Headphone Measurements Explained - Total Harmonic Distortion Plus Noise Part 1

The primary goal of any audio transducer is to reproduce the signal it's receiving faithfully: high fidelity. Of course, no transducer is perfect and all alter the sound to some extent. Those alterations are distortions of the original signal. In other words, if you put in a pure 1kHz tone, harmonic distortions may cause some of that energy to apear at 2kHz, 3kHz, 4kHz etc; and non-linear behaviors (diaphragm brake-up; hair touching the driver; driver housing rattle) may cause broader band, noisy signals to apear almost anywhere in the audible spectrum.

There are numerous types of distortions and various measurement approaches that allow you to tease out the types of distortions present. The Total Harmonic Distortion plus Noise measurement method used at InnerFidelity is poor at differentiating what types of distortions are present, but very good at portraying the amount of distortion of all types present.

How THD+noise is Measured
The principle method of taking a THD+noise measurement is quite simple really. You drive the device under test (DUT; a headphone in this case) with a pure tone (called a probe tone), say 1kHz. Then, in the case of headphones, you take the return signal from the measurement head and run it through a narrow notch filter tuned to 1kHz to remove the the probe tone, leaving you with any energy in the system apart from the probe tone. This process is repeated over and over throughout the audio spectrum up to 7kHz drawing a plot of how much the headphones distort the original signal over the audio spectrum.

(The reason the plot stops at 7kHz is because at that frequency the third harmonic is 21kHz and will go out of the measurement range of the instrument by 22kHz. Once the third harmonic leaves the measurement window the measured distortion will abruptly lower even though distortion in the system hasn't actually changed. Therefor measurements of THD+noise are not internally consistant beyond 7kHz.)

Let's take a close look at the causes of both harmonic distortions and noisy distortions.

Harmonic Distortion and the Transfer Curve
Audio amplifiers have a gain curve and headphones have a transfer function, but they really all mean about the same thing. (In fact after scouring the web for a half hour I'm not sure there's an agreed upon term here...please feel free to comment about what you think is the right term.) It's a way to depict how "linear" a device is. Let's talk our way through a few transfer curves.

Headphone101_InterpretingTHDnoise_Illustration_FlatGainCurve

Diagram of a basic linear transfer curve.

In the illustration above, the incoming signal is in green at the bottom of the diagram. The square above it is the device under test, and the red line is its transfer curve. (For the moment let's just assume we're talking about an audio amplifier as it's a bit easier to understand this stuff for amps than it is for headphones.) As the input signal from the bottom moves left to right in sinusoidal manner, its position is projected up to the transfer curve. When viewed from the output side, this sinusoid can be seen moving up and down. Because the transfer curve is a straight line, the output signal is undistorted.

You'll also note the small frequency response plot in the lower right of the illustration. This shows the output spectra. In the case of a perfectly linear device at 0dB gain (output amplitude the same as the input amplitude) with a 1kHz input, we will see a single output tone at 0dB at 1kHz. Keep an eye on this plot as it will begin to show the harmonic distortion products as we move into non-linear transfer curves.

As I mentioned, in an audio amplifier this is sometimes called a gain curve. Not surprisingly, when we change the gain of an amplifier by adjusting the volume, we change the slope of the transfer curve.

Headphone101_InterpretingTHDnoise_Illustration_HighGainCurve

Turning the volume up increases the slope of the gain curve.

When we turn the volume of an amplifier up, we increase the slope of the gain curve. You can see in the illustration above that by increasing the slope of the gain curve we increase the amplitude of the output signal while the input signal remains the same size. The output harmonic spectra plot still shows only a single peak (because the transfer curve remains linear) but it is now at about a +3dB level.

Headphone101_InterpretingTHDnoise_Illustration_LowGainCurve

Turning the volume down decreases the slope of the gain curve.

Likewise, turning down the volume decreases the slope of the transfer curve, lowering the output level. And again, because the curve remains linear, the output shown in the harmonic spectra is a single tone but now about -3dB in level relative to the input.

