Presto Change-o: Measurement Transformations

Back to Basics
In my SOL Republic Tracks review I made the statement, "Even though the 300Hz square wave is severely misshapen due to the mid-range notch..." which stimulated a short dialog in the comments section. It's a terribly interesting subject, and I'd like to amplify on it a bit.

Serious_Listener responded:

"But the notch is in the frequency response--that is, amplitude relative to frequency. Square waves don't aren't a measure of amplitude; they indicate waveform--the characteristics of wave we're hearing. One is a measure of quantity, and the other is a measure of quality or specific features, without regard to quantity. They simply aren't related."

Then Ultrabike chimed in:

"Think of the headphone as an acoustic filter whose frequency response is shown in Tyll's plot. It has a severe notch that covers roughly 200 to 700 Hz. This means that sound around those frequencies will be attenuated relative to the frequencies between 20 to 200 Hz and 700 Hz to 1kHz. Things are a bit attenuated above 1kHz (relative to the lower frequencies - hence a bass heavy can)

"The square wave is used to excite the headphone. A 300 Hz square wave will have a fundamental at 300 Hz, and odd harmonics decreasing in magnitude by 1/n where n is the nth harmonic. This means that the 600 Hz harmonic will be severely attenuated by the headphone."

The point of Ultrabikes comments above is that the frequency response and square wave response are indeed related. Because a square wave contains many frequency components in precise ratios, the shape of the square wave will change with changes in frequency response. The same thing is true of the impulse response.

Why bother with multiple measurements then? Well, the problem is that the data measured lives in three domains: time, frequency, and amplitude. Any single two axis (two domain) graph will only visually reveal information in the two displayed axis, and will hide information in the remaining domain. You can think of it sort of like the image of the shapes at the top of this page. When looking at the solid at the left of the image from the top, front, or side, you'll see a circle, triangle, or square. But the actual solid itself is something quite different than a sphere, cube, or cone. When we look at headphone graphs, we have to look at a number of differently measured characteristic plots and mentally integrate them to get a full picture.

For example, frequency response shows information in the frequency versus amplitude domains. Impulse response shows only time versus amplitude domains. I wish I could measure the acoustic phase response because then we could observe phase (time shift) versus frequency. (I currently measure the electrical phase on the impedance plot, which has little to do with acoustic phase. Acoustic phase is very difficult to measure as there's also a time delay in the loop between the driver in the ear that would have to be compensated for each headphone individually.)

Square wave response is a mix between time, and both amplitude and phase. Both frequency response and phase delays will have an effect on the square wave shape.

So, if the information contained in the frequency response measurement is close to the same thing as the square wave response, we should be able to predict what the square wave looks like from the frequency response plot, right? Abso-friggen-lutely! And Head-Fi member Soaa- has done exactly that!

His amazing graphs on the next page ...


ultrabike's picture

Wow! Caught by surprise here! I'm very honored to be mentioned in one of your articles Tyll :) I've been an avid reader and admire your work.

I feel that in general it is possible to obtain the frequency response phase from the impulse response. However, the devil is always in the details. Linear phase (time delays) are usually viewed as desirable. It may be possible to "calibrate" this for visualization...

Once again, very good job!

13mh13's picture

Tyll ... not sure why you didn't mention the stuff on HR:
I take it you wrote a lot of that....

You use a lot of proxies (supporting content written/noted by others) to support your position ... but humans are very keen to notice this stuff (evolutionary psychology).
With sqr waves, I think you make solid arguments (I have no effing clue what Serious_listener REALLY is trying to convey)
You should have more confidence in yourself. Take a Tony Robbins course.
Stuart Smalley has even better advice:
So, have more confidence, and say:
"I'm good enough. I'm smart enough. And doggone it, people like me."

You can use proxies, but they work AGAINST you in most cases.


mikeEN's picture


I'm a great fan of your site. Keep up the good work.

In this piece you mention phase or time-delay-distortion as a potential problem you'd like to be able to measure.

But surely acoustically based phase errors most be exceedingly small when it comes to headphones. Most are single unit designs with a fairly fixed listening distance and physically very small transducers.

Most of the potential causes of phase distortion we know from traditional loudspeakers are absent, - or am I missing something?


zobel's picture

I had the same question Mike. I think acoustic phase (or group delay) would be very linear with single driver headphones. The acoustic center of the drivers varies only slightly with frequency, and there is no phase shift due to crossover filters.

I had another question. How would one know if the spike (or overshoot) at the start of the square waves is a result of reflected sound from the ear, or actual driver overshoot?

ultrabike's picture

I'm not certain, but I have a feeling that the frequency response phase might get bumpy and weird on the high frequencies of most headphones... probably more so on closed headphones...

Not sure about how to differentiate between driver overshoot and reflected sound from the impulse response alone though... as long as things are approximately linear things are all lumped up there.

AstralStorm's picture

The best plot in fact is a full range waterfall impulse response plot (properly called Cumulative Spectral Decay plot).
Shows energy at given frequencies over time.
(not frequency response, that would make no sense)

Here's how it looks:
Standard frequency response graph should be a almost-steady state version of it - e.g. average of white noise over time.

It is more important when measuring headphones (esp. large) than IEMs, since they can actually retain some sound-wave energy and resonate it back later. Yet more important for speakers, but hard to get reproducible results.

The only other plot you may need is perhaps a phase and resistance diagram.

adranaflores's picture

Great work. Keep sharing with us.

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