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How To Calculate Audio System Noise

orchardaudio

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I am working on an article and looking for some feedback. Thanks.
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Product specifications often list the signal-to-noise ratio, sometimes written as "SNR" or "S/N," but what does it mean? While the math behind SNR is technical, the concept is not, and this value can impact a system's overall sound quality.

SNR compares the level of a signal to the level of noise. In other words, it compares the ratio between the relevant (wanted) and the irrelevant (unwanted) information. It is most often expressed as a measurement of decibels (dB). Higher numbers generally mean a better sound quality, since there is more useful information (the signal) than there is unwanted data (the noise).

A decibel is a logarithmic ratio in which every 20 dB represents a factor of 10. For example, going from 80dB to 100dB means increasing by 10 times. As an example, if you have an amplifier with a gain of 20dB that means it multiplies its input signal by a factor of 10, one (1) volt on the input becomes 10V on the output.

1650220349636.png


I created the table to the left to show the most common dB ratios.

As an example when you are comparing two products with a signal-to-noise ratio (SNR) of 120 and 100 dB. The 120dB product has 10 times less noise than the one with 100dB spec.

For other dB values and ratios, this is a very good calculator to use: http://www.sengpielaudio.com/calculator-db.htm

Noise is additive that is why it is important to understand the ratios to help you figure out the total noise in your system.
























Total System Noise
Let's take a look at a typical system that has 3 components, DAC, preamplifier, and power amplifier. For this example, let's say all 3 components have an SNR of 120db. However, just this information is not enough we also need to know the gain of each stage and the output voltage for each device at which the SNR specification is given. For this example let's assume the below values:
DAC - Output Voltage 4VRMS
Pre-amp - Output Voltage 4VRMS, Gain -10dB
Power Amp - Output Voltage 34.64VRMS (150W into 8Ω), Gain 26dB

To figure out the total system noise, we will start with the DAC and work our way down. The DAC has an output voltage of 4V and an SNR of 120dB. This means that its total output noise is 4uV (0.000004V) or 4V / 1,000,000. Similarly, the pre-amp has a total noise of 10uV and the amplifier has a total noise of 28.28uV.

Now we need to take the gain of each device into account. The gain of any device applies to both signal and noise. In our example, the noise from the DAC is amplified by the pre-amp and then again by the amp. The noise from the DAC, pre-amp, and amplifier are uncorrelated, this means that the addition of the two noise sources has to be done using the square root of the sum of the squares (SRSS) method.
Taking it step-by-step:
Noise at output of pre-amp = 4uV (self-induced)
Noise at output of preamp from DAC = 4uV * -6dB
Noise at output of preamp from DAC = 4uV * 0.5 (-6dB is a ratio of 1 to 2, use calculator referenced above)
Noise at output of preamp from DAC = 2uV
Total noise at output of preamp = sqrt (4^2 + 2^2) = sqrt(16 + 4) = sqrt(20) = 4.47uV
Noise at output of power amp = 34.64uV (self-induced)
Noise at output of power amp from pre-amp and DAC = 4.47uV * 26db
Noise at output of power amp from pre-amp and DAC = 4.47uV * 20 (26dB is a ratio of 20 to 1)
Noise at output of power amp from pre-amp and DAC = 89.4uV
Total noise at output of power amp = sqrt (34.64^2 + 89.4^2) = sqrt(1200 + 7992) = sqrt(9,192) = 95.87uV

The system's SNR can now be figured out as that is the ratio of the total noise to the final output voltage.
SNR = 34.64V / 95.87uV
SNR = 34.64V / 0.00009587V = 382,184
SNR = 111.2dB

As a comparison let's take a look at PecanPi Streamer + Starkrimson Mono Amp as a system. In this case, the PecanPi Streamer functions as both DAC and pre-amp

PecanPi Streamer – Output Voltage 5VRMS, SNR 130dB
Starkrimson Monoblock Amp – Output Voltage 34.64VRMS (150W into 8Ω), SNR 121dB, Gain 16.8dB

Taking it step by step:
Total noise at output of streamer = 5V / 3,160,000 (130dB) = 1.58uV
Noise at output of amp = 31uV (self-induced)
Noise at output of amp from streamer = 1.58uV * 16.8dB
Noise at output of amp from streamer = 10.92uV
Total noise at output of amp = sqrt (31^2 + 10.92^2) = sqrt(961 + 119.2) = sqrt(1080.2) = 32.87uV

The system’s SNR can now be figured out as that is the ratio of the total noise to the final output voltage.
SNR = 34.64V / 32.87uV
SNR = 34.64V / 0.00003287V = 1,053,848
SNR = 120.5dB

Summary

Since the gain of a device multiplies both signal and noise, devices with high gains will have a significant effect on total system performance. Keep the number of gain stages to a minimum as each gain stage adds noise and multiplies the noise from the previous stage.
I have created a calculator for two, three, and four device systems. It is at this link.
 
