many as 14 duplicates in a row.
How often do duplicate samples appear in music CDs? More often than I would have guessed. Here are a few histograms I created after analyzing the occurrence of duplicate samples in a variety of commercial audio CDs. The there is a ‘|’ in the center of the plot which denotes ‘zero’. Moving to the right, the values of the samples increase; moving to the left, they decrease. Each ‘-’ indicates samples with a value proportional to the lateral displacement were present in the CD track. If the line of ‘-’ ends before the ‘:’ is reached, it means the track was recorded at a low level. If the line of ‘-’ goes to the ‘:’ it means the track was probably normalized. If a sample with the absolute maximum/minimum value was found, the last ‘-’ is replaced with a ‘+’.
Asterisks rising above the dashes indicate increasing numbers of duplicates at each sample value. Asterisks descending below the dashes indicate longer strings of duplicate samples. I won’t explain the numbers at this time because understanding the histogram is complicated enough.
Code:
Track05 5:53.07-0.00(3)
681490/136677/818167 330726/56918/387644 6/12/28970 25.8/5.8
751865/129168/881033 363458/54504/417962 15/15/33079 28.6/3.5
: *** :
: **** :
: ****** :
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: ************ :
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: ---------|-------- :
: ************* :
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: :
Code:
Track03 4:27.20-0.00(3)
80206/9395/89601 38757/4242/42999 7/7/2773 0.0/37.6
79613/11620/91233 38231/5195/43426 8/8/3105 0.0/35.8
: ** :
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: ************** :
: ********************** :
: ***************************** :
: ****************************************** *:
:* ** ******************************************** * *:
:* * ** ************************************************* *:
:+-------------------------------|------------------------------+:
:* * ** ************************************************* *:
: * ** :
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: :
Code:
Track01 5:53.60-1.92(3)
46270/21923/68193 21543/8791/30334 15/12/2056 0.0/36.4
45233/22353/67586 21121/9088/30209 14/12/2029 0.0/38.4
: ** :
:* ** *:
:* ******* *:
:* *************************** *:
:* ***************************************** *:
:* **************************************************** *:
:** ********************************************************** **:
:****************************************************************:
:--------------------------------|-------------------------------:
:****************************************************************:
:* ** *:
: ** *:
: :The final question: Are duplicate samples audible? Yes, but it first takes some education. From the plots I knew there were lots of duplicates but I didn’t know where they occurred in the music. Of course, duplicates associated with clipping occurred at or near the peaks, but what about all the others? To find out I find out, I rerecorded a number of CDs after converting every duplicate sample to zero. The zero samples highlighted the location of the duplicates and, with a little practice, I was able to recognize duplicate samples in unmodified recordings.
The next step was to eliminate the duplicates by modifying the CD tracks in a benign way. After a few tries I think I succeeded. I fine-tuned the algorithm by ear and I think it sounds very good.
The final step will be to implement the algorithm in hardware. I am working on a DAC that will do that and a few other interesting ideas I’ve been working on.
You see, there is more to life in digital audio then jitter.