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The Signal Chain: How do noise and distortion propagate through my system?

DonH56

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This article is to explore how noise and distortion change through our system, and how different components can influence the final output.

<Note 1: This article assumes some knowledge of performance terms, what frequency response and harmonics are, and so forth. There are other articles, many of which are linked in my signature, that can provide the background for these terms.>

<Note 2: This article is very long. There are not any good shortcuts to understanding the problem and complexity of signal chain analyses. Trying to reduce years of education and practice into a few posts on the ‘net is challenging. I struggled for a while to decide what program(s) were appropriate – ADS, SPICE, Mathcad, Matlab, etc. – but chose Python since I need to improve my Python skills. This took several days to write as I was going through the Python/NumPy/SciPy learning curve along the way, then more days to create the examples and write the text. The advantage is that changes can be made quickly to vary different parameters, and the program can be used by anyone willing to download Python (free open-source program) and learn the language. I am NOT a Python expert.>

<Note 3: I used fairly simple models for distortion and noise. This is not for a specific design but rather a general introduction into distortion and noise changes through a signal chain. Distortion is modeled using a tanh() function appropriate for a bipolar transistor differential pair’s exponential gain characteristics. A MOSFET pair would provide square-law behavior, a “softer” curve. The tanh() function provides only odd-order harmonics so a small offset adds even-order distortion since perfect matching (and thus a perfect tanh() transfer function) is impossible in the real world. Noise is modeled as a Gaussian noise source typical of real circuits. Distortion and noise levels are adjusted independently. This is much simpler than the real world but again is just an introductory example. All voltages are peak with 1 Vpk being full-scale in the model and gains applied appropriately through the chain.>


A common question is “If I have DAC A, preamp B, and amp C, where should I set the volume and gain controls for the best performance?” Usually this results in some handwaving and a few recommendations based upon (hopefully) similar systems because we do not have all the required information for an exact answer. That information includes distortion, noise, and gain details of each block in the signal chain as well as the listener’s preferences (loudness, frequency response) and sensitivity to things like noise and distortion. Many blocks in the chain are internal and undefined to the customer, and others may have just a single set of specifications that do not completely describe the block (component). An example signal chain is shown below.

1650835720720.png


Figure 1: Example Signal Chain​

This example shows a source (DAC), preamp, and power amp. The preamp and power amp may be combined in an integrated amp or receiver (or AVR). The preamp has input and output gain blocks (Ain, Aout) with a volume control (attenuator, G1) between. The power amp has an input volume (gain) control (which is not always user-controlled) and final amplifier (Amp). Each of these signal blocks has its own transfer function that includes noise and distortion. Generally all we as consumers know are the specs at the output of each component, limited to a few specific test cases. Analyzing the complete chain is a lot of guesswork, sometimes educated, other times not so much.

Common performance metrics include THD, SNR, and SINAD (THD+N). Total harmonic distortion, THD, is the ratio of the harmonic distortion to the signal itself and varies with the signal (larger signals are more distorted). By convention the sum of the first ten harmonics are used to calculate THD (so, “total” = 10). There are other types of distortion such as intermodulation distortion (IMD) that happens when two different signals are distorted to create non-harmonic terms but those are not included in this simplified model. For a discussion on THD and IMD, see this thread https://www.audiosciencereview.com/...armonic-and-intermodulation-distortion.25436/ The equations relating THD and IMD are this post: https://www.audiosciencereview.com/...odulation-distortion.25436/page-2#post-878301 Signal-to-noise ratio, SNR, does not include harmonic distortion so really is just signal to noise. There are some caveats when measuring THD and SNR, however…

The first catch is that there will always be a little noise at the harmonic frequencies, and there is no easy way to exclude that bit of noise. (There are mathematical schemes that let instruments exclude most of the noise.) Distortion is very low when the signal level is very small, so at low levels noise tends to dominate the THD measurement. Another catch is that SNR includes non-harmonic distortion like power supply noise, clock noise, crosstalk, and so forth. Again, there is no easy way to exclude these terms, so “noise” in the SNR measurement includes everything but the fundamental signal and its harmonics. The signal-to-noise-and-distortion ratio, SINAD, includes THD and noise (THD+N) – it is the ratio of the signal to everything else. This article only considers harmonic distortion and random noise from the blocks; no other error sources are included. In the real world, the nature of the noise and distortion also affects the sound, along with those other sources not included in this simplified analysis.

I first created a model of a single block including noise and distortion as independent parameters, and a set of plots to help see the results. This also makes it easier to see how distortion and noise affect each other in the measurements. The distortion model is a tanh() function typical of a differential amplifier using bipolar transistors (this varies with device and amplifier topology; I used what I am most familiar with). The added noise is normal Gaussian noise typical of most circuits and is uncorrelated (not related to) the signal. While not strictly true, the model assumes noise is a constant and distortion increases with signal level. This is a reasonable if simplified approximation of how most real circuits operate.