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ex audiophile

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I applaud your effort but you lost me at

"It is a unit for expressing the ratio of the magnitudes of two electric voltages or currents or analogous acoustic quantities equal to 20 times the common logarithm of the voltage or current ratio"

So keep in mind that you'll be writing to a very small niche group of experts who already understand this stuff. If you're trying to reach the broader enthusiast audience I would encourage some simplification, err, a lot of simplification!
cheers
 

MCH

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Hi Orchard,
I don't know to what audience your explanation is aimed but i would guess that at "newbie enthusiasts". As a newbie that learned these concepts in these pages not long ago, i hope you find my feedback useful.
Your explanation is not the easiest to understand, perhaps you could use two lines to explain what signal to noise ratio is and its implications in audio instead of jumping directly to the concept of db.
There are a couple of ^ missing and where it should read xx^2, reads xx2
You should explain also the concept of attenuation and its effect in SNR, as i guess people dont listen to music all the time at max volume, and specifically digital vs analog attenuation.
Correct me if i am wrong, and remember i am a newbie, but if you are using a preamp between your dac and power amp, the result is not going to be the same if your volume control is analogue than if it is digital. What are the implications in your calculations?
Additionally, and related to this. Your second system is outputting 36 volts and your first only 28 volts, so to listen at the same volume you will have to turn down the volume of the second system. What are the implications in the SNR so that we are comparing apples to apples?

I hope it helps and thank you for writing this.
 
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Lambda

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DAC – Output Voltage 4VRMS
it would be way easier to follow along and calculate if you stay in the logarithmic domain and use dBV

lets say the DAC has -120dBv noise now you go in an preamp with 15dB gain and also -120dBV self noise
-120dBV + 15dB gain -105dB

-105dB and -120dB in sum are -104,8dBv
 
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orchardaudio

orchardaudio

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Hi Orchard,
I don't know to what audience your explanation is aimed but i would guess that at "newbie enthusiasts". As a newbie that learned these concepts in these pages not long ago, i hope you find my feedback useful.
Your explanation is not the easiest to understand, perhaps you could use two lines to explain what signal to noise ratio is and its implications in audio instead of jumping directly to the concept of db.
There are a couple of ^ missing and where it should read xx^2, reads xx2
You should explain also the concept of attenuation and its effect in SNR, as i guess people dont listen to music all the time at max volume, and specifically digital vs analog attenuation.
Correct me if i am wrong, and remember i am a newbie, but if you are using a preamp between your dac and power amp, the result is not going to be the same if your volume control is analogue than if it is digital. What are the implications in your calculations?
Additionally, and related to this. Your second system is outputting 36 volts and your first only 28 volts, so to listen at the same volume you will have to turn down the volume of the second system. What are the implications in the SNR so that we are comparing apples to apples?

I hope it helps and thank you for writing this.

Thank you. I am saving the digital vs analog volume control stuff for another article. I did however update the article so both systems end up with the same 150W (34.64V output).
 
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MCH

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Thank you. I am saving the digital vs analog volume control stuff for another article. I did however update the article so both systems end up with the same 150W (34.64V output).
I think there is one extra 0 in your last snr calculation
Edit: no, actually should read 37.8 instead of 378
 

NTK

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I think the numbers @orchardaudio presented are a little bit "unfair". The Orchard Audio system of the PecanPi Streamer and Starkrimson amp was carefully gain matched, whereas the system with the DAC/pre/power amp was not.

The PecanPi/Starkrimson system has a total max gain of 16.8 dB. The rated output of the PecanPi is 5 Vrms. With 16.8 dB gain, the max voltage output to the speaker is 34.6 Vrms, which gives a maximum power of 299 W into 4 ohms.

On the other hand, the max combined gain of the DAC/pre/power amp system was 12 + 26 = 38 dB. We can easily remove the pre-amp and the DAC will still be able to drive the power amp to well beyond clipping (4 Vrms with 26 dB gain = 79.8 Vrms, which is 1592 W into 4 ohms).

If the power amp has a volume pot, we can passively attenuate its input to reduce its effective gain to 18.8 dB. The noise at the speaker terminals calculation in this case will be (thermal noise from the vol pot is neglected):

Contribution from DAC = 4 uV * 10^(18.8/20) = 4 * 8.7 = 69.5 uV
Contribution from power amp = 34.64 uV

Total = sqrt(69.5^2 + 34.64^2) = 77.65 uV

Final SNR = 20 log10(77.65*10^-6 / 34.64) = 113 dB.

Now the SNR of 113 dB for the DAC/power amp combo does not look nearly as bad compared to the PecanPi/Starkrimson combo of 120 dB. And with the DAC/power amp combo, I have the freedom to drive it into clipping. In the studies by Geddes and Lee, and Voishvillo, for amps that behave well under clipping, the audible effects, if any, were rather benign (contrary to popular belief).

