what information an impulse response graph provides about headphones?Submitted by MEGA_AEOIU792 t3_10kgr8r in headphones
audioen t1_j5r8ty5 wrote
It is actually the same information as the (complex) frequency response. This is just the time-domain representation of it. More specifically, the Fourier transform of the impulse response is the (complex) frequency response, and the inverse Fourier transform of the (complex) frequency response is the impulse response.
I guess usual smoothed magnitude spectrum has elided phase information, while phase information is in some sense still visible in the impulse response. It is thus the more complete record of the system's behavior. That is why I put the word "complex" in parenthesis above, it means that the ratio of the imaginary and real part of the complex number gives the phase angle. In my opinion, the phase should be processed to group delay plot which shows how much the system delays sound across the response's frequency range. I agree in that I don't think the raw impulse response is easy to read at all.
Group delay is not often an interesting plot with headphones because they are not supposed to have much group delay to begin with. Group delay is more of a property of electronics and digital filters. However, sometimes group delay plots of headphones have big spikes that show that phase is wildly inconsistent at some frequency, and this is often something like a resonating structure in the headset cup. It would also be apparent in frequency response as a narrow spike at that location, but often frequency response plots are heavily smoothed which hides these defects.
As an example, Hifiman Ananda has something wrong in its group delay plot: https://www.audiosciencereview.com/forum/index.php?attachments/hifiman-ananda-group-delay-measurements-open-back-planar-headphone-png.122835/ -- the plot can be quite noisy which probably comes partially from how the headset sits on the fixture and how reflections go inside the headset cup. However, curve fragments going up and down all over, especially in low frequencies above 200 Hz just isn't normal. So this is example of headset with messed up phase that indicates a sound quality problem.
No-Tune-9435 t1_j5rzc8c wrote
This is an online audio fallacy / myth. Time domain and frequency domain are only equivalent if both are infinite.
I wrote out a longer response in replay to a different comment, but an easy way to see the issue with what you’re starting is to ask how bass response would appear on the above chart (which is completely flat past ~0.002 seconds. One cycle at 200 hz is 0.005 seconds. One cycle at 20 hz is 0.05 seconds! How could you possibly infer anything about the bass response of that unit from that graph? How then can you say that graph is telling us only and exactly what the FR is?
If we want to get real technical, you’d also have to address how certain time domain translations do NOT alter the frequency domain (see shift property of the Fourier transform). That is, time delays do not alter the frequency domain. Relative timing information is very likely lost due to these two effects (representing the freq domain on a finite spectrum and not accounting for time delays).
Please stop propagating this misunderstanding that time and frequency are 100% equivalent
audioen t1_j5soc56 wrote
I think you are just wrong, and you do not seem to know what flat frequency response looks like in impulse response graph. It is one sample long singular spike, followed by perfect silence afterwards, forever. You could say that studio monitor speaker systems strive to reproduce just such an impulse, and the closer it is to a very narrow spike, the better the acoustic system. Even this headset looks like it is not that far from perfect impulse apart from some ringing afterwards which suggests it has some resonance peaks and probably highpass filtering because the impulse goes below zero, and that would indicate it cancels some of the sound it produced earlier after a time delay, which is how highpass filters generally work. I don't know, it is really hard to try to read the frequency response off impulse response.
It is completely obvious to me that time delays do not have impact on the magnitude spectrum. They have an effect on the complex spectrum because phase (and group delay) are different and are encoded in the ratio of the imaginary and real parts of the complex number which is usually not shown because phase angle is difficult to relate to anything we actually hear. In that case, group delay would show the added fixed delay just fine, though.
I can only assure you that from mathematical point of view, the impulse response and complex frequency response can be converted to each other without loss. Whether Fourier analysis is good model of human hearing is perhaps a thornier question, as this isn't quite how our ears work, but I think it is still plenty useful as a construct.
[deleted] t1_j5sqmgp wrote
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SoNic67 t1_j5tfqxd wrote
Limit the mathematical model only to a couple of terms of the series, and see how "lossless" your transformation is.
TaliskerBay22 t1_j5tvung wrote
I do not understand, your comment can you explain? Only thing I am saying is that the impulse response of the headphone in this graph is known to infinity as the assumption that it will continue to be 0 after the end of the plotted data is certainly a valid one. So we can pad with 0s this graph as much as we want. On the other hand the time 0 in this impulse wrong seems to be arbitrary. If somebody can pass me the data of the graph I can certainly do an FFT and get the Freq response.