First is a plot showing only distortion; there is no noise added to the signal. For this model, target THD was 100 dB at full scale input (Vin) around 1V. You can see how THD rises smoothly until the amplifier begins to clip just before 1 V. No special low-level distortion such as crossover distortion is included in the model. The SNR is set by the mathematical computational limits of the program (Python) and computer (Windows PC) and is basically a flat line at the limits of the mathematical noise floor until the signal clips. Since there is effectively no noise, SINAD is the same curve as THD (though changed in polarity since THD is usually expressed as a negative number and SNR/SINAD as positive).

1650835797396.png

Figure 2: Distortion Sweep without Noise​

The next plots show noise alone with no added distortion, and without clipping. The SNR target was 120 dB at full scale. At low signal levels, the noise is a large fraction of the signal, so SNR is low. As the signal level (volume) increases, the noise stays the same, so SNR rises. Since there is no distortion, SINAD follows the SNR curve. Notice THD is not 0%; this is because there is a little noise in each harmonic bin (the noise is “everywhere” and so a little bit ends up at each harmonic frequency) so the THD curve looks like the SNR curve (in the opposite direction since the sign is reversed). Because THD only looks at ten harmonic frequencies, and SNR looks at all the other frequencies, THD is lower (better) than SNR.
1650835826068.png

Figure 3: Noise Sweep without Distortion​

Here are the plots when both harmonic distortion and noise are included. This is more interesting and similar to what real measurements often show. At low signal levels, distortion is very low, and noise dominates the measurement so the little bit of noise in the harmonic bins (at the harmonic distortion frequencies 2…10 times the signal frequency) leads to rising THD. Note that the THD level is still lower than the total noise, since only ten frequencies are included in the THD plot so there is less energy than the total noise overall all other frequencies. At around 40 mV in this example, distortion begins to exceed noise, and the THD curve begins to reflect the actual distortion of the signal and not just noise. SNR peaks around 120 dB as intended, and THD around -100 dB, at full scale output just before the signal clips. SINAD follows the SNR curve until THD begins to exceed the noise level, at which point the SINAD curve bends down a little as distortion begins to dominate. Hard clipping sets in above full-scale output and everything “crashes”.
1650835850698.png

Figure 4: Distortion + Noise Sweep​

Finally, this first section ends with a couple of figures showing the signal behind the plots above. First is for Vin = 0.5 V, below clipping and near maximum (best) SINAD. The effective sampling rate is >1 MS/s (sampling frequency is 1.024 MHz) and bit resolution is “infinite” (limited only by the computer). A 1 kHz signal is applied. The upper plots show the entire acquisition (measurement) time, and a single cycle of the output signal (Vo). The bottom plots show the complete FFT spectrum of 512 kHz and restricted to 25 kHz to show the audio band. The noise floor is essentially constant and enough to create 120 dB SNR at full scale, and only a small bit of 2nd- and 3rd-harmonic distortion terms are visible. For this input, THD = -107.8 dB, SNR = 114.0 dB, and SINAD = 106.9 dB. When comparing values in dB, a difference of 10 dB between SNR and THD changes the SINAD by about 1 dB. If both THD and SNR are equal, the resulting SINAD will be about 3 dB worse than the individual numbers.
1650835869556.png

Figure 5: Output Plots for Vin = 0.5 V (unclipped)​

Below is the same set of plots for Vin = 1.26 V, past the 1 V clipping point. Notice how the top and bottom of Vo are clipped (squared off), and now a “spray” of harmonics are visible in the frequency spectrum (FFT). At this level, THD = -21.1 dB, SNR = 45.0 dB, and SINAD = 21.1 dB.
1650835882505.png

Figure 6: Output Plots for Vin = 1.26 V (clipped)​

Hopefully this provides some insight into how noise and distortion will influence the output performance, and how they are related. At very low signal levels noise tends to determine the SINAD, and at high signal levels distortion (THD) dominates. There is an implicit trade that carries through the signal chain: for the highest SNR, use the highest signal level; for the lowest distortion, use the smallest signal level. Finding the optimum signal level through a cascade (chain) of blocks requires knowing the noise and distortion performance of each block as well as the listener’s desired maximum output level.
 
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DonH56

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Next consider the signal chain shown previously and redrawn below using Gn for each gain (or loss) term through the chain. The gain is the ratio of output voltage to input voltage, Vo/Vin, for each block in the chain. Volume controls in this example are passive attenuators G2 and G4 that reduce the signal to the next stage.

