# R2R vs String vs Delta Sigma

I'm reading some conflicting literature that suggests R2R has higher "resolution" than Delta Sigma and some literature that says just the opposite. This is both confusing and frustrating.

Can someone offer a more lay explanation of this? These are reputable manufactures and I doubt they're posting erroneous data sheets/tech notes.

Furthermore,I'm mostly interested DACs in the context of audio, which is why I started looking into this, so perhaps there are different performance metrics that would/would not be relevant in the 20-20K spectrum and change how one might evaluate one vs another.

Finally, i'm not wrapping my head around R2R and string differences.

• Delta-sigma has a variable resolution as opposed to the fixed resolution of pretty much everything else. The catch is delta-sigmas require really high oversampling rates to get meaningful resolution (I think it was 4x oversampling per bit or something). So technically speaking both your sources are right, but under different circumstances. Plus it's impractical to build r2r/string dacs for 24 bits as you'd need a ridiculously long resistor string.
– Sam
Jan 1 '17 at 7:14
• @Sam yeah, ridiculously long and at the business end you'd need 0.00001% tolerance resistors. Jan 1 '17 at 7:40
• @Sam, that 4x oversampling per added bit is not about $\Delta\Sigma$ per se. Otherwise 3 MHz or 6 MHz sample rate would be far insufficient for audio ADC or DAC. $\Delta\Sigma$ also employs feedback and noise shaping which, in addition to the oversampling, increases resolution in the baseband. Jan 2 '17 at 6:21

R2R was the original technology. You have to add more stages for more resolution, and the matching of the MSB to the LSB over a wide range is ciritical and performance limiting. They tend to give you 8 to 16 bits resolution. They have O(log(N)) resistors, where N is the number of levels they can resolve.

Resistor string tries to get round the matching problem by using all the same resistors, for integration of monolithic DACs. As they are monolithic, they can afford to integrate the 1000s of resistors required. They have O(N) resistors, many more, so are mostly found in the range 6 to 10 or (rarely) 12 bits.

There is a hybrid of the above two types, which typically decodes the top 4 bits to switch from a string of 16 equal value resistors, then uses R-2R for the lower significance bits. It reduces the resistor ratio on the part by a factor of 16, which improves the matching.

Sigma delta uses time and code, not resistors for resolution. Any number of bits of resolution can be targetted, so their performance is noise, drift, linearity limited. Resolution usually starts at 16 bits, though fewer is set for high speed operation, and goes up through 20 to 24, and a 32 bit part has just been announced.

Sigma delta does not have perfect linearity. Second order effects like voltage non-linearity of output impedance of current sources, systematic code-dependent differences in the timing of a transition, and other subtle things like those can mess up the linearity and spurious-free dynamic range in the 20 bit region, check the SFDR specs for any converter you intend to use.

• Don't R-2R ladders use 2N resistors - hence the name? Feb 25 '18 at 22:46
• @ScottSeidman They use 2N resistors if you define N to be the number of bits, but I defined N in my answer text to be the number of levels, so, no. The reason they're named R-2R DACs is that all the resistors are of value R and 2R. Feb 26 '18 at 7:23
• Obviously, R-2R refers to the values of resistors in the ladder -- I just haven't wrapped my head around referring to the number of levels instead of width of the word. I've just never thought about the problem that way. Feb 26 '18 at 16:13

You have to separate between the physical resolution and the effective resolution. The physical resolution is the number of voltage (or some other physical quantity) levels that can be produced. The effective resolution is determined by the signal-to-noise-and-distortion ratio (SINAD). The SINAD can be directly converted to effective number of bits (ENOB) using a simple formula.

A string DAC, for example, is perfectly monotonic, so each level is unique and the levels always increase. This means that they can have a high physical resolution. However, such DACs tend to have non-uniform steps (the difference between each level), and therefore distortion is generated. This distortion will compromise the ENOB. A Delta-Sigma modulator seeks to get around this problem by using a single bit, a switch, but switching it very rapidly and using a low-pass filter to average the output. By doing so you can get a fairly high ENOB in a certain frequency band. More modern Delta-Sigma DACs have multiple output levels (so-called multi-bit Delta-Sigma), and uses dynamic element matching (DEM) to remove the distortion due to non-uniform step-sizes.

The best I have measured was a 24-bit word-width Asahi Kasei Microdevices audio (multi-bit Delta-Sigma) DAC that produced an ENOB slightly in excess of 16 bit. The best hybrid R-2R and string DAC I have measured was a 20-bit word-width Analog Devices DAC that had an ENOB slightly in excess of 15 bit.

From above . . . "I'm reading some conflicting literature . . . "

Writing has a magical quality; man naturally assumes anything written is true.

“Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense.” . . . Buddha

Translation for our purposes . . . your ears are the only worthwhile sound "measurement instrument."

"Calibrate" your ears by attending live performances, symphonies, jazz, operas, etc.

Absolutely the most musical DAC chips ever made; the Burr Brown PCM 63.  The PCM 63 is the pinnacle of audio digital to analog converter engineering.

The PCM 63 combines all the advantages of a conventional DAC (excellent full scale performance, high signal-to-noise ratio and ease of use) with superior low-level performance, especially around bipolar zero,where the DAC output should change from positive to negative or vise versa. Two DACs are combined in a complementary arrangement to produce an extremely linear output. Inside are two 20 bit R2R DACs with a common laser trimmed resistor network.

DACs utilizing Laser trimmed resistor networks are VERY expensive to manufacture. The MBAs and accountants decided a new DAC architecture was needed. After all, the important thing is profit, not excellent engineering or mindless musicality. Note the MBAs and accountants have a 1960s vintage transistor radio on their desk that "sounds fine for listening to their favorite talk show." The solution? Throw a million transistors at the pesky DAC problem. Thus the bit stream or noise shaping DAC was born. The very word NOISE in a DAC description should be the clue that MASH DACs are non musical.

The Burr Brown PCM54HP R2R DAC with laser trimmed resistor network is far more musical than any noise shaping DAC. In fact, the musical dynamics are very similar to the PCM 63.

Yes, they have great specifications; they sound horrible. Indeed, the engineers who designed the PCM 63 must have been disgusted with the MASH DAC chips.

BSEE NCSU + 24 hours EE graduate school including two graduate courses acoustics . . . acoustic radiation, acoustic instrumentation, psychoacoustics, etc.

References:

Burr Brown PCM63 Data sheet

Fundamentals of Acoustics by Lawrence E. Kinsler & Austin R. Frey 2nd edition . . . Chapter 11 Hearing and Speech

Digital Integrated Electronics by Taub and Schilling

• Buddha to you too. Feb 24 '18 at 7:17
• This isn't head-fi, here. And OP didn't even specifically mentioned audio usage.
– dim
Feb 24 '18 at 7:37
• You've given no actual technical basis for your claims. Feb 24 '18 at 20:25
• There is no "Thread start". This is not a forum.
– pipe
Feb 24 '18 at 21:18
• The reason I wanted to iterate that it is not a forum is because you seem to treat it as such, and for me it is a very important distinction, and the reason I don't like traditional forums. On Stack Exchange we are looking for well researched and focused answers. You seem to be looking for an argument.
– pipe
Feb 25 '18 at 19:27