Tuesday, June 10, 2014

Recap on DSD information loss theory

In classic 1x DSD delta sigma modulation, canonically one has 1 bit at 64Fs.  Seen from the perspective of 1Fs, 44.1kHz, which itself is a bit more than twice the "standard" human hearing limit of 20kHz (most adults have less than 16kHz), all these bits are equal, regardless of order.  Therefore, the total information at 1Fs is 65--there are 65 possible states, simply the total number of "on" bits from 0-64.  Expressed as "dynamic range" this (64/1) is 36dB before noise shifting.  The dynamic range rises 6dB for each octave lower than 1Fs/2, so at 20kHz it is about 43dB.

Meanwhile, classic Redbook CD 44.1 kHz 16 bit has 65536 possible states at 1Fs--the same 98dB of dynamic range it has at all in band frequencies.  Just by these numbers, this means that standard 44.1k 16 bit has 65536/65  or about 1008 times more information than DSD.  No DSD or Sigma Delta system that I am aware of has more potential information per second in the audio band than Redbook CD.  Higher Resolution PCM systems do have easily calculable more information than Redbook and should be preferred even over Redbook CD, I think (though the complete argument for this--the encoding of frequencies believed too high to hear--is too long for this essay).

Does this matter?  Most of the frequencies in the Redbook vs DSD comparison are not beyond human hearing, and they determine the fine contour of waveforms, even if not hugely present.  Of course, the amount of information present matters!  I'm not saying it can be easily heard, but it is a form of dynamic distortion far larger than those is PCM, and DSD is supposed to be better, a successor to PCM.  So it should not have additional kinds of distortion, albeing this one a kind of "smoothing" distortion that is relatively pleasant…but is information losing.  When information is lost, the results are not good, even when you can't hear it all in one listening, more information makes successive listening session more interesting, and even the same listening session when the information loss itself has a predictable nature.

There are counterarguments.  (a) we are less sensitive to state limitation in the extreme highs, where it is particularly true (ironically maybe because of the "higher bandwidth") that DSD has the least state information.  (I don't buy this argument, I think clusters of high frequencies on a transient basis are found in many places even if they don't show up in averages.)  (b) Even within band, the ordering of the same count of bits matters, even at 1Fs.  (I can't see an easily defended argument for this.  1Fs is 44.1 kHz, more then twice the range of human hearing.  So modulations above 44.1kHz shouldn't be relevant, and that's all I would expect the ordering of bits to change.)

However, being somewhat unsure about (b), I came up with a factor of 16 to describe how much the ordering of bits might matter.  This was the minimum to satisfy the possibility I might hear the benefit of DVD-Audio recording over the sigma delta modulator in my Denon 5900, which is quite good.  So the factor of 16 has been quasi-empirically determined, and the actual adjustment factor may be higher (or nonexistent!).

So by this standard, using my 16x factor, the DSD information loss over Redbook drops to 1/64 instead of 1/1024.  I use this 16x factor below, but bear in mind the raw information loss numbers are far larger and may be more relevant.

Pure 1 bit 10x DSD as used in the new PS Audio gets to 1 / 6.4 times as much information as Redbook.  Quad DSD eels out 1 / 25.6.  But actually most Sigma Delta converters have far better information recovery than canonical DSD anyway, by virtue first of having more bits, then higher speeds also.  Still, however, none has more information than Redbook.  So for example the prehistoric sigma delta Burr Brown 1720's in my Denon 5900 are 3 bits run at 256Fs (quad dad rate, btw).  The 3 bits add 8x information.  This brings it to 1/2 times as much information as redbook, more than 3 times as much as the new PS Audio 10x DAC.  It's obvious that it's easier to get more information with bits than high rates of oversampling.  But strangely the audio world is moving in the opposite direction.

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