First, consider how noise and distortion propagate through the chain to generate the output signal. Noise is uncorrelated (unrelated) to the signal and each block generates its own noise, uncorrelated to any other block, that “adds up” along the way. Since the noise is random, we cannot just add the noise, but rather combine it in root-sum-square (RSS) fashion. We must also take gain into account since the preceding stage’s noise is amplified (or attenuated) at each step in the chain.
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Figure 7: Signal Chain​

The DAC’s noise is affected by every block along the chain. The preamp’s output only sees the power amplifier so it’s output noise has fewer terms on the way to the speakers. We assume that passive attenuators (volume controls) G2 and G4 add no noise or distortion of their own (not strictly true, even for passive controls, but simplifies the analysis). Then we can generate the output noise Nout from all the active sources (XXn) in the signal chain as:
  • Nout = sqrt[(DACn*G1*G2*G3*G4*G5)^2 + (G1n*G2*G3*G4*G5)^2 + (G3n*G4*G4)^2 + G5n^2]
The DAC’s output noise, DACn, is multiplied by the gain through the entire signal chain, squared, and added to the total. The power amplifier’s output noise, G5n, appears directly at the output so is just squared and added to the sum. Taking the square root of the sum provides the resultant output noise. For each active stage (this example assumes the passive volumes controls add no noise or distortion, a simplistic assumption but often relatively true) the noise at the output of the chain is thus:
  • DAC noise = DACn * Gain to output = DAC noise * (G1*G2*G3*G4*G5)
  • Preamp input stage (Pre1) noise = G1n * (G2*G3*G4*G5)
  • Preamp output stage (Pre2) noise = G3n * (G4*G5)
  • Power amp (Amp) noise = G5n
Now the output noise is given by:
  • Nout = sqrt[ (DAC noise)^2 + (Pre1 noise)^2 + (Pre2 noise)^2 + (Amp noise)^2 ]
The output signal is simply the DAC’s output multiplied by all the gains: Vout = Vdac*G1*G2*G3*G4*G5. Then the SNR in dB is found as 20*log10(Vout/Nout).

Distortion is correlated to the signal, but since each stage in the chain is independent, the distortion does not simply add linearly. Each stage will add a little delay and phase shift so that the distortion is not exactly in phase with the input signal. To calculate the output distortion Dout we again RSS the distortion from each stage (XXd):
  • Dout = sqrt[(DACd*G1*G2*G3*G4*G5)^2 + (G1d*G2*G3*G4*G5)^2 + (G3d*G4*G4)^2 + G5d^2]; or,
  • Dout = sqrt[ (DAC distortion)^2 + (Pre1 distortion)^2 + (Pre2 distortion)^2 + (amp distortion)^2) ]
In dB, THDout = 20*log10(Vout/Dout), similar to finding the output SNR.

To find the output SINAD, we must combine noise and distortion:
  • SINADout = sqrt(Dout^2 + Nout^2)
Now we can plug in datasheet numbers, make some assumptions about the gain blocks, and use our own choices for volume controls to look at the performance at each step and the overall performance.
 
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DonH56

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Here are the assumptions for an example system:

Stage​
Description​
Gain (dB)​
Max Vout (V)​
SNR (dB)​
THD (dB)​
DAC​
DAC output​
User​
1​
120​
-120
(0.0001%)​
G1​
Preamp input (Pre1)​
10​
2​
120​
-110
(0.000316%)​
G2​
Preamp volume​
User​
-​
-​
-​
G3​
Preamp output (Pre2)​
5​
5​
110​
-100
(0.001%)​
G4​
Amp volume​
User​
-​
-​
-​
G5​
Power amplifier (Amp)​
25​
50​
100​
-80
(0.01%)​

Your system will almost certainly be different, but these represent values for what I have around the house. Variables we control are the DAC’s output level and gain (volume) settings for the preamp (G2) and power amp (G4). Note the gain of the preamp’s input stage, 10 dB (3.16x), means that at 1 V output the DAC will be clipping the preamp’s input stage. The maximum DAC output the preamp can handle is about 0.632 V (2 V max/3.16 gain). This is not an uncommon situation in the real world, when some element in the chain has the ability to overdrive the rest.

A power amp with 25 dB of gain needs ~2.8 V to provide full output. The entire chain has maximum 40 dB gain, a voltage factor of 100 (10^40/20 = 1^2 = 100), which means an input (DAC output) signal of 0.5 V will drive the power amplifier to maximum output if all controls are at max (0 dB attenuation). Also note the power amplifier has the worst performance and so will usually determine the performance of the entire chain. Again, this is not uncommon – compare typical DAC, preamp, and power amplifier specs. For better or worse, speakers are usually much worse for distortion, but are not part of this analysis. I only want to focus on the electronic components and speakers are one variable too many.