Now let's imagine we have a solid-state amplifier that has +/-5 Volt power supply rails, an input a signal that's 10 Volts peak-to-peak, and we turn the volume up just a little above unity gain. Well, if the amp only has 10 volts to work with between the +5Volt and -5Volt supplies it's simply unable to produce a signal that's bigger than 10Volts peak-to-peak.

Headphone101_InterpretingTHDnoise_Illustration_HardClipCurve

Transfer curve with flattened bottom and top similar to what might be found at the power supply limits of a solid-state amplifier.

In the diagram above you can see that the portion of the output signal reproduced on the central linear portion of the transfer curve keeps it's shape, but once the input signal reaches the upper and lower limits of the transfer curve it simply clips off the bottom and top of the waveform.

Now it's a bit complicated, but when you begin to flatten the top and bottom of a sine wave you begin to get odd harmonics. A square wave is essentially a fundamental tone and its odd harmonics at the appropriate phase and amplitude relationship. (More on this topic here.) So you can see in the harmonic spectra plot that you have the fundamental tone at 1kHz, but you also have harmonic tones at 3kHz and 5kHz—the odd harmonics.

The thing to take away from this plot is that as soon as the transfer curve is anything but a straight line, you will begin to produce harmonic distortion and a series of overtones stimulated by the probe tone.

It's also worth noting at this point that if you've been around long enough to remember the introduction of solid-state audio electronics they used to make a big deal of the term "Linear Amplifier". The term linear in that phrase refers to the gain curve of solid-state amplifiers which can be made much more like a straight line than the tube amplifiers common at the time.

Speaking of tube amplifiers, they too can exhibit signal clipping, but unlike the solid-state amplifier's hard clipping and abrupt end to the transfer curve, tube amplifiers loose gain at the extremes much more gradually.

Headphone101_InterpretingTHDnoise_Illustration_SoftClipCurve

Soft clipping of an "S"-shaped transfer curve similar to that found in tube amplifiers.

The plot above shows the situation where the device under test loosing gain (reducing slope) as it reaches the extremes of its response. This would be typical of some tube amplifiers, and tends to be less abrasive sounding than the hard clipping of transistor circuits.

But what about headphones? Do they exhibit these types of response? Yes! For example, at very low frequencies as the headphone attempts to reproduce large, low frequency excursions, they begin to run out of steam on both the compression and rarefaction extremes. This causes the transfer curve on many headphones at low frequencies to exhibit the "S"-shaped response seen in the plot above. Another area with headphones that can cause a similar response is when, with large excursions, the voice coil leaves the area in the magnetic gap with maximum flux density. As the driver reaches its extremes, the coil windings are away from the flux reducing motive power and limiting movement at the extremes.

So far we've talked about symmetrical, S-shaped transfer curves that produce odd-order harmonics. It's called symmetrical because it exhibits a symmetrical loss of gain in both the positive and negative going directions. But there is another type of transfer curve we can consider: a simple "C"-shaped curve.

Headphone101_InterpretingTHDnoise_Illustration_AsymetricalGainCurve

An asymmetrical transfer curve has different gain profiles between the positive and negative going portions of the waveform.

In the above plot we see a simple "C"-shaped curve as the transfer curve. This curve will produce more gain (steeper slope) for the positive half of the incoming waveform than it does for the negative half (less slope). This will result in the positive half of the output waveform being stretched and the negative half being squished. This type of distortion will produce even-order harmonics, and you can see in the harmonic spectra plot 2kHz, 4kHz, and 6kHz harmonics are being created from the probe tone.

This type of distortion is often seen in single-ended, class-A amplifiers with single output devices. It can also occur in headphone when there are asymmetrical forces at play. For example, there is a limited amount of acoustic space behind the driver. As the driver moves outward away from the structure behind the diaphragm they have less effect than when the driver moves inward toward the structures. This produces an asymmetry in response between the compression and rarefaction stroke of the driver.

