The meaning of "linear"

[MaxwellsEq]

if the non-linearity errors are known, measured and small enough to have no impact on the performance of the system in its designed operating range parameters, than that system can be labeled as linear

We will have to agree we see things differently at this point

Personally, I couldn't label it as linear, since the non-linearities haven't disappeared. In a complex chain they can additively and subtractively interfere with each other to generate new non-linearities which may be significant - who can tell in advance. I own some kit which measures extremely well, perhaps even to the point where non-linearities are inaudible, but as an engineer, I still know they are non-linear.

There's something contingent on labelling it linear. By that I mean that we all, for example, universally agree that electronic noise below -100dB across the frequency band is contingently inaudible to humans so is "in practice" linear.

On the other hand, I've never experienced a linear cartridge or loudspeaker, necessarily or contingently!

 

[BelgianJoey]

So why are SNR signals typically given at a low value like 1W? During a long time I was thinking too that this was completely irrelevant. But looking at the graphs Amir produces when testing amplifiers I see now that this makes sense after all. 1W is a power that any amplifier can produce. The SNR of an amplifier typically improves first with increasing power until distortions and eventually clipping occurs when maximum power is approached. So a value at 1W or the ASR dashboard at 5W is relevant to compare SNR values of amplifier with different maximum power.

 

[MaxwellsEq]

So why are SNR signals typically given at a low value like 1W? During a long time I was thinking too that this was completely irrelevant. But looking at the graphs Amir produces when testing amplifiers I see now that this makes sense after all. 1W is a power that any amplifier can produce. The SNR of an amplifier typically improves first with increasing power until distortions and eventually clipping occurs when maximum power is approached. So a value at 1W or the ASR dashboard at 5W is relevant to compare SNR values of amplifier with different maximum power.

I think there is continued development on "what is good enough" for an amplifier driving speakers. It's possible to design an input stage and main voltage-gain stage that measures very well. The challenge comes in the final stage which moves the "motors" (classical) or "transformer/capacitor" "electrostatic). High power needs both high voltage (to cope with the high-impedance bits and high current to cope with low impedance bits. Amir's selection of 5W feels more relevant with modern low-efficiency speakers than the (I think) historical 1W. But, assuming the amp is comfortable at 5W, the noise floor should be largely unchanged than a 1W measurement, so the SNR at 5W will be better, and so on until the amp meets its upper limits.

 

[egellings]

For amplifiers... Linearity is the ability of a amp to maintain the signal integrity from the input of the amp to the output and... it also means the amplifier output wattage doubles for each halving of the load resistance.

If you plot input vs. output of a linear amplifier on X-Y graph paper, you get a straight line.

 

[MaxwellsEq]

If you plot input vs. output of a linear amplifier on X-Y graph paper, you get a straight line.

So, given the formal (mathematical/engineering) meaning of linear forces us to state there are no necessarily linear playback systems, perhaps a better definition of "linear for audio systems" shorthand is that we can identify a relevant straight line and measure which systems are very close to that line, and which are not

 

[JRS]

I remember the superposition principle as being a key criterion, that is the sum of the whole is equal to the sum of it's parts. In other words for a system to be linear we can operate on the sum of two inputs a and b and arrive at the same result as if we operated on the parts a and b separately and then summed those. Much easier to write mathematically . A function is linear if f(x1) + f(x2) = f(x1+x2) and the result is the same today as it will be tomorrow, in other words the function is time invariant. The beauty of linearity is we can decompose a complex input into a sum of simple parts for which solutions are known and get answer to complex input. Thats a BFD in engineering and makes for a more predictable universe.

 

[MaxwellsEq]

I remember the superposition principle as being a key criterion, that is the sum of the whole is equal to the sum of it's parts. In other words for a system to be linear we can operate on the sum of two inputs a and b and arrive at the same result as if we operated on the parts a and b separately and then summed those. Much easier to write mathematically . A function is linear if f(x1) + f(x2) = f(x1+x2) and the result is the same today as it will be tomorrow, in other words the function is time invariant. The beauty of linearity is we can decompose a complex input into a sum of simple parts for which solutions are known and get answer to complex input. Thats a BFD in engineering and makes for a more predictable universe.

That's what I believe as well.

 

[Newman]

So, given the formal (mathematical/engineering) meaning of linear forces us to state there are no necessarily linear playback systems,

Sure, but that doesn’t mean their linearity is zero.

If “linear” requires A to be constant in “Output = A x Input”, then linearity increases as the error bars on A get smaller, over the range of outputs that are of interest.

perhaps a better definition of "linear for audio systems" shorthand is that we can identify a relevant straight line and measure which systems are very close to that line, and which are not

Correct.