The first trial is to simply sweep the DAC's output voltage with all the volume controls set to max (0 dB attenuation). This should show where the best overall performance lies for the chain. Below are SNR, THD, and SINAD plots as the DAC’s output voltage changes from 0.1 V to 1.0 V in a few steps. You can see how the DAC improves all the way to 1 V output, but all the other curves dive after 0.5 V as the preamplifier clips. The overall output performance is set primarily by the amplifier as expected, primarily by its THD, until the preamp clips and overwhelms everything else. Also note the preamp’s input stage, Pre1, actually shows higher SNR at the output than the DAC even though they are rated the same (120 dB). This is because the DAC’s noise is amplifier by the preamp’s input gain, G1, so at the output DAC noise is higher than the noise from Pre1.

1651421746996.png
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Figure 8: Sweep DAC Output, Max Volume​
 
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DonH56

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The next test was to fix the DAC’s output at 0.5 V and sweep both attenuators together, like a ganged (coupled) volume control, from 0 dB (no attenuation) to -10 dB (the half loudness point). Without clipping any stage, this is more “classical” performance, with THD improving (falling) and SNR reducing as the signal level drops (remember positive attenuation reduces the signal amplitude so the signal level is dropping as you move to the right on the plots). The first volume control (G2) is after the DAC and preamp input stage Pre1 so they are not affected by reducing the volume (increasing attenuation reduces the volume). And as expected from the first post, output SINAD rises (improves) as the signal drops due to lowering THD, then falls again as noise instead of distortion begins to dominate SINAD.

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Figure 9: Ganged Attenuator Sweep​
 
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DonH56

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An interesting test is to sweep the attenuators in the opposite direction with the input signal (DAC output) fixed at 0.5 V, keeping the overall gain and thus output signal the same. This shows why “where” you change the attenuation (volume) matters and why increasing the level at one point and decreasing it someplace else in the chain does not always yield the same result. In this case, the preamp’s gain G2 is swept from 0 dB to -20 dB while the amp’s gain G4 is swept from -20 dB to 0 dB at the same time.

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1651424231298.png
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Figure 10: Alternate Attenuator Sweep​

For this example, performance remains high and it is hard to see any change in the output. The plots below show only the output so you can see how, even though the output amplitude stays the same, SNR, THD, and SINAD do change. SNR drops (worsens) a little, THD improves (reduces) a little, but overall SINAD is lower when we reduce the preamp’s volume while raising the amp’s volume to compensate and keep the output signal the same. Which control to adjust for the best performance in any given system depends on the parameters of each block in the system.

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Figure 11: Alternate Attenuator Sweep (Output Plots)​
 
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DonH56

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The last trials dig a bit more into the difference between adjusting preamp volume (G2) versus amp volume (G4) since that begins to address the common question of “Where do I set the volume controls in my system for the best performance?” As is hopefully clear now, the answer depends on knowing a lot of details about the system. And to be clear, I am not volunteering to run this program for every user and system on ASR (or anywhere else)! I have a demanding day job and (believe it or not) life outside ASR. :)

For this example, DAC output voltage (the input to the signal chain) is swept from 0.1 V to 0.6 V (remember clipping occurs in the preamp just a little over 0.6 V). The first trial set the preamp volume (G2) to -10 dB, leaving the amp volume control (G4) at 0 dB. The second sweep flipped that, with preamp gain (G2) fixed at 0 dB and amp gain (G4) at -10 dB. Only the output plots are shown; the intermediate plots are too busy to be useful, and besides would make unlucky 13 pictures in this thread.

1651425785134.png
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Figure 12: Sweep DAC with G2 or G4 Fixed at -10 dB​

For this system, output SNR is better when amplifier volume (G4) is adjusted rather than preamp volume (G2), at least for this setting (-10 dB). Output THD is essentially the same for either choice since it is limited by the power amplifier. The result is that overall output SINAD follows the SNR curves and again best performance is obtained by adjusting the amp rather than preamp volume control – for this system and at this volume level.
 
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DonH56

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That is a lot of posts with a lot of detailed information to digest. And it barely begins to describe the trades possible, and the effort a designer puts into optimizing the gain chain for any given system. At the end of the day, it provides (hopefully) a look into why “Where should I set my DAC’s gain, preamp’s volume, amplifier gain, etc. for the best performance?” is so very hard to answer.

A few final comments:

My program is somewhat ugly and user-hostile so is not posted here. It was created as much as a learning experience for me as for generating the data for this thread. I have been using Python for several years at work but am certainly not a great programmer. I need to extend a program I wrote to calculate bit error rate probability density functions when using forward error correction, something involving much uglier math than this, to cover an upcoming standard and thus it was a good time to explore a few other things in the Python world (mainly some additional math functions and plotting capability). Compared to something like Matlab it is amazing how much can be done in a general-purpose language like Python, but there were a few frustrating things along the way.