Another asymmetrical phenomena exists due to the diaphragm being firmly fixed at its edges (for most dynamic headphone diaphragms), which causes the useful driving surface of the diaphragm to get smaller as the diaphragm moves away from center. But because of the bulging shape of the taurus between the fixed edge and internal dome, the change in driving area with excursion is different depending on whether it's moving outward or inward.

Headphone101_InterpretingTHDnoise_Illustration_CompoundGainCurve

Most system will exhibit some amount of both symmetrical and asymmetrical bent in its transfer curve.

The truth is most systems will exhibit some amount of both symmetrical and asymmetrical behavior and will produce both even and odd harmonics. In fact, it's been suggested that when looking at the harmonic series, the best sounding products will exhibit both even and odd harmonics and they will appear in nice, smooth, ever decreasing series. Unfortunately, InnerFidelity's THD+noise measurement is unable to show this.

Harmonic Distortion Summary
All the above dialog about the shape of the transfer curve of the device under test and how, when not linear, it will produce harmonic overtones fall into the "THD" part of the THD+noise measurement. Put in a pure 1kHz tone, and you'll get some of that energy moved into a higher frequency that is a multiple of the original probe tone.

However, there are other mechanisms that might cause probe tone energy to be moved into other frequencies that have nothing to do with the transfer curve of the device under test. These spuria fall under the heading of "noise" in the measurement.

Noise Sources in THD+noise Measurements
Many headphone enthusiasts have experienced the dreaded "hair in the driver" syndrome. This is where a hair works it's way through the various coverings in front of the drive and is beginning to touch the diaphragm as it moves causing a buzzing noise—very bothersome if you've ever experienced it. The thing about this buzz is that it creates a broad band noise that isn't limited to the harmonic structure of the probe tone—it is a noise source and is measured by the THD+noise measurement.

Another example of a noise source might be if a small part is loose and rattles. In this case, the rattling might be frequency sensitive: at 100Hz the part might rattle, but at 1kHz the probe tone frequency might be too high to stimulate the rattle. The THD+noise measurement will show this as elevated distortion at those frequencies where the rattle occurs.

A third example, and one that is very important with headphone measurements is the problem of diaphragm break up, which usually occurs at frequencies above 1kHz. Diaphragm break up happens when the normal pistonic movement of the diaphragm moving forward and back begins to exhibit another type of movement—sometimes rocking motions, sometimes folding motions (think taco) of the diaphragm. These oscillatory mode of the diaphragm movement will produce tones in addition to the probe tone. They will be mathematically related to the probe tone, but won't necessarily be in the harmonic series. None the less, they will be measured and will appear as features on the THD+noise plot.

Power Handling and THD+noise Plots at 90dBspl and 100dBspl
The last thing to note about THD+noise measurements at InnerFidelity is that the THD+noise measurement is done at two different sound pressure levels in the headphones: 90dBspl and 100dBspl. Most people consider about 85dBspl as a reference listening level—if you're going to mix a record, you should do it at about 85dB. It's a pretty solid level, and louder than I normally listen for enjoyment. So, the 90dBspl THD+noise plot is fairly indicative of the worst case amount of distortion you'd get during normal listening. But, we have to remember that there will be dynamic peaks in the music much louder than the average level, so it is reasonable to test for distortion at higher levels, which is why InnerFidelity measures THD+noise at 100dBspl as well.

The thing to know about the THD+noise plot is that it is the amount of signal present apart from the probe tone measured relative to the amplitude of the probe tone itself. If I measure 1% THD+noise at 1kHz at 90dBspl, and the same 1% at 100dBspl, there will actually be 10dB more THD+Noise in the second measurement because it's measured relative to the 10dB higher in level probe tone.

Most headphones will exhibit mildly worse THD+noise numbers at higher volume levels, but some will exhibit much worse figures at some frequencies as the level goes from 90dBspl to 100dBspl. In that case we can say the headphone has poor power handeling, and is on the ragged edge of misbehavior during the high volume peaks of the music.

