In the digital audio world, we are often confused by many terms flying around, and one of it is R2R or Sigma Delta. Today, with my limited experiences and understanding of it technicalities, will attempt to cover some of it and hope that you, our fellows enthusiasts, will find it helpful in your journey.
R2R or Resistor ladder is as ancient as the first primitive flash based conversions that started out since the beginning of digital audio. By using a references voltage to convert binary values input into voltage values outputs as one point of references for the processing, this is called Quantization steps. According to the gold standard that we have known and accept for a while, the Nyquist theory Quantization in a 16 bit depth at twice the maximal frequency that human can observe, which is 22.05 X 2 toward a standard of 44.1Khz. Yet, human hearing is often understood that 20-20Khz is our limits, and using 22.05 as the standard, let’s just say that is a headroom. How so ? Because headroom is needed due to the basis of all things, the variations, or tolerances, nothing is perfect, there has to be room for the variants.
When talking about tolerances and variants, basic electronic components are under the standard of precision’s tolerances. A 1 ohm resistor isn’t exactly 1 ohm, but it could be 0.9 ohms, and or 1.1 ohm. This is 10% precision tolerances. Then another property to look for is Temperature Coefficient. This just means that materials will expand when heated up and contracted when cooled down. So, when given the 1 ohm value, with 10% tolerances, more than often that is speaking under normal room temperature or somewhere around 75F, or ambient temperature, which human is comfortable at working in. When taking this factor into the equations, we will have the precisions to drift away from temperature obviously from cold operational into warmed up.
All of those inaccuracy are “errors”, and when Quantization errors are encountered, the result is the decrease of amplitudes, which in turns is the determining factor of dynamic and micro-dynamic deliveries. There are always ways to overcome such limitations , and let’s just call that being a “cheating”. It ranges from “more progressive errors corrections algorithms” to the more advantages of Over Sampling (OS).
The best R2R should be the resistor with silicon wafer based technology, or IC chip! Not discrete array, because these newer technology can produce resistors in IC package as accurate as 0.005%. This precision tolerances is not the matching tolerances that some makers would misleading and marketing to be 0.0004% precision matching. There is no such references, unless plausibly that someone is claiming that is because matching this way is collectively and selectively picking 1 out of 10 thousands resistors. Even so, there is no proof!
Why OS ? Because simply put, you read a complex book once and try to tell the story , you will have so many errors, but if you read it back 8x before telling, there are less errors. Based on the same theory, instead of stopping at 2X the sampling as Nyquist stated, we would want 4x/8x/16x....etc....the only limitations is the technologies of silicon and chip makers. The most recent top of the line DAC are all capable of 256x, which is basically 8X.
What are all of those multiplication factors coming from ? Nyquist theory stated that 16/44.1 at twice the sampling 2X is the standard. So, 16 bits by 2 factors and we have 32. When taking into the saying from AKM datasheet that is capable of 256x OverSampling, that is 256/Nyquist bit depth standard, which is 256/32 , then we have 8X. Why do they use the number 256x instead of 8X ? Because 8X is the factors OverSampling further than Nyquist standard, and taking 256x in speaking term is basically that is how far away from Nyquist theory the maker AKM can achieve with AK4497EQ
So by implementing OS after the resistor arrays to OverSampling the values achieved, even with errors, the OverSampling provide a more accurate predictions of frequencies based on the mathematical models, and also correct the errors while doing so.
R2R is typical having a darker, warmer, more blooming textures or soften textures, and higher noise floor, despite it being Over Sampling or Non OS. However, with OS, R2R can achieve further resolutions, dynamic range
Sigma Delta This is basically the term of taking the first and compared to the second take of the same thing, as I said, reading the book the second time and comparing to the first time....etc...This is OverSampling, and with digital filters, interpolations, algorithms, we simply call it modulators. Each modulators have it own sound signatures , and that is uniquely related to the way the calculations and errors corrections work as a whole. This results in AKM with Velvet engine being different than ESS Sabres
Sigma Delta at it nature is OverSampling, which is superior technically to Non OverSampling as discussed above. Then there are different ways to achieve it as much as R2R, it can be built Discretely or on a Wafer based Chips. The modern technologies and investments are toward Wafer based Chips, and that is for good reasons as stated, the resistors and silicon can achieve 0.005% precision tolerances, at a smaller package, and computer with robotic controlled environments. It is a fool proof model. The only thing left is the competitions between engineering pedigrees from modulator, and other areas such as Maximum Dynamic range etc....
So, the more accurate, the higher the dynamic range, the lower the floor noises, and the higher the resolutions. Sigma Delta usually is known as having a digital signatures as cold, non engaging, snappy transients, great clarity and great soundstage.
Finally, when it come to personal preferences, there is neither wrong nor right, just know that there are only 2 topologies for processing, either Nyquist standard or NOS, then OS. Just do not worry about it being R2R or Sigma Delta. Because these are confusing. Just as long as you are enjoying something, that is all that mattered. Then, when it comes to technicalities, Sigma Delta is the superior method in many aspects, so much as that in Studio, all recordings are being done on a very high sampling rate, which then will be downsampled to Nyquist, and or being sold as High-Res.
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