I made a host of assumptions about noise and distortion through the chain, left out a lot of other factors, and generally simplified the heck out of this problem to make it easier to implement the program and present to non-engineer readers. Those who know the technical details behind these posts have no need for them and will find them simplistic. Many folk with no technical background, or expertise in another field, may find them overwhelming. Thread, meet eye of the needle.

I’ll reiterate that I have not the time (nor really the interest, truth be told) to run this analysis for anyone else (OK, maybe if you pay me, but even then I haven’t the time right now). It is meant to illustrate a problem and, while there may be a solution for a specific case, the solution is likely different for each case (system). It’s an introductory tutorial, please treat it as such.

HTH - Don
 
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MattJ

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Egads man!! Just when I thought I could enjoy my damn system! J/K
Many thanks for this, so many things to check now!
 
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DonH56

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Here are a couple of examples building from the first example. Recall the parameters of the system:
  • DAC output = 1 V max, 120 dB SNR, -120 dB THD (0.0001%);
  • Preamp input stage Pre1 = 2 V max, 120 dB SNR, -110 dB THD (0.003%), 10 dB gain;
  • Attenuator (preamp volume control) nominally set to 0 dB;
  • Preamp output stage Pre2 = 5 V max, 110 dB SNR, -100 dB THD (0.001%), 5 dB gain;
  • Attenuator (amp volume control) nominally set to 0 dB; and,
  • Power amp Amp = 50 V max (~150 W/8 ohms), 100 dB SNR, -80 dB THD (0.01%), 25 dB gain.
First repeat the first DAC output voltage sweep but with 0.1 V steps for finer resolution and plot the results through the chain (below). Note Pre1 will clip with about 0.63 V input (DAC output) and since it is the first gain stage that determines the limits of the entire chain.

1651803719391.png
1651803724673.png

1651803733847.png
1651803741333.png


The plots clearly show from the dramatic drop in performance how the different stages are clipping. For this example, each stage is first tested independently, so it is easy to see where each stage clips. The DAC does not clip until its output exceeds 1 V (its maximum output). Pre1 clips at just over 0.6 V from the DAC as expected, Pre2 clips for 0.8 V from the DAC, and the amplifier at 0.5 V from the DAC. The overall gain is 40 dB, or 100x, so 0.5 V from the DAC will cause the amp to produce 50 V, the maximum it can deliver without clipping. You can see how THD and SNR hit their peak values just before clipping, with SINAD (which includes THD and noise/SNR) actually flattens and even drops a little as distortion begins to dominate over noise. Notice that, when the amp clips, the DAC’s SINAD has not peaked yet, Pre1 has peaked and is starting to fall, Pre2 is basically flat, and of course the amp’s SINAD is also falling as distortion rises. We’re getting full amplifier output but only using half the DAC’s range and overdriving the preamp. Blah.

The final plot shows the output voltages from each stage vs. DAC output so you can see how they limit (flatten) past the maximum output each stage can deliver. (N.B. The DAC, Pre1, and Pre2 are tied to the left y-axis, and Amp output to the right y-axis to make it easier to see them all in the same plot.) They do not instantly clip because (a) the distortion function I used “tails” off similar to a real amplifier, and (b) as the signal clips its energy (as measured by an FFT) continues to grow a bit (because a square wave has more energy than a sine wave (higher create factor, see e.g. https://en.wikipedia.org/wiki/Crest_factor).

Next, insert a 6 dB attenuator (or add a volume control adjusted to half level) at the output of the DAC, before the preamp’s input stage (Pre1). This is a common way to keep full DAC resolution without overloading the preamp (and thus rest of the chain).

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Now THD and SNR are rising smoothly all the way to the point at which the DAC and amplifier clip, with a bit of range yet through the preamp. Thus we get a little better performance up to where the DAC and amp clip, with Pre1 and Pre2 SINAD essentially “flat” from around 0.5 V DAC voltage until clipping. Output performance is still dictated by the amplifier (THD, SNR, and SINAD) but now we have a much wider (~twice the) range of volume control with good performance. This is generally a more desirable gain distribution, achieved with a simple change at the input.

HTH – Don
 

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Blumlein 88

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Back when CD became mainstream I had a Conrad-Johnson PV-5 and MV-50. The PV-5 could put out up to 25 volts, but the performance was designed for best results at 2.5 volt output. The MV-50 reached rated power at 2.35 volt input. Geared toward LP use it wasn't too badly put together. Or even with tape machines for line level input. With CD being standard at 2 volts and many brands going a little higher basically CD players would drive the amp to rated output with no gain. I did the sensible thing and built a passive volume control and only switched in the PV-5 for phono use. The line stage of the PV-5 had 28.5 db gain which meant with CD levels you had to turn the volume down low, noise performance suffered, channel tracking was rather poor and the volume control very, very touchy to adjustments. For a friend with the same preamp and similar power amp I built a fixed 18 db attenuator which made the combo livable and in the fat part of the performance curve. The C-J's when gain staged rightly were all very quiet devices for tube gear. Like 90 db SNR.