Conversely, we can sometimes see headphones where the 100dBspl THD+noise plot is below the 90dBspl plot. In this case we've got a headphone that is very well behaved as volume gets louder and we can say the headphone has good power handling capability.

THD+noise Plot Summary
The THD+noise plot sweeps a probe tone from 20Hz to 7kHz through he device under test. This probe tone will stimulate various distortion mechanism in the device under test, which will move some of the probe tone energy elsewhere in the frequency spectrum. The resulting signal is sent to an audio analyzer that first filters out the probe tone with a narrow notch filter that tracks the probe tone as it is stepped through the spectrum. The audio analyzer then measures the RMS amplitude of the remaining signal, which contains all the total harmonic distortion and noise products of the probe tone.

In part two of this article we look at a series of THD+noise plots to look at the various features of these plots and what they mean.

COMMENTS
bluecap's picture

Might as well be measuring it with Flintstones technology. http://www.innerfidelity.com/content/flaw-tylls-headphone-measurements

innerfidelity_login_id's picture

Instead of rushing to be the first to make a snarky comment, you should actually READ THE ARTICLE on how the THD+noise measurement is being performed. You would see that the argument you make in your linked 'discussion' is invalid given the testing methodology.

I understand the point you are making in the linked thread, but literally the first graph shown makes clear why it is now irrelevant.

bluecap's picture

I read it. I just think my way is better. Blocking out the frequency you're trying to measure and measuring the remaining noise is interesting and elegant, but it doesn't really tell you what frequency the headphone is really putting out when fed a particular frequency.

Tyll Hertsens's picture
Dude, you're deluded, headphones are a minimum phase device, meaning they're moving right along, in-phase, with the driving signal, throughout their normal operating range. In other words, 1kHz in, is 1kHz out, lagging possibly be a few tens of degrees in phase.

You need to give it up. You're a troll and don't even know it. Your premise is invalid. Spend some time grocking shit.

bluecap's picture

Just like in the thread you want to assume that a headphone fed x hz makes that sound. It's just not true, especially not for iems. I gave you an easy way to test it. But you'd rather assume instead of taking a little time to test it.

Your measurements are based on a false premise

bluecap's picture

P.s. I'm thinking about a particular 12 driver iem that I'm almost certain had the problem I described.

bluecap's picture

P.S.S. rhymes with Dana Carvey. I wonder if all the extra drivers are causing a problem in this respect and resulted in a "veiled" sound. Test it if you want.

zobel's picture

I imagine that higher frequency distortion and noise would generally be similar in scale and shape as the graph up to 7 KHz, unless something weird went on with the drivers right?

I also imagine that inter-modulation distortion would parallel the THD + noise in terms of overall quantity in most cases. That would be another good test though. IM distortion is always present, and as you no doubt know can be graphed like this;
http://www.neumann-kh-line.com/neumann-kh/glossary.nsf/root/19025D8C4F12...
This is the best method I've seen so far for IM distortion graphing.

When you measure the THD + noise with microphones in the dummy head at the ear drum position, isn't that graph influenced by the raw SPL / frequency curves of the headphone being measured? Would a HRTF curve need to be applied to the measurement to illustrate the audibility of the distortion? Again, so much easier in loudspeakers, where what you see is what you get.

Glad you do these THD/noise tests, just as they are they are easy to read, and offer fair comparisons with all cans of a type being tested, regardless of correcting for the HRTF or not. Have you ever correlated THD/noise with the impulse response? I bet the quietest impulse response (or a quick smooth waterfall chart) would go hand and hand with low THD/noise, and low IM too, but that is guessing on my part.

Thanks Tyll! Keep on!

RazrLeaf's picture

I don't think he understands enough physics to know what you're saying.

