# Digital Audio Converters (DACs) Fundamentals

#### amirm

Staff Member
CFO (Chief Fun Officer)
Author: our resident expert, DonH50

The purpose of this thread is to provide a quick introduction to digital-to-analog converters (DACs), the magical things that turn digital bits into analog sound. Previous threads have discussed sampling theory, aliasing, and jitter. Now we’ll get down to the hardware and take a look at some basic DAC architectures.

The two criteria most often used to describe a DAC are its resolution (number of bits) and sampling rate (in samples per second, S/s, or perhaps thousands of them, kS/s). If we think about producing an analog output both of these are important. The resolution determines the dynamic range (in dB) of our DAC, and sampling speed determines how high a signal can be output. As discussed in those earlier threads, resolution sets an upper limit on signal-to-noise ratio (SNR) and spurious-free dynamic range (SFDR, the difference between the signal and highest spur). For an ideal (perfect) DAC, we can find:

SNR = 6.021N + 1.76 dB; and,
SFDR ~ 9N; where N = the number of bits.

While not terribly difficult to prove, I shan’t delve into the math here. For that, and the definitions of numerous other specifications (and how they are tested), pick up a copy of the IEEE’s Standard 1241 for ADCs. Plugging in, for a 16-bit DAC, SNR ~ 98 dB and SFDR ~ 144 dB. Figure 1 shows the output in the first Nyquist band from a perfect 16-bit DAC with a 1 kHz input signal sampled at 44.1 kHz. The SFDR (noise floor) is “only” about 130 dB because I did not use an infinite number of data points (samples) in the simulation (only 65,536). The signal-to-noise-and-distortion ration, SINAD, is essentially ideal, so it’s close enough (and I do not have infinite time to wait for a simulation of infinite points to finish!) SINAD is the same as SNR for an ideal DAC, and is used to calculate actual performance for the rest of this article (matching the IEEE Standard).

For sampling rate (frequency) Fs the maximum input signal frequency to an ADC is strictly < Fs/2 to prevent aliasing (folding of the signal around Fs/2 to a lower frequency; see the sampling thread for more about why). It is important to note that, while a DAC cannot be driven by a signal >= Fs/2, it can produce output signals higher than that. In fact, the signals from Fs/2 will be replicated at all multiples of Fs/2, producing a wide range of output signals. It is probably easiest to think of this as the opposite of aliasing in an ADC; at the output of a DAC, images occur at multiples of Fs/2. There are of course other less desirable outputs, also broadband, such as glitches, ringing, distortion, clock feed through, and other spurious products. For now, just remember everything equal to and above Fs/2 is squashed back into the audio baseband (d.c. to Fs/2) by an ADC, while a DAC generates a series of images of the baseband multiplied up by Fs/2.

Figure 2 shows what the output images look like from the simulation above. Note how the signal is replicated (imaged) around each Fs/2 segment (the even multiples are mirror-imaged so we see signals just below and just above multiples of Fs/2, i.e. at M*Fs/2 +/- 1 kHz where M = 0, 1, 2,…). Whether an ADC or a DAC, aliasing/imaging means frequency information is lost – there’s no way to tell the intended frequency, input or output, after it’s sampled and aliased. However, RF systems take advantage of those DAC images by using bandpass filter to select an upper image and eliminate a mixer.

Note that the amplitude rolls off in the images, and there are notches (nulls, zero inflection points) at each multiple of the sampling frequency (44.1 kHz). It’s in the math which I’m trying to minimize, so trust me – it’s correct. Technically, it follows a sinc (sinX/X) envelope. Furthermore, it’s easy to show that at the Nyquist frequency (Fs/2) the output is actually down -3.54 dB. We don’t measure this roll-off in commercial audio DACs because peaking filters (or other techniques not discussed here) are used to flatten the response. Analog output filters are used to roll off any signals and noise above about 20 kHz so you will not see these images at the output of a commercial audio DAC. Not without some serious test equipment, anyway…

Most people here probably understand how a conventional Nyquist DAC functions, at least to a hand-waving level. A 16-bit DAC resolves 2^16 (65536) levels, each of which is generated by the DAC’s internal circuits. For a 1 Vpp output, that means the least-significant bit (lsb) is only ~15 uV. Any errors in the DAC’s levels will cause distortion at the output. With a fully unary 16-bit DAC, using 65,536 individual (unit) cells to create the levels, each cell need only have error < 50 %. However, such a large number of cells requires a lot of area and thus is expensive to make. That many cells also present a significant load at the output, reducing bandwidth. In contrast, a binary DAC needs only 16 cells, with weighted values [2^15, 2^14,… 2^0]. While this seems much simpler, the most-significant bit (MSB) must match the lower bits to the 16-bit level (0.00153 %) to prevent errors when switching from lsbs to the MSB. Achieving such high precision is very challenging, requiring calibration and trimming that is again expensive and difficult to maintain over time, temperature, power supply variations, etc.