Hence the term in audio, “linearity” meaning how close to a straight line, or in math, how small are the error bars for the ideally-constant factor A.

 

[FrantzM]

Hi

Hi!

I'm new to ASR, but thought the folks here might be able to help me understand what various groups mean when they say an amp or speaker is "linear."

I'm a (biomedical) engineer, and so when I hear the term, I understand it to mean that a system's output is *linearly* proportional to the power input; and a corollary - that it's frequency response is *independent* of the power input.

But I think I've heard many, knowledgeable, people call a system linear if it had a *flat* frequency response. Linear as in, it looks like a horizontal line?

Which brings me to the second part of my question - what *is* the term used for talking about linearly from an EE "signals and systems" perspective? It would seem pretty interesting to me to learn what amps and speakers have a similar frequency response or noise characteristic over a range of input power. E.g. how do Class A/B amp topologies typically perform, compared to Class D? Usually, the data I've seen (with a few caveats) for SNR have been for a low/reference power level, like 1W.


Thanks for humoring me!
David

Hi
IMHO, among the best answers are these posts:
This one #10, by @IAtaman
That one # , by @JRS

It is also important to understand that a system can be considered linear within certain limits. A pragmatic approach. Boundaries must define those are the values within reach , a range , where the system responds to the notion of linearity.
Audio reproduction being about human, our non-linear perceptions must at time s be taken into account to define a system as linear. thus my response to this reply...

To me, "linear" means at different loudness levels, the “system” frequency response keeps its shape/curve.
Quite easy to check with REW, taking a few measurement sweeps at 5dB intervals.

Problem is that our hearing itself is ... non-linear. We don't hear the same across our hearing range... Let's stick to 20 Hz to 20,000 Hz... The famed Fletcher-Munson curves do paint a picture of our non-linear hearing apparatus.... If a system remains "linear" across the range of SPL, it will not be perceived as "linear" by our hearing apparatus. Here

The ideal would be a system that follows those curves. First you EQ your system to flat , across the hearing range then, , hope that a mechanism compensates for this non-linearity so that the overall system is truly linear to our ears. The best I have found so far is Audyssey Dynamic EQ. A requirement, henceforth for me in any audio system.

Peace.

 

[MaxwellsEq]

Sure, but that doesn’t mean their linearity is zero.

If “linear” requires A to be constant in “Output = A x Input”, then linearity increases as the error bars on A get smaller, over the range of outputs that are of interest.


Correct.

Hence the term in audio, “linearity” meaning how close to a straight line, or in math, how small are the error bars for the ideally-constant factor A.

The OP asked for "The meaning of linear". In mathematics and engineering (i.e. everything else in the world that is not a home audio system), linear does not mean "approaches a line which is constrained".

However, in the audio playback world, I guess we have to accept linear means: "as close to a flat line as possible within the constraints that we've decided are relevant".

 

[FrantzM]

The OP is an engineer. I believe he already knows what linearity means in Mathematics and Engineering.
My take on linear in Audio: Getting as close as possible to an accepted or reasonably studied, curve. We, the subject in audio do not have a linear hearing, thus following a curve of our perception as closely as possible is, in this context, "linearity". An example: It is quite possible to devise a flat measuring headphones, perhaps with the help of DSP. I mean flat no curve. For most subjects , audiophiles or not, such headphone would be perceived as not linear, not pleasing, not Hi-Fi.

Peace.

 

[Newman]

The OP asked for "The meaning of linear".

You are mistaking the title for the question. The OP actually asked “what various groups mean when they say an amp or speaker is "linear."”

 

[fpitas]

So why are SNR signals typically given at a low value like 1W? During a long time I was thinking too that this was completely irrelevant. But looking at the graphs Amir produces when testing amplifiers I see now that this makes sense after all. 1W is a power that any amplifier can produce. The SNR of an amplifier typically improves first with increasing power until distortions and eventually clipping occurs when maximum power is approached. So a value at 1W or the ASR dashboard at 5W is relevant to compare SNR values of amplifier with different maximum power.

All that, and in real use you'll be using about 1W average, even though peaks may be considerably higher.

 

[dcuccia]

OP here! I'm blown away by the fantastic responses, thank you! Hope to have time this weekend to reply to specific comments. Briefly, yes, I know what linear means from an engineering perspective, and the spectrum of comments and opinions here help me feel like maybe I'm not alone in the ambiguity of how the term gets used.