Rillion's picture

bluecap,

I have measured a number of headphone responses using two different approaches: looking at the resulting RMS response to individual frequencies and also by feeding the headphones a broad-band noise sound (white noise) and then doing a power spectrum analysis of the recorded headphone response (this accounts for all frequencies simultaneously). The results of the two approaches is that they match to within a decibel with the headphones that I have tried, at least above 100-200 Hz depending on the headphones. The response to broad-band sound in bass is exaggerated, however--I don't know if this is due to the headphones, my measurement gear, background sounds, or other issues. Pink noise might provide better signal to noise in the bass so might provide more reliable results in bass.

When comparing two headphones on different ears, keep in mind that the left ear can hear things quite differently than the right. Our brain does some sort of averaging so we don't normally notice this.

Don't become discouraged from trying to understand this stuff. It is easy to fool yourself (it has happened to me many times). Keep questioning everyone, including yourself, and eventually you will figure out the reasons for puzzling results.

bluecap's picture

Did you do that test with iems? Link the results

Rillion's picture

Nope, that's a valid point. Nor do I have access to the wide range of headphones that Tyll has. If you think about how drivers work, it is hard to imagine why there would be pitch errors, but "hard to imagine" does not mean "impossible." I'll keep an open mind about it until I see some hard data with iems. I don't have the gear to do those measurements because my microphones block the entrance to the ear-canal.

In the meantime, you might try checking if your left and right ears perceive certain pitches differently using the same speaker.
The ear canal resonances probably occur at somewhat different pitches in each ear. Here a couple of websites that you might find useful or interesting:

http://newt.phys.unsw.edu.au/jw/hearing.html

http://www.audiocheck.net/blindtests_index.php

Rillion's picture

Oh and I believe your criticism has more to do with using a tone-sweep for the Frequency Response plots than the %THD+noise plots. As others have pointed out, Tyll's methodology should show a large distortion in the THD+noise plots if there were pitch errors that move the pitch outside the output notch.

Keep in mind that very few people have "perfect pitch" which would allow them to clearly distinguish amplitude changes from small pitch changes. There is a test of how well you can distinguish pitches in the website of the second link that I included in my last message.

Alondite's picture

Your posts in that thread might be the most flat-out wrong bit of drivel that I've ever read. I can't tell if you're some pure-subjectivist crusader or are just painfully ignorant.

If you are feeding headphones a [X]Hz source with a [X]Hz notch filter, and your headphones are putting out a [Y]Hz fundamental frequency, then you're going to get a massive spike at [Y]Hz (not to mention spikes at harmonics of that frequency, which would clearly not be harmonics of the source fundamental).

That's literally what Tyll's THD+N test is measuring. I don't understand why that's so difficult to comprehend.

tony's picture

I just happened to be in the Lab with the Noise-Vibration-Harshness lads, I was showing them some of your headphone measurements. I asked if I could insert a 31 band Eq into my stuff to flatten out my Sennheiser HD580s. They quickly took me to the Deep End of the Pool. phew! People that understand this stuff (enough to work with it) seem able to describe these relationships but my understanding diminishes by 50% for each 8 hour period afterword.

Their answer is YES!, I can EQ my little system but I need to take care, I can easily have blood trickling down my cheeks & neck.

Some good news is that the lowest priced cars are gonna be decidedly un-rattly sounding, these NVH guys are discovering/eliminating all the little buzzy things that make Shit Boxes sound like Shit boxes. And: these NVH guys run very similar graphs of Cars to those you run for headphones. They had to keep reminding me that it's not about Audio but rather Physics. Sound guys are Physicists. Get the physics right and the sound will be right. Lots of Math at work here.

One of todays little lessons : lowering the resonance point requires either adding mass or adding dampening or both!

I hope you're not gonna make us take a test to maintain our site subscription, I'll have to pay one of the NVH guys to take it for me.