Typical real-world DACs are segmented into unary and binary sections. For instance, a 16-bit DAC with the upper six bits unary and remainder binary requires only 1/2^6 (1/64) the precision of a binary design in the upper bits, or error < 0.1 %. This is still quite challenging but much easier to achieve and maintain than 0.0015 %, and provides a compromise between performance and complexity. There are 64 MSB cells with a weight of 1024, and then 10 binary lsbs bits weighted [1, 2, 4,…512]. When all the lsbs are ON, they sum to 1023, thus the 10 lsbs fill in all the steps between the MSBs. The net number of steps is 64*1024 = 65,536 as expected for a 16-bit DAC (2^16 = 65,536).

Errors have been discussed before but it’s time to touch upon them again. Jitter is discussed in other threads so we’ll only look at amplitude errors now. Figure 3 again shows the 16-bit unary DAC, but this time 1-lsb standard deviation random (normal distribution) errors have been added to the thresholds. Note that the noise floor looks the same but is higher, and SINAD has degraded from 98 dB to 87 dB. For each additional increase in noise by 1 lsb (standard deviation) we’ll lose another 6 dB in SINAD.

An interesting experiment is to add a slight (1-lsb peak) bow to the DAC thresholds. This models a gradient with the endpoints (min/max DAC values) having zero error and the center code having a 1-lsb error with a smooth (parabolic, x^2) curve from the ends to the center. This is a very slight error, but because it is not random it generates a significant distortion term, in this case second harmonic distortion (2HD) The frequency response in Figure 4 uses the same parameters as in Figure 1. The noise floor is as before for the ideal DAC, but notice the 2HD spur is only 96 dB down, or about 30 dB above the noise floor. This is well below what we can hear, of course, but highlights how even very small errors can be a problem if they are correlated (deterministic) rather than random. The 2HD spur will rise an additional 6 dB for each 1-lsb increase in error (e.g. a 2-lsb bow will introduce a 90 dB 2HD spur).

A segmented DAC may have higher distortion because segmentation introduces patterns (correlation). This is the trade for reduced complexity (and lower power, etc.) For example, the 6/10-bit segmented DAC has 2^6 = 64 upper (MSB) cells each weighted by 1024. In between, the 10 lsbs fill in the 1023 steps between the upper cells, creating a continuous range of steps from 0 to 65,535. However, since the codes occur in segments between MSB steps, those 1024 lsb steps are repeated (64 times) over the DAC’s full-scale range. Adding 1-lsb errors creates a repetitive pattern, leading to correlated errors and thus higher distortion spurs. Since there are 64 segments, we expect a series of spurs, not just a single spur such as we saw with a simple bow error. And, that’s just what Figure 5 shows. The individual errors are still random, still 1-lsb standard deviation, but because the segments are repeated a series of spurs results. The largest is only 83 dB down; still in audible, but again it’s clear that just a few lsbs of error can create significant distortion in the DAC’s output. Furthermore, the combination of higher noise floor and spurious products has reduced SINAD to 80 dB, equivalent to about a 13-bit DAC (we have lost three bits of resolution).

One thing not discussed is the output filter. Ideally, assuming a 20 kHz output bandwidth, then any spurious products must be suppressed to below the DAC’s noise floor, or at least below the SINAD (i.e. by 98 dB for a 16-bit DAC). Looking at the distortion plots one might think this isn’t terribly challenging as long as the DAC is pretty linear. However, there are those pesky images to consider. Remember, any output is imaged (or “mirrored”) around Fs/2. This means a 20 kHz output signal has an image at about 24 kHz for a 44.1 kS/s DAC. While unlikely in practice, designers must assume that users may in fact output a full-scale 20 kHz signal, requiring that 24 kHz image to be suppressed by more than 98 dB. That takes a very, very steep (high-order) filter. Unfortunately, very high-order filters tend to have high phase shift, ringing, amplitude ripples, and other artifacts at frequencies well below their cut-off, i.e. well into the audio band. This is the oft-discussed filtering problem non-oversampled (NOS) DACs face.

