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

1. Digital-analog converter schemes

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

Read the infographic description below

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

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

R2R DAC (or binary-weighted resistor digital-analog converter) has non-linearity issues due to 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 a pure digital module, which doesn't need to adjust during production. It simplifies manufacturing and decreases the cost of a digital-analog converter device.

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

DSD DAC has 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 a converter's analog filter. Read details >

DAC design comparison in brief

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

The analog filter aim is alias removing of digital to analog conversion. Analog filter has 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 to intermodulations.

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

Read details here >

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

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

Of course, real DACs are more complex devices, than are shown in 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 a better quality of certain DAC type.

Read below how to work these schemes in details.

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

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

Read the details in this article below.

How DAC types sound

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

The author will not consider here record quality and different mixing/mastering issues, that also is a 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 has several stages:

• recording;
• mixing;
• mastering;
• conversion to different formats.

How to musical test samples are produced

In the picture above only some options for producing music test samples are shown.

For some of the test samples, several 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 a difference in recording tools (microphones, microphone pre-amps, analog to digital converters, etc.) and its settings.

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

The main technical problem of DAC type comparison is the various inner working of the devices.

In the picture, the impact of DAC's internal modules to sound quality was shown for various converter types.

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 a single device model.

Another example: a PCM DAC has alias issues of the oversampler, but a competing DSD DAC contains 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 the sound of DAC types as abstract units. But it is possible to compare the sound of real instances of digital-analog converters, despite its inner workings.

General requirements for DAC

In simple, a 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 at elementary resistor DAC:

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

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

Analog filter interpolates intermediate points between times of digital samples.

Voltage in point A (before analog filter) is:

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

where:

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

- R1, R2 are values of resistors.

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

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

A resistor, as a real electronic component, has some value deviation. It causes 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):

The resistors' values define the voltage level before the analog filter.

In 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 the problems of the DAC. To solving the issue, a low sample rate is upsampled and digitally filtered before analog filtering (part B of the picture above). Read the details below.

How analog filter works

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

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

Analog filter is interpolator: math that creates a seamless signal between reference points of digital samples.

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

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

Let's look at the filter bands:

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

Filter bands:

• Passband - filter pass signal thru
• Stopband - the band with maximal suppression
• Transient band - band between pass and stop bands

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

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

When says 'low-frequency filter' it mean filter with passband in low-frequency area.

A real analog filter is not steep and demands a 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 to the non-steep transient band of the analog filter.

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

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

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

Ladder DAC non-linear distortions

When the resistors in the scheme of the r2r ladder DAC is ideal (zero tolerance or zero resistance deviation), voltage before the analog filter is altered linearly for a 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;

Thus:

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.

Thus:

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.

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

DAC non-linear distortions. The level error depends on input PCM code

R2R ladder DAC precision

Warning: Calculations below are intended for approximate estimation only.

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

NS_L = 20 * log₁₀(1/2^N-1).

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

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

Each of bit resistors causes an error in voltage before analog filter following resistor precision.

Voltage error may be estimated by formula:

V_err=V_in*R2/(R_Nbit+R2)-V_in*R2/(R_Nbit*(1+r_err/100%)+R2),

where:

V_err - absolute voltage error for Nth bit;

V_in - input logic voltage value on a bit;

R2 - common DAC resistor connected with the ground;

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

r_err - R_Nbit resistor error in percent.

According to the formula, the most significant impact on absolute voltage error V_err causes resistors of the highest bits.

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Maximal voltage level before analog filter (when all bits in logic '1') is V_in*(1-2^1-N) and may be accepted as equal V_in.

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

To compare with noise, the error is normalized in dB:

V_{err dB} = 20 * log₁₀( V_e / V_in).

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

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

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

As example, bit #14 causes error V_{err dB} -78 dB. It is 32 dB over the noise level.

If we 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 the author would suggest comparing the deviation with the quantization noise level. Because the lowest bit is changed in time.
Thus errors V_{err dB} should cause dB error below -110 dB.

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

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

Bit #14 cause error V_{err 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 the tolerance, resistor value depends on temperature. Temperature is defined by environmental temperature and current that pass through the resistor.

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

Let's consider 24-bit ladder DAC now:

Here we can view values the same for 16-bit ladder DAC. Because the same resistor values are used in the 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

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

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

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

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

For this DAC type resistor value deviation doesn't cause non-linear distortions. It only impacts to total amplitude value of an 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 doesn't contain oversampler and sigma-delta modulator modules with their issues.

DSD DAC

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 has the same issues too.

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

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

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

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

But on the other hand, the 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 be damaged by overload. It does not matter of DSD DAC though.

NOS DAC. Non-oversampling digital-analog converter

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

It is the usual DAC without oversampler.

Read details about NOS DAC >

Conclusions

1. In the general case, R2R DAC is tougher in design and adjusting, than sigma-delta-modulator based PCM DAC.
2. Minimal DSD DAC has 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 doesn't guarantee better/worse quality.

Frequently Asked Questions

What is R2R DAC?

R2R DAC is DAC bases on a resistor matrix, which forms output voltage according to a sample code of digital signal.

As example, a digital audio signal, which comes from a computer.

What are the advantages of R2R ladder DAC?

At first sight, R2R DAC has simpler design advantage. I.e. lesser element lesser distortions.

There is no sigma-delta modulator, oversampler, that can cause ringing.

However, simplicity requires the greatest precision of the components, constructive and montage. It may not (or almost may not) be achieved for home devices. At least, it requires a big time of manufacturing, including adjusting.

And, in ordinary conditions of work, the temperature and the humidity variate parameters of such converter scheme. The variations cause distortions.

How does a R2R ladder works?

R2R DAC contains a resistor matrix, that converts input code (value of sample) to voltage. After it, a sequence of voltage values passed through an analog filter, that smooth output voltage. To be exact, the smoothing is calling as 'interpolation' and filter remove aliases of the 'stair-step' signal.

What is a weighted resistor DAC?

Weighted resistor DAC (or R2R, ladder) is DAC based on a resistor matrix. The matrix contains several resistors.

1 resistor is fed by 1 bit of input code (of a sample of a digital audio signal). The resistors have different electrical resistance (weight).

The resistance value causes different values of the output voltage.

The sequence of the voltage values forms an analog audio signal from the sequence of codes (PCM digital signal) at the input of the DAC.

Author: Yuri Korzunov,

Audiophile Inventory's developer.

April 30, 2020 updated | since February 14-24, 2018

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

References