References:
 
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orchardaudio

orchardaudio

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I think the numbers @orchardaudio presented are a little bit "unfair". The Orchard Audio system of the PecanPi Streamer and Starkrimson amp was carefully gain matched, whereas the system with the DAC/pre/power amp was not.

I agree and at the same I believe what I have show also represents a real world scenario which only makes things worse as the volume has to be put way down.

My amps specifically have lower gain than the so called "standard" 26dB for this exact reason.
 

restorer-john

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I think the numbers @orchardaudio presented are a little bit "unfair".

Orchard Audio only posts in either his own product promotion threads, or ones that would serve to further his own interests.

Unfortunately, this thinly-veiled advertisement masquerading as an 'article' is just par for the course.

His 'analysis' highly flawed and completely off the mark with typical voltages. Let's talk about why, using his examples:

  • We'll take the D/A output at 4V RMS as gospel with a residual of 4uV. That's double the standard of 2V, but we'll assume XLR 4V.
  • The preamp may have a residual of 10uV at its rated output of 10V RMS with a gain of 12dB. It is hardly typical for a normal HiFi preamplifier as they have typical rated outputs of 1.0-2.0V, but can often swing voltages well above 10V if need be.
  • A 150W @ 8R amplifier with a 26dB gain and an S/N of 120dB would have a residual of 34.64uV.
A normal power amplifier with such a gain would have a sensitivity of between 1-2V for full rated power. Let's say 2.0V (as his example has ~3dB less gain than most commercial power amps) which is very common. There is no situation where 10V will be input into such an amplifier.

The preamplifier will be attentuating the D/A's output by 50% to drive the power amp to full rated power. The preamp's voltage gain will actually be -6dB, not 12dB. An 18dB difference. The volume control will be positioned right down at the low end. Preamplifiers also rate their residual noise at the full volume (wide open with shorted input) position. Now, depending on the topology, that noise will be attenuated somewhat, not at all, or massively attenuated depending on the design.

What @orchardaudio clearly isn't aware of, is that many preamplifiers use 4 gang volume pots which attenutate both input and the signal to the output buffer/line stage, in order to significantly attenuate the residual noise as the volume/gain is reduced.

Here's a block diagram of one of my preamplifiers which, incidentally has a <1.5uV residual noise rating and a 105dB S/N (wide open):
1650240910867.png
 

SIY

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Things I would add:

1. The importance of specifying bandwidth.
2. The difference between current and voltage noise.
3. The way of adding correlated vs. uncorrelated noise.
4. The difference between noise and noise floor.
 

DonH56

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Perhaps a note explaining how dB differs between volts and watts as well...
 

HammerSandwich

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And an example of what this all means in the end. For example, 96uV is -89.4dBW, which would give 0.6dB SPL at 1M with "90dB sensitivity" speakers.
 

restorer-john

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And an example of what this all means in the end.

Most traditional systems (from the source, pre and power stages) operated with what we would think these days as having too much gain, and consquently, a higher noise floor than we have come to want in 2022.

That was purely to do with the requirement to deal with line sources with outputs from 100-150mV through to 4V (now) and still be able to allow for a decent range on the volume, without it being scrunched up at one end.

Now we have guys working out their entire gain 'staging' from end to end based on 0dBFS (2/4V). That's also unrealistic and robs them of a whole lot of usable power when they are playing anything other than a 0dBFS test tone. Music isn't like that.

We have people strapping a D/A straight into a low gain power amp (hypex/purifi etc) where the power amp will likely never be able to achieve rated power swings on any normal content. They are selling themselves short of all the power they have paid for.

A system needs plenty of gain to account for low level recordings. Not so much that residual becomes a problem, but enough to give you a decent, well controlled range with plenty in reserve for the inevitable CD or digital track recorded back when 6 or 8dB below full bit level was regarded as a good recording. I have plenty of excellent CDs from the early 1980s where the abolute level was around 10dB down from 0dBFS. Someone with a low gain chain isn't going to be able to play tracks like that at a high level with just a D/A and matched low gain amp.
 
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HammerSandwich

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I can't argue with that, though I believe some alternatives are equally valid. Simple examples:

Especially for the PC-only user, it's trivial to normalize digital files to 0dBFS or whatever limit you prefer. Sure, you still want plenty of gain, but it needn't be in the preamp.

And with cheap watts available, I feel it's totally reasonable to buy an NC502 or similar, then plan your sources so the power amp simply cannot clip. You see it as wasting power that's already paid for. I see it as buying enough headroom to sacrifice 1dB (~80W here, but audibly minor) while ensuring the system's always clean. If it's okay to ensure headroom while recording, why not on playback?
 
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