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R 2R Ladder DAC vs Sigma-Delta PCM DAC vs DSD DAC | Read Now

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yuri korzunovPCM digital analog converter may be based on either resistor matrix (R2R) or sigma-delta modulator (SDM). Last one is most popular. But some people prefer R-2R converters. Also in the article non-oversampling NOS DAC will considered. DSD DAC is alternative of PCM DAC types. Read article about comparison of the digital analog converter types, its advantages and disadvantages by audio software developer Yuri Korzunov.

  1. R2R ladder DAC versus sigma-delta DAC versus DSD DAC
  2. DAC design comparison in brief
  3. Feature comparison: ladder, sigma-delta PCM, DSD DACs
  4. How DAC types sounds
  5. General requirements for DAC
  6. Digital-analog converter schemes
  7. NOS DAC. Non-oversampling digital analog converter
  8. Conclusions


Watch and share: Hi Res Audio [How it works. Sound quality. 7 Myths]

R2R ladder DAC versus sigma-delta PCM DAC versus DSD DAC

R2R ladder DAC vs sigma-delta PCM DAC vs DSD DAC comparison

R2R ladder DAC vs sigma-delta PCM DAC vs DSD DAC comparison

Read the infographic description below


PCM DAC based on sigma delta modulator have 2 key advantages:

  • linearity of input/output voltage characteristic of digital to analog conversion;
  • simplicity of design and producing.

R2R DAC (or binary weighted resistor digital analog converter) have non-linearity issues due available resistor tolerance. The non-linearity cause distortions. Also it can cause audible products by ultrasound, that degrade sound quality.

Sigma delta modulator may be tough in design. But it is pure digital module, which don't need to adjust during production. It simplify manufacturing and decrease cost of digital-analog converter device.

However, R2R PCM DAC don't contains sigma-delta modulator and have no tolerance to overload issue.

DSD DAC have no issues with R2R non-linearity and overload tolerance. DSD recording (original or pre-converted from PCM) may be noise shaped differently. The noise shaping may be more or less optimal for analog filter. Read details >

Watch and share: What is DSD audio? Video explanation



DAC design comparison in brief

DAC type comparison

DAC type comparison

Minimalistic R2R DAC (part A of the picture above) contains resistor matrix (ladder). Each of the matrix resistors have value deviation. It cause non-linearity.

The analog filter aim is alias removing of digital to analog conversion. Analog filter have gradual suppression growth with frequency increasing. And the analog filter can not deep filter all aliases. These aliases can cause audible distortions generated by ultrasound due intermodulations.

Analog filter have minimal suppression in low frequency area. To suppress aliases in the low frequency area, oversampling and digital filtering (more steep than analog one) are used (part B of the picture above).
However, oversampling cause own aliases. And issues are possible with filtering of these aliases.

Read details here >

Non-linearity of the resistor matrix may be solved via digital sigma delta modulator (part C of the picture above). Because such modulator is linear device. But sigma-delta modulator have issues with broken stabulity due overload.

When input digital audio stream is DSD (1-bit sigma-delta modulation) instead PCM, minimalistic DSD DAC contains pair of resistors and analog filter (part D of the picture above).

Of course real DACs are more complex devices, than are shown at the picture. There are matters of power supply quality, temperature stability, deviation of logical level voltage, etc. DAC concepts (A, B, C, D parts at the picture) give potential design abilities only. And they do not guarantee better quality of certain DAC type.

Read below how to work these schemes in details.



Feature comparison: ladder, sigma-delta PCM, DSD DACs

Features Ladder (R2R) PCM DAC Sigma-delta PCM DAC DSD DAC
Way of voltage generation by code Resistor matrix Sigma-delta modulator 1 level circuit
Analog fitering of output signal Yes Yes Yes
Amount of reference voltage values Bit number - 1 1 or more
[if multibit sigma-delta modulator is there]
Linearity issues of digital to analog conversion Non-linearity Linear Linear
Non-linear distortions in analog circuits Yes Yes Yes

When we consider conversion to analog of sigma-delta modulation, 1 level may mean 2 levels actually (positive and negative).

Read details in this article below.



How DAC types sounds

Often author read in the discussions, that some people prefer one DAC type to other. They have practical experience in sound quality of a DAC types.

Author will not consider here record quality and different mixing/mastering issues, that also is matter of DAC-sound estimation. Because it may be technically impossible to achieve full identity of single phonogram copy in different formats.