I don't have any fundamental problem discussing about a flat frequency response as "linear" to point out that there aren't big dips or peaks. In fact, I suspect our collective experience with positive/negative interference and Q factor somehow tie to the "feeling" of a linear system.

But yes, clearly, voltage-dependent clipping introduces non-linear responses (e.g. harmonics) at the upper end of the amp or speaker's range (x-max is a decidedly important factor!). It was also well pointed out, while maybe not as obvious, that fixed noise characteristics which don't scale with the input also create non-linearities at the lower end. And maddeningly our non-linear eardrum/sensory system makes even a perfect system (from an SPL level) perceived as a nonlinear one (I will have to learn more about Audyssey Dynamic EQ!). It's interesting how SINAD combines noise and distortion, thanks for that tip!

My comment-question about amps was related to my fairly basic understanding (need to go back to Horowitz & Hill to remind myself...) that the different amplifier topologies have different clipping characteristics/failure modes. IIRC, Class D often have a hard-cutoff, and it will just "clamp" the voltage to the max capable output, whereas A/B fail more "gracefully" where the magnitude of damping increases as voltage/power increase - i.e. rounded off sine waves, vs "flat" tops. As I was getting interested in the field, I wondered how the community talks accurately/quantitatively about these types of differences in this difference of behavior at the limit.

I've also heard conversations like, "this amp sounded particularly great at low volumes" (e.g. compared to competitors), somehow indicating that it was better at reproducing lower frequencies, maybe often tied to having a high power. I've always wondered if this was more about the (non-linear) mechanical properties of the speakers, with some sort of mechanical friction/drag at low speeds that is harder for an underpowered amp to overcome. If this is the case, that would be a big deal for determining a proper setup. Are there analyses of speaker sensitivity or MTF that are performed over a range of voltages/power levels? Not sure if this is snipe hunting, but curious.

In any case, thanks again for so many great thoughts and interactions. What a great group!

David

 

[robwpdx]

Hi!

I'm new to ASR, but thought the folks here might be able to help me understand what various groups mean when they say an amp or speaker is "linear."

I'm a (biomedical) engineer, and so when I hear the term, I understand it to mean that a system's output is *linearly* proportional to the power input; and a corollary - that it's frequency response is *independent* of the power input.

But I think I've heard many, knowledgeable, people call a system linear if it had a *flat* frequency response. Linear as in, it looks like a horizontal line?

Which brings me to the second part of my question - what *is* the term used for talking about linearly from an EE "signals and systems" perspective? It would seem pretty interesting to me to learn what amps and speakers have a similar frequency response or noise characteristic over a range of input power. E.g. how do Class A/B amp topologies typically perform, compared to Class D? Usually, the data I've seen (with a few caveats) for SNR have been for a low/reference power level, like 1W.


Thanks for humoring me!
David

As an engineer you know models. We make approximate models of complex analog and biological systems. One of my most insightful professors stressed, state your model and what it leaves out.

When someone states "linear" in audio, examine the model (measurement hopefully) and what it leaves out. From an engineering standpoint, an engineer would consider statics and dynamics. Beyond that in audio, it is often a marketing meme.

 

[MaxwellsEq]

One of my most insightful professors stressed, state your model and what it leaves out.

+1

An representative example is boundary conditions when measuring harmonic distortion at the upper end of the frequency range. The 2nd harmonic of 100Hz is 200Hz, the 3rd is harmonic is 300Hz. The 2nd harmonic of 1kHz is 2kHz. So if we measure the harmonic distortion of 100Hz within a bandwidth of 20KHz, we can measure potentially hundreds of harmonics. We see fewer for 1kHz, but it's probably plenty. What about 5kHz - an important area of hearing? The 2nd harmonic is 10kHz and the 3rd Harmonic is 15kHz and the 4th is 20kHz. Given thier levels, we probably can't hear above the 2nd harmonic, so is it worth measuring up to, say 50kHz in order to capture the 10th harmonic? Instinct says no, so we use a model that leaves out higher harmonics of 5kHz. This is even more so for 10kHz and 20kHz.

So, if we use a model that leaves out harmonics above 20kHz and produce a single number measure for 10kHz we will get an artificially low numeric value for harmonic distortion, at that frequency. As your professor pointed out, we need to declare what the model leaves out. Bandwidth-limiting harmonic distortion measurements produces artificially low measured values for higher frequencies with the error getting worse as the frequency increases. Given the low levels of music signals at these frequencies and our limited hearing, it probably doesn't matter - but it should be made clear that this is what is left out of the model when producing a numerical value for harmonic distortion.