It surprised me that for a good 15 years at least audio companies stuck by the old high gain preamps instead of adapting for the higher output of CD players. Pre's either needed little gain or should have had a dedicated CD input with an attenuator built in. In time you did see such things from at least some brands of gear.
 

MarcosCh

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Thank you very much for putting this together. I was waiting for someone to do exactly this. Very informative, relevant and easy to understand.

Do you plan to expand the article to digital volume control and perhaps differences between using digital vs analog attenuation. Or even further, the role of the bit depth/dynamic range of the digital signal. Sorry if you find it too unrelated but i think it would bring the full picture of a real 2022 system together beautifully.

PS: i am not sure if these threads are meant to be answered. If not i will delete this post no prob.
 
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DonH56

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One last (or more likely “more”) test case: what if we buy a much better amp, one that still has 100 dB SNR, but now THD is also -100 dB (0.001%) instead of -80 dB (0.01%) at full output? Since the output performance has been dominated by the amplifier (*), we’d expect this to yield a significant improvement.

As a reminder, here is the performance as-is including that 6 dB pad (attenuator, volume control) after the DAC so we do not overdrive the preamp:

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This is not bad, but you can see how the amp and total output curves essentially overlie and the performance is entirely determined by the amplifier (specifically it's -80 dB THD) to be lower than the rest of the components in the chain (this is most obvious in the SINAD plot).

Now, swap out the amp to our fancy new -100 dB THD model (albeit with the same 100 dB SNR spec) and see the difference:

1651937923008.png
1651937928324.png
1651937933371.png


The amplifier still sets (limits, if you prefer) the performance, but now the curves are closer, with SINAD exceeding 90 dB for about half the range. Zooming in to ignore the big drop after clipping, we can see that the DAC and Pre1 stages’ SINAD are still very high, with Pre2 (preamp output) and amp performance are closer. Furthermore, the overall chain is good enough that, with individual blocks contributing more equally, the total output SINAD is around 95 dB instead of 100 dB. Unlike the previous case where the amp reached 80 dB or so, since it was so much “worse” than the other components, now the amp is no longer completely dominating the performance. Other stages, in particular the preamp’s output stage (Pre2), contribute enough to reduce the net (total) output performance.

1651937953859.png


Whether this is “the” goal is debatable. Not having things clip at drastically different levels through the chain is almost always a good thing, as you can use the full dynamic range of each block. But having different SNR/THD/SINAD levels (curves) is a more personal choice. You can choose a DAC that has performance much greater than the amplifier, arguably “wasting” the DAC’s performance due to the amp, but you can upgrade the amp later. Alternatively, perhaps it is better to select (and pay for) a DAC and preamp with somewhat less performance, and spend the money on a better amp since that is the main limiter in performance (always with the caveat that the speakers will likely set the final distortion level). Just one of many trades going into a purchase decision.

Onwards - Don

(*) In a real system, the performance is usually (far) dominated by the speakers. This whole thread is focused on just the electronics.
 
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DonH56

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Thank you very much for putting this together. I was waiting for someone to do exactly this. Very informative, relevant and easy to understand.
Thanks! I am not all that happy with it personally; too much info, and trying to cover it without leaving out or glossing over things vs. writing a book (or 1000 forum posts) is walking a knife edge. Hopefully it is enough for most.
Do you plan to expand the article to digital volume control and perhaps differences between using digital vs analog attenuation. Or even further, the role of the bit depth/dynamic range of the digital signal. Sorry if you find it too unrelated but i think it would bring the full picture of a real 2022 system together beautifully.
I thought about trying to deal with quantization in general, including digital volume controls, but decided against adding that to this thread. I even have a quantization term in my Python functions to quantize to any number of bits, but in the end quantization noise looks like noise for all practical purposes, and so is covered by just using SNR to set the noise level. If you check the link in my sig, there are a number of other articles dealing with sampling and quantization, and I decided they are best left as separate subjects. This thread is long enough as it is, and I have to sleep sometime...

That said, a thread about digital attenuation might be interesting... Note that there are actually (at least) three main attenuators in use today:
  1. Analog control using a potentiometer (resistor);
  2. Pure digital control that mathematically reduces the digital signal going into the DAC (effectively reducing the bit depth); and,
  3. Digitally-controlled analog attenuators (these are common in preamps and AVRs).
The last does not quantize the signal itself, it stays analog, but provides discrete (quantized) steps in volume.

PS: i am not sure if these threads are meant to be answered. If not i will delete this post no prob.