The next thread will explore oversampling and DAC architectures that take advantage of oversampling to improve their performance and simplify their filter requirements.

#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
Bringing this up again in response to earlier questions about conventional DAC architectures. This is one of the threads Amir copied over. It hits upon some of the issues with conventional DACs and what sort of errors can be introduced. It is long and boring but does have pictures to break up the textual babbling. I had thought to generate a thread on conventional DACs but (a) I've been busy and (b) this one does a pretty good job albeit may be hard to follow at first read.

HTH - Don

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#### Cosmik

##### Major Contributor
This is the oft-discussed filtering problem non-oversampled (NOS) DACs face.
So they believe in the theory of digital audio, but they don't believe in the the theory of digital audio..?

#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
So they believe in the theory of digital audio, but they don't believe in the the theory of digital audio..?

Not sure I understand? Image filters are not part of the sampling theorem, nor counter it, but are required in practice. Thay are placed after the DAC and can add artifacts depending upon their design. (Note these are not the FIR filters often discussed that are placed before the DAC itself, or as part of it in the case of some oversampled designs.) Many of the same issues with filters occur on the ADC side where you need antialiasing filters before quantization.

#### garbulky

##### Major Contributor
So if I'm reading this right a filter can induce phase shift, amplitude ripples. What standard measurements can be used to measure or prove that. Do you have an example?

Reason I ask is for the Schiit DACs their claim to fame is a new filter which is supposed to help phase shift and be more time accurate. I was wondering if there was a way to test to prove or disprove this claim.

#### RayDunzl

##### Grand Contributor
Central Scrutinizer
I was wondering if there was a way to test to prove or disprove this claim.

I can play a sweep tone, with or without phase modification (via external DSP) and examine the output of my DAC.

With no modification the reported phase is a flat line, except at the extremes, which I might attribute to the 25ft analog cable between DAC and PC.

With modification the phase line reflects the change.

Wanna see it?

OP

#### amirm

Staff Member
CFO (Chief Fun Officer)
Reason I ask is for the Schiit DACs their claim to fame is a new filter which is supposed to help phase shift and be more time accurate. I was wondering if there was a way to test to prove or disprove this claim.
The amplitude changes show up in my frequency response measurements.

#### garbulky

##### Major Contributor
I can play a sweep tone, with or without phase modification (via external DSP) and examine the output of my DAC.

With no modification the reported phase is a flat line, except at the extremes, which I might attribute to the 25ft analog cable between DAC and PC.

With modification the phase line reflects the change.

Wanna see it?
Yes!! Thank you. Is there a way to quantify this result? Also, is there any audible threshsholds to that that you know?

#### garbulky

##### Major Contributor
The amplitude changes show up in my frequency response measurements.
So like when it goes down at the high frequency extremes that's the ripple?

#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
Ringing is visible in most time-domain plots (look at the impulse response) of delta-sigma converters.

Ripple looks like ripples on a pond. The roll-off at HF is normal and is what the filter is designed to do.

Filter theory and practice is beyond the scope of this thread so it may be best to start a new thread for that. This one is more focused on the basics of DACs and some of the errors that contribute to nonlinearity. Generally speaking higher-order filters roll off faster so better reject (suppress) HF images but exhibit more phase shift and ringing in the time domain. All life is a compromise...

#### Jakob1863

##### Addicted to Fun and Learning
Seems to be a misunderstanding.
Don56 just explained that in reality a DAC - that does not incorporate any oversampling/upsampling digital filter - would use (at least in the old days) a high order (very steep) analog filter at the ouptut as a reconstruction filter.
In the first generation of CD-Players, that were not using the Philips oversampling approach, were equipped with high order Chebyshev filters that introduced phase distortions and asymmetrical impulse response (which by itself might have been not so bad at all - considering what todays filter try to achieve sometimes) .
But that was based on the demand to get a reasonable flat passband up to 20 Khz and still a reasonable attenuation at around 24 kHz.

A "modern" NOS-Dac approach might ease some of the demands and could therefore avoid some of the drawbacks of high order analog filters.