Tony in Michigan

xnor's picture
Tyll, in the 90 dB SPL measurements it seems to be the case that in some (many?) measurements the THD+N is limited by some noise source to roughly 0.3%. Let's take the Focal Spirits as an example. Besides the observation that peaks in the frequency response will result in higher SNR in the THD+N plots, why is the THD+N line so high at 1k Hz? They all seem to reach the magic minimum of 0.3% at 200 Hz, so I guess the 90 dB SPL calibration is not done at 1k Hz?
xnor's picture
They reach the minimum at about 120 Hz, not 200 Hz.
Tyll Hertsens's picture
I'll see if I can take a noise only measurement to find the noise floor.
xnor's picture

Thanks. There does seem to be some hidden variable. Maybe it's the time of day you make the measurements. Maybe some mains hum that changes with the load/conditions. Maybe the speaker in the box produces noise when it is on and shuts off automatically after a few minutes...

Three Toes of Fury's picture

Great write up Tyll.....the example transfer curves are immensely helpful in explaining the details of harmonics.

Keep up the great edumakation!

Peace & Living in Stereo

3ToF

(PS: goes without saying, but ignore the trolls, lots of us out here in headphone land absolutely appreciate your knowledge transfer. Folks are welcome to offer constructive feedback or alternate thoughts on a subject but it should be done so constructively)

miled's picture

This is an awesome answer to a question I had over on /r/headphones a couple days ago but couldn't find any info on:

https://www.reddit.com/r/headphones/comments/2ywqcc/do_headphone_drivers...

castleofargh's picture

can't wait for the next one. when getting bigger, that series will become a very nice place to point to instead of repeating the same things again and again on forums ^_^.

maybe insist on the fact that clipping isn't the normal behavior of an amp? because what I read as "tubes when pushed too far, will sound nicer" which in my book translates into "I might not even realize I need another amp for those cans when I use a tube". I'm afraid others may read it as "tubes rox". maybe insist on the fact that there is no reason why any amp should be pushed into clipping outside of uninformed purchase for a too demanding headphone pushed loud.

gatucho's picture

Hi Tyll. Is there a reason for the THD to increase practically linearly (in the log scale) in lower freqs? This seems rather artificial to me.

xnor's picture

With lower frequencies the driver needs higher excursion. Max excursion is very limited for some headphone drivers. Halving the frequency theoretically requires 4 times the excursion.

Tyll Hertsens's picture
My understanding is headphones work as a pressure transducer and doesn't follow the same rues as speakers. Anyway, I'm sure you've got a decent grip on these things, just wondering if you're sure about your number there. And might the rules be very different between sealed and open ear cup designs.
xnor's picture

You are right of course. The reason we see the increase in distortion is much more complex. I just wanted to show a simple linear relationship between halving the frequency and increasing distortion.

Yeah, the number is actually that of a sealed/infinite baffle speaker. A headphone obviously usually works by some level of seal between eardrum and driver, so this number will be different.

Still, an exponential rise in distortion (vs linear decreasing frequency) is not something out of the ordinary.

Tyll Hertsens's picture
I'm thinking you get an ever rising distortion when it's getting excursion limited, and a more humped response (goes up and then levels off) when you're being limited by leakage....or something like that...and most likely a mix of both.
xnor's picture

Yeah, less seal means more excursion.
The surrounds of the diaphragms will therefore exert a stronger opposite force trying to re-center the diaphragm.
If this "suspension" is not symmetrical then you will usually see more 2nd order distortion, because the large excursion in combination with this suspension will introduce a DC component in the diaphragm position.
There's more distortion coming from the off-center position of the voice coil, the moving diaphragm experiencing resistance from the air, etc.

gatucho's picture

Thanks, that's a good explanation of the exponential response. But,isn't the actual distortion behaviour too close to the ideal?

gatucho's picture

I mean too close to the ideal to suspect that something else may be at play. Normally when the experimental results follow the theory THAT close, there is a very specific reason; specially when dealing with what would otherwise be a very complex mechanical system. I would only expect this kind of "too nice" behaviour from electrical systems, for example.

Pages

X