Audio track production have several stages:

How to musical test samples are produced

How to musical test samples are produced


At the picture above only some options of producing of music test samples are shown.

For some of test samples several of stages may be excluded. Or single master-record (final stuff of music production) may be converted to different formats.

Single acoustic stuff may be recorded in 2 formats at once. There is difference in recording tools (microphones, microphone pre-amps, analog to digital converters, etc.) and its settings.

Therefore, DAC type comparison can includes comparison of audio file converter quality or recording tool difference at least.


Main technical problem of DAC type comparison is various inner workings of the devices.

At the picture DAC type comparison impact of DAC internal modules to sound quality was shown.

There are many variables, that need to be taken into account when digital-analog converters are compared.

As example, in a ladder DACs, resistors with different tolerance may be installed. It can lead to different non-linearity and cause different sound. Even for different items of single device model.

Other example: a PCM DAC have alias issues of oversampler, but a competing DSD DAC have a worse analog filter. It is possible to suggest which one of the digital-analog converters is better sounding? Probably, no.

So it is technically impossible to compare sound of DAC types as abstract units. But it is possible to compare sound of real instances of digital-analog converters, despite its inner workings.



General requirements for DAC

In simple, digital-analog converter should provide:

  • digital value conversion to analog voltage level with given precision,
  • limited level of distortions (in 0 ... 20 kHz frequency range),
  • given magnitude and phase linearity deviations of frequency responses.



Digital-analog converter schemes

Let's look to elementary resistor DAC:

Elementary R2R DAC

Elementary resistor DAC

The scheme contains pair of resistors (R1 and R2) and analog filter. Resistors define the voltage in point A. When digital "0" at input, 0V present at point A. When digital "1" at inpit, voltage, defined by R1 and R2, present at point A.

Also digital "0" may be converted to negative value, to avoid DC bias at analog output. Though there are ways to remove the bias.

Analog filter interpolate intermediate points between times of digital samples.

Voltage in point A (before analog filter) is:

V=[Bit #0 Voltage]/(R1+R2)*R2;


- [Bit #0 Voltage] is logic levels "0" or "1";

- R1, R2 are values of resistors.

Thus voltage precision at point A depend on physical logic level precision and resistors' tolerance.

Resistor tolerance is limit of resistance value deviation (in percents).

Resistor, as real electronic component, have some value deviation. It cause the voltage level deviation and non-linear DAC distortions if there are several bits (read below).


In multibit ladder DAC, additional resistors are added (1 resistor per 1 bit):

R-2R ladder PCM DAC

R2R Ladder PCM DAC

The resistors' values define voltage level before analog filter.

At the picture above (part A) we can see elementary R2R ladder DAC. Analog filtering at low sample rates (44100, 48000 Hz, as example) is one of problems of the DAC. To solving the issue, low sample rate is upsampled and digitally filtered before analog filtering (part B of the picture above). Read details below.



How analog filter works

At the picture DAC analog filter (part A) spectrum before analog filter is shown.

It is spectrum of "stairs", that is drawn on PCM pictures usually.

Analog filter is interpolator: math that create seamless signal between reference poits of digital samples.

Ideal analog filter must cut all frequency range above [sample rate]/2 to restore of original analog spectrum (see the picture below, part C).

DAC analog filter

DAC analog filter


Otherwise, aliases from this frequency range (above [sample rate]/2) can generate audible products due non linear distortions in DAC's electrical circuits (see the picture above, part D).

Let's look to a filter bands:

Filter bands (analog and digital): pass, transient, stop

FIlter bands

Filter bands:

  • Pass band - filter pass signal thru
  • Stop band - band with maximal suppression
  • Transient band - band between pass and stop bands

These band have no exact borders. As rule bands are defined by minimal (for pass band) / maximal (for stop band) allowable filter gain.

Filter gain is output/input level ratio for given frequency.

When says "low frequency filter" it mean filter with pass band in low frequency area.


Real analog filter is not steep and demands wider transient band (between pass and stop bands) comparing digital filters. So it is too difficult to provide steep transient between output frequency ranges: below and above [sample rate]/2 (see picture "DAC analog filter", part C).

Thus audible products of intermodulation distortions can be caused due non-steep transient band of analog filter.

To improve the filtering, upsampling is implemented. It shift original [sample rate]/2 position ([oversampled sample rate]/2) to area of deeper supression of analog filter.