#### RayDunzl

##### Grand Contributor
Central Scrutinizer
Yes!! Thank you. Is there a way to quantify this result?

REW calls it "degrees". See the right hand scale below.

The software compares what/when it sent vs what/when it receives. I haven't found fault with its analysis yet.

Top line, frequency response, bottom line, phase. REW swept sine wave digits -> switch -> Behringer DEQ2496 -> miniDSP OpenDRC-DI ->Benchmark DAC2 -> (digits -> analog) Krell KCT Zone 2 output -> 25 foot cheezecable -> PC sound card - (analog -> digits) -> REW software analysis.

Flat signal (uncalibrated level), flat (within reason and conditions) phase:

Put in an IIR filter creating a moderately broad dip at 1kHz, causes phase change around the filtered frequency:

-9dB dip with a Q of 6 at 1kHz requested.

This is amateur work, so take that into account.

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#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
Phase is measured in degrees (or radians). A full cycle is 360 degrees.

Actually, any non-oversampled DAC (new or old) requires an anti-image filter. Oversampling moves the images up in frequency so they are easier to filter without affecting the audio frequency range.

#### Cosmik

##### Major Contributor
Not sure I understand?.
I'm saying that some people go so far as to trust the theory of digital audio as far as the DAC but then claim not to trust those same techniques to create the filter.

#### RayDunzl

##### Grand Contributor
Central Scrutinizer
Actually, any non-oversampled DAC (new or old) requires an anti-image filter.

I've inherited a Pro-Ject dac box s fl which claims no oversampling and no filter... (I think)

#### garbulky

##### Major Contributor
Thank you gentleman for the info. I assume the suggestion to take it to another thread was directed at me. I'm not familiar enough with what counts as dac fundamentals and what doesn't and so belongs elsewhere. So I'll try somewhere else.

#### captain paranoia

##### Active Member
What standard measurements can be used to measure or prove that.

You don't need measurements. You just need a filter design program. High order (steep brick wall) filters do that. The proof is in the maths...

But, if you really must, inject a signal into the input of the reconstruction filter, and measure the phase and amplitude response.

#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
I've inherited a Pro-Ject dac box s fl which claims no oversampling and no filter... (I think)

If you can live with the images you do not need a filter, but most of the DACs I've seen without filters exhibit images. That feeds extra ultrasonic energy to the system, which may or not filter it before it hits your tweeters. Some designs used an output transformer that also provided some filtering. There are also glitches, fast transients, that are generated at each step and those too can contribute ultrasonic energy. The frequency plots in the first post clearly show the images. I should have added plot with a tone right near Nyquist up around 20 kHz and you'd see a spur at ~24 kHz of almost the same amplitude. That image spur is the reason for sharp image filters.

#### DonH56

##### Master Contributor
Technical Expert
Forum Donor
Thank you gentleman for the info. I assume the suggestion to take it to another thread was directed at me. I'm not familiar enough with what counts as dac fundamentals and what doesn't and so belongs elsewhere. So I'll try somewhere else.

No, you might have brought it up, but many jumped on, including myself. It is a related subject but filter design is pretty vast all on its on. The problem is trying to keep the threads in this part of the forum a bit more focused so they serve as a reference for the future. When they grow to ten or twenty pages of quasi- or unrelated content they lose value as people no longer want to wade through all the "other" stuff even if it is relevant. Some of these articles assume at least some technical background so it is hard to grasp starting from ground zero. Be like me trying to understand a medical article, even an introductory one; too many assumptions and terms that would be beyond me.

#### captain paranoia

##### Active Member
I've inherited a Pro-Ject dac box s fl which claims no oversampling and no filter... (I think)

"High quality D/A converter without oversampling and digital filter"

My emphasis. There's no way they will not have put in an anti-alias/reconstruction filter. But it's an analogue filter, not digital.

The objection (by some) to digital filters is those used after the upsampling, in order to further ease the job of the analogue anti-alias/reconstruction filter.

A pure upsampler just inserts zeros between the samples (e.g. 4x upsample would be [sample1, 0, 0, 0, sample2, 0, 0, 0, etc]) and, by the magic of sampling theory, the aliases are still moved out to the new Fs. But you can apply a digital filter to the output of the upsampler, to 'smooth' the output of the upsampler.

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