No problems posting now! It was opened up for comments when I finished the initial examples I wanted to cover, and our fearless and sleepless moderator team (@AdamG247 in this case) removed the warning and opened the thread. Like any good engineer, I couldn't leave it there, and keep thinking of new things to add.

You have put so much work into this Don! Much appreciated. Just saw it and promoted to home page.

Thanks Amir! This was a biggie, and a topic I have mentioned for several years without getting the gumption to actually do the work and write it. It ended up taking about 10x the work, natch, and it is still very long and sometimes hard to follow, but it's hard to reduce a fairly complicated subject into a few posts. It only scratches the surface, but it needed to be scratched. :) I am hopeful that the last few examples help clarify its application in the real world.

Edit: I almost never look at the home page, usually just click on "new posts". I took a look and decided starting with my notes wasn't the best intro for the home page so added an introductory line to the first post. Need to remember that in the future, always a good idea to state the thesis up front.
 
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(*) In a real system, the performance is usually (far) dominated by the speakers. This whole thread is focused on just the electronics.
And therein, as they say, lies the rub. Transducer gain (alongside room propagation loss), hearing threshold and peak levels are things that IMO should not be left out of this discussion, and arguably even play a central role. It is crucial to not only know what kind of dynamic range you are getting out of your electronics but also where it ends up in terms of absolute sound levels. They say the two most useless things in aviation are altitude above you and runway behind you, and about the same is true for dynamic range either below the hearing threshold or above your absolute worst-case highest listening volume.

Connect some 83 dB / W / m speakers in a large, well-damped room, and amplifier power / gain is going to be a major concern.
Connect some >105 dB horns in a smaller, more reflective space, and you'll be fighting hiss left and right but a handful of watts of power is likely to be perfectly fine.
Connect a headphone amplifier with cans, volume set for 110 dB SPL out at full DAC output, and anything with a dynamic range of over 90 dB will do. (Rule of thumb for vanishing noise in headphones = <20 dB SPL. For white noise in speakers it's ~4 dB SPL.)

Ending up with 95 dB of dynamic range while speaker playback levels top out at 120 dB SPL is the type of situation that you rather want to avoid. Conversely, if you have set up your peak speaker playback levels in the low 90s, you should find that even worst-case required noise / distortion performance ends up being quite modest, with the difference between a DAC at 100 dB and another at 120 dB being pretty much exactly zilch.
 

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And therein, as they say, lies the rub. Transducer gain (alongside room propagation loss), hearing threshold and peak levels are things that IMO should not be left out of this discussion, and arguably even play a central role. It is crucial to not only know what kind of dynamic range you are getting out of your electronics but also where it ends up in terms of absolute sound levels. They say the two most useless things in aviation are altitude above you and runway behind you, and about the same is true for dynamic range either below the hearing threshold or above your absolute worst-case highest listening volume.

Connect some 83 dB / W / m speakers in a large, well-damped room, and amplifier power / gain is going to be a major concern.
Connect some >105 dB horns in a smaller, more reflective space, and you'll be fighting hiss left and right but a handful of watts of power is likely to be perfectly fine.
Connect a headphone amplifier with cans, volume set for 110 dB SPL out at full DAC output, and anything with a dynamic range of over 90 dB will do. (Rule of thumb for vanishing noise in headphones = <20 dB SPL. For white noise in speakers it's ~4 dB SPL.)

Ending up with 95 dB of dynamic range while speaker playback levels top out at 120 dB SPL is the type of situation that you rather want to avoid. Conversely, if you have set up your peak speaker playback levels in the low 90s, you should find that even worst-case required noise / distortion performance ends up being quite modest, with the difference between a DAC at 100 dB and another at 120 dB being pretty much exactly zilch.
Well as a first step ignoring speakers to simplify the situation to allow better understanding by those reading the post I think is a fine idea. Once they get that down, then yes going forward with speakers and room makes plenty of sense.
 

formula 977

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Well as a first step ignoring speakers to simplify the situation to allow better understanding by those reading the post I think is a fine idea. Once they get that down, then yes going forward with speakers and room makes plenty of sense.
You are far too kind.
Going from the OP's absolute mathematic information to relative speculation under indeterminate circumstances as analogsteph is pushing, deliberately ignoring the context, will prove nothing.
 
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And therein, as they say, lies the rub. Transducer gain (alongside room propagation loss), hearing threshold and peak levels are things that IMO should not be left out of this discussion, and arguably even play a central role. It is crucial to not only know what kind of dynamic range you are getting out of your electronics but also where it ends up in terms of absolute sound levels. They say the two most useless things in aviation are altitude above you and runway behind you, and about the same is true for dynamic range either below the hearing threshold or above your absolute worst-case highest listening volume.