Oversampling and analog filter

Oversampling and analog filter


Also digital filter, implemented in oversampling, may be steeper than analog one.


Steeper oversampled digital filter to better removing of excessive aliases

Steeper oversampled digital filter to better removing of excessive aliases


And steeper digital filter can better remove excessive aliases, than analog filtering.


Alias removing: digital vs analog filter

Alias removing: digital vs analog filter



Ladder DAC non-linear distortions

When the resistors in the scheme of the r2r laddier DAC is ideal (zero tolerance or zero resistance deviation), voltage before analog filter is altered linearly for sequential altering of binary code at DAC input (0000[0], 0001[1], 0010[2], 0011[3], etc.).

But errors in bit-resistor values, cause non-linearity.


Example #1:

Bit voltage for Bit0 is 1 V (Volt);

Bit voltage for Bit1 is 2 V;


Input code 00: 0+0=0 V;

Input code 01: 0+1=1 V;

Input code 10: 2+0=2 V;

Input code 11: 2+1=3 V.

Sequence 0, 1, 2, 3 V is linear.


Example #2:

If Bit0 resistor cause 0.1 V error, it produce 1.1 V instead 1.0 V,

and Bit1 resistor cause -0.2 V error, it produce 1.8 V instead 2.0 V.


Input code 00: 0+0=0 V;

Input code 01: 0+1.1=1.1 V;

Input code 10: 1.8+0=1.8 V;

Input code 11: 1.8+1.1=2.9 V.

Sequence 0, 1.1, 2, 3.1 V is non-linear.

Laddier DAC non-linearity due resistor precision

Laddier DAC non-linearity due resistor precision


In other words, the non-linearity is error altering by input PCM code. Let's see dependency of "Total error" value of several bits by input PCM code at the picture below).


DAC non-linear distortions. Level error depend on input PCM code

DAC non-linear distortions



R2R ladder DAC precision

Warning: Calculations below are intended for approximate estimation only.

If R2R DAC have N bit input, its approximate noise level is:

NSL = 20 * log10(1/2N-1).


For 16-bit DAC the noise level 96 dB is expected.

But really the level about -110 dB due averaging and distributing in Furie transform discrete frequency positions.


Each of bit resistors cause error in voltage before analog filter in accordance with resistor precision.

Voltage error may be estimated by formula:



Verr - absolute voltage error for Nth bit;

Vin - input logic voltage value on bit;

R2 - common DAC resistor connected with ground;

RNbit - resistor in bit circuit (receive the logic voltage);

rerr - RNbit resistor error in percent.

Scheme of ladder DAC bit error measurement

Scheme of ladder DAC bit error measurement


According the formula, most significant impact to absolute voltage error Verr cause resistors of highest bits.

Maximal voltage level before analog filter (when all bits in logic "1") is Vin*(1-21-N) and may be accepted as equal Vin.

It works when bit #[N-2] give 0.5*Vin level and bit #[N-1] change polarity of output voltage.

To compare with noise, error is normalized in dB:

Verr dB = 20 * log10( Ve / Vin ).

In the table, caused error is shown for each of resistors from bit #8 to #14.

16 bit ladder DAC

16-bit ladder DAC. Caused maximal error, dB
Bit resistors' tolerance 0.05%.

Bit numbers begin from 0 and ends 15.
Bit #15 is sign (+/-). It change voltage polarity, but don't have own resistor.
For bit #14 RNbit=R2
Bit number Caused maximal error
Verr dB in dB(Vin)
14 -78
13 -81
12 -85
11 -91
10 -96
09 -102
08 -108

Resistors with 0.05% tolerance is precise enough at modern technology level.

But we can see that 0.05% tolerance cause errors with level values above noise, that was accepted at -110 dB above.

As example, bit #14 cause error Verr dB -78 dB. It is 32 dB over the noise level.

If refer to measurements practice, deviation of generated voltage value should cause errors 3...10 times (not in dB) lesser than lowest bit (#1) value -96 dB. I.e. higher bits (#1 and above) should not mask work of lowest bit (#0).
But author would suggest compare the deviation with quantization noise level. Because the lowest bit is changed in time.
Thus errors Verr dB should cause dB error below -110 dB.

Let's look to voltage error with 0.0005% resistor tolerance:

16-bit ladder DAC. Caused maximal error, dB
Bit resistors' tolerance 0.0005%.