Connect some 83 dB / W / m speakers in a large, well-damped room, and amplifier power / gain is going to be a major concern.
Connect some >105 dB horns in a smaller, more reflective space, and you'll be fighting hiss left and right but a handful of watts of power is likely to be perfectly fine.
Connect a headphone amplifier with cans, volume set for 110 dB SPL out at full DAC output, and anything with a dynamic range of over 90 dB will do. (Rule of thumb for vanishing noise in headphones = <20 dB SPL. For white noise in speakers it's ~4 dB SPL.)

Ending up with 95 dB of dynamic range while speaker playback levels top out at 120 dB SPL is the type of situation that you rather want to avoid. Conversely, if you have set up your peak speaker playback levels in the low 90s, you should find that even worst-case required noise / distortion performance ends up being quite modest, with the difference between a DAC at 100 dB and another at 120 dB being pretty much exactly zilch.

Speaker performance and room interaction is way beyond the scope of this article, let alone hearing loss and such. As I said, this is focused on the electronics, and the issues of optimizing the signal chain up to the speakers. That topic comes up often enough that I thought it worthwhile to make an attempt, realizing it would not be enough for some like you (and likely too much for others). What I wanted to show is how the chain affects the performance at the output; whether that performance meets the needs of a given system is a choice best left to the listener IMO, and is the sort of thing that appears in numerous other threads. I had hoped this thread might show the impact of trades such as "Does it matter if my DAC has 120 dB SINAD if my amp only has 80 dB?" Maybe I missed, but I did my best in the time I had. Perhaps it is a worthless thread, but the topic kept arising, so...

An article relating amplifier power to speaker output might be useful, but there are so many variables, so many online calculators, so many threads on the subject, and speaker choice is so personal that it hardly seems worthwhile, or maybe just impractical. At least to me; you are more than welcome to tackle the subject! Just not in this thread, please, as it is long enough as it is.
 

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Speaker performance and room interaction is way beyond the scope of this article, let alone hearing loss and such. As I said, this is focused on the electronics, and the issues of optimizing the signal chain up to the speakers. That topic comes up often enough that I thought it worthwhile to make an attempt, realizing it would not be enough for some like you (and likely too much for others). What I wanted to show is how the chain affects the performance at the output; whether that performance meets the needs of a given system is a choice best left to the listener IMO, and is the sort of thing that appears in numerous other threads. I had hoped this thread might show the impact of trades such as "Does it matter if my DAC has 120 dB SINAD if my amp only has 80 dB?" Maybe I missed, but I did my best in the time I had. Perhaps it is a worthless thread, but the topic kept arising, so...

An article relating amplifier power to speaker output might be useful, but there are so many variables, so many online calculators, so many threads on the subject, and speaker choice is so personal that it hardly seems worthwhile, or maybe just impractical. At least to me; you are more than welcome to tackle the subject! Just not in this thread, please, as it is long enough as it is.
Definitely not a worthless thread. I've already seen others point people to this thread twice in the last few days. Whether you get many responses here I think people will be reading it benefiting from it for some time. Your efforts are worthwhile, and not in vain.

It already has just short of 1000 views.
 
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DonH56

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Definitely not a worthless thread. I've already seen others point people to this thread twice in the last few days. Whether you get many responses here I think people will be reading it benefiting from it for some time. Your efforts are worthwhile, and not in vain.

It already has just short of 1000 views.
Thanks. My intent from the beginning was to focus on just the electronics; speakers introduce a host of complications I did not want to include in this thread.

The main problems I have with speakers are their widely (wildly) varying sensitivity and my lack of a good distortion model. Working in voltage, as I have done, you could use an online calculator like http://myhometheater.homestead.com/splcalculator.html to estimate the power needed to achieve your target SPL given your speaker's sensitivity and average impedance. From there you can calculate the voltage and work the SNR specs through the chain, and maybe distortion if you have a target in mind and know what the speakers are doing. And your room's noise floor, your sensitivity to the noise, etc. In any event the speaker's sensitivity and distance to the user could become just another attenuation term at the output of the amplifier, assuming the speakers add no noise of their own. That seems reasonable for SNR.

As for speaker distortion, whilst I have a fair understanding of the mechanisms, I have no simple model for speaker distortion nor sufficient data to make any sort of general estimate to plug into my program. The use of a tanh() function with ideal feedback for the electronics was a bit tedious but straight forward and, with the added offset to include even-order distortion, matches the sort of distortion I am familiar with in the real world. Not being a speaker designer, I am not sure what components contribute most to distortion nor their magnitude, and did not have (or was unwilling to take) the time needed to research it. I did flip through my old grad texts but that gave me ugly equations and a bunch of parameters I'd need for each speaker. It's complicated and I know just enough to know I don't know.

An article on speaker effects and such would be interesting but beyond the scope of this thread and beyond my field of expertise.

FWIWFM - Don
 
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