Bit numbers begin from 0 and ends 15.
Bit #15 is sign (+/-). It change voltage polarity, but don't have own resistor.
For bit #14 RNbit=R2
Bit number Caused maximal error
Verr dB in dB(Vin)
14 -118
13 -121
12 -125
11 -131
10 -136
09 -142
08 -148

As author know, resistors with 0.0005% tolerance are most precise at moment of the article writing [1].

Bit #14 cause error Verr dB -118 dB. It is 8 dB below the quantization noise level.

So 16 bit ladder DAC may be implemented on 0.0005% tolerance resistors.

Unfortunately, except tolerance, resistor value depend on temperature. Temperature is defined by environmental temperature and current that pass thru the resistor.

Also bit input voltage is switched by electronic keys. These keys also cause voltage error that depend on temperature too.


Let's consider 24 bit ladder DAC now:

24 bit ladder DAC

24-bit ladder DAC. Caused maximal error, dB
Bit resistors' tolerance 0.05%.

Bit numbers begin from 0 and ends 23.
Bit #23 is sign (+/-). It change voltage polarity, but don't have own resistor.
For bit #22 RNbit=R2
Bit number Caused maximal error
Verr dB in dB(Vin)
22 -78
21 -81
20 -85
19 -91
18 -96
17 -102
16 -108

Here we can view values the same for 16 bit ladder DAC. Because same resistor values are used in highest bits.
However, for 24-bit R2R DAC these errors should be compared with -144...-150 dB quantization noise.



PCM DAC with sigma-delta modulator

Aforementioned ladder DACs have output non-linear distortion issues due resistor value deviation and temperature.

To solve the issue we can reduce number of bit resistors. It allow to build DAC easier way and reduce impact of resistance temperature stability.

Using of intermediate sigma delta modulator is way of the reducing of the resistor number.

PCM signal is oversampled and converted (into digital domain) to sigma delta modulated one. Analog filtration at output remove modulation noise of the sigma-delta modulator. At output audio signal, restored from digital form, is present.

PCM DAC based on sigma-delta modulator

PCM DAC with sigma-delta modulator


For this DAC type resistor value deviation don't cause non linear distortions. It only impact to total amplitude value of analog signal.

However, oversampler with digital filter have alias issue and sigma-delta modulator have issues with overload tolerance.



DSD DAC vs PCM DAC sigma-delta based

DSD DAC don't contains oversampler and sigma-delta modulator modules with their issues.



In DSD DAC resistor value deviation don't cause non-linear distortion. Input voltage modulation can cause non-linear distortions though. But it may be fixed via filtering of power line and other. PCM DAC have same issues too.

Main feature of DSD recording or PCM to DSD conversion is noise shaping (pushing quantization noise energy out of audible range).

Noise shaping define low border of frequency range where DSD-modulation-noise spectrum have significant level growth. At the picture below the "low border" is most left point of "modulation noise" shape at the "frequency" axis.

Sigma-delta modulation and analog filter

Sigma-delta modulation and analog filter

The higher by frequency modulation noise border (at the picture above, part B) provide better DSD noise suppressing for same analog filter.

Because noise is pushed in frequency range, where analog filter have higher suppression.

But on the other hand, higher border can cause lesser tolerance to overload at sigma-delta modulator input. I.e. there is more probability, that sigma-delta modulated stuff will damaged by overload. It is not matter of DSD DAC though.


Article: R 2R Ladder DAC vs Sigma-Delta DAC vs DSD DAC


NOS DAC. Non-oversampling digital analog converter

Non-oversampling DAC is way to rid ringing artefact and other distortions, that caused in digital filtering inside DAC.

It is usual DAC without oversampler.

Read details about NOS DAC >




  1. In general case, R2R DAC is more tough in design and adjusting, than sigma-delta-modulator based PCM DAC.
  2. Minimal DSD DAC have no oversampling/digital filtering and sigma-delta modulator stages. Thus it is easier than any kind of PCM DAC.
  3. Real DACs are more complex devices, than its concepts (A, B, C, D parts at the "DAC type comparison" picture). The concepts give to engineers potential design abilities only. DAC type on its own don't guarantee better/worse quality.


Author: Yuri Korzunov, , February 14-24, 2018,

Audiophile Inventory's developer.


Read also: Power Conditioner. Do You Have Audio Quality Benefits? [Explained] >




  1. Resistor tolerance



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