SoNic67 t1_j5uqz7w wrote
I am talking about people that say the FFT is equal to impulse response because... math.
FFT implies infinite bandwidth.
TaliskerBay22 t1_j5vuc8d wrote
The maximum frequency of a fast Fourier transform is 1 over the sampling interval or the sampling frequency. As shown in this example https://uk.mathworks.com/help/matlab/ref/fft.html
SoNic67 t1_j5vxaxg wrote
Nope.
blutfink t1_j5v6ctl wrote
> Time domain and frequency domain are only equivalent if both are infinite.
We’re talking about discrete FTs here. There is nothing infinite about those.
> One cycle at 200 hz is 0.005 seconds.
This is completely irrelevant. You can easily pack a signal that contains components below 200 Hz into a time window much shorter than 1/200 s.
If your understanding of the “myth” is based on this flawed intuition, you may want to reconsider your argument.
No-Tune-9435 t1_j5vfwi9 wrote
I’m getting bombarded with a lot of “nuh uh”s, your comment included, despite my mathematical explanation above. Would you care to mathematically demonstrate your claim that they are linked?
blutfink t1_j5vmj53 wrote
What exactly is it you doubt? That the impulse response and the frequency response are linked? This is taught in any undergrad course on the subject. For instance, see this online textbook, last sentence on the page. (“Decayed” here is often named “windowed” in other texts.)
It’s also very easy to convince oneself of the fact. Just fire up MATLAB, Octave, Mathematica, etc. Manipulate a vector to emulate an impulse response, calculate its DFT (typically using the FFT algorithm), then calculate the element-wise magnitude — et voilà, that’s the associated frequency response. If you apply the inverse DFT on the output of the DFT (before you calculate the magnitude), you get the original impulse again.
No-Tune-9435 t1_j5vrj9e wrote
Have you done that with a 20-20k FR and looked at the time domain?
blutfink t1_j5vvd3w wrote
Of course. As in professionally, for decades. Note that for that to work, you need the full, complex-valued FR of a system. The amplitude response does not contain phase information.
That’s why people add the impulse response as supporting information: A plot of the complex-valued response (either 3D or as two real-valued graphs) is not intuitive for humans.
Here is a quick explanation of how the responses are calculated in REW.
No-Tune-9435 t1_j5vwfa2 wrote
Perfect. What ~time gate would that signal have sufficiently decayed? (per your DTFT link)
blutfink t1_j5vz4wj wrote
Proper decaying/windowing is performed via element-wise multiplication with a window function. All we want is that the signal is smoothly approaching zero at the edges. Since this is approximately the case for typical impulse responses of audio systems, it won’t change the result that much if you don’t window it at all.
To help your intuition, convince yourself that the FT of a centered Gaussian bell curve is itself a Gaussian bell curve. Note that the “low frequencies” in this graph are near the center.
No-Tune-9435 t1_j5vzhjj wrote
You didn’t answer my question at all though. If we want to represent an FR from 20-20khz, what time window would be appropriate for the impulse response? Or have you actually not done this math before?
TaliskerBay22 t1_j5w0hju wrote
Look here is an example, the time domain lasts for a ms and the Fourier response extends from 0 to some tens of KHz. Plug it in Matlab and tweak it as you like
% Define time-domain signal
dt = 0.000001; % time step (s)
t = 0:dt:0.001; % time vector (s)
x = cos(50000.*t-0.0004).*exp(-(t-0.0003).^2/0.00006^2); % time-domain signal
% Perform FFT
X = fft(x); % FFT of time-domain signal
f = (0:length(X)-1)/(dt*length(X)); % frequency vector (Hz)
% Plot results
figure;
subplot(2,1,1);
plot(t,x);
xlabel('Time (s)');
ylabel('Amplitude');
title('Time-domain signal');
subplot(2,1,2);
Amplitude=abs(X);
semilogy(f(1:length(X)/20),Amplitude(1:length(X)/20));
xlabel('Frequency (Hz)');
ylabel('Amplitude');
title('Frequency-domain signal (FFT)');
blutfink t1_j5w0fke wrote
In the extreme case? Infinitesimally short time. The FT of a Dirac pulse is a flat, constant response from 0 Hz to infinity Hz (or, in the case of DFT, Nyquist frequency).
As I said, the FT of a Gaussian is a Gaussian. Really sharp in the time domain means really wide in the frequency domain. If you do not understand why that is, your intuition about FTs will be flawed.
[deleted] t1_j5w2sn4 wrote
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blutfink t1_j5w2dc2 wrote
Maybe this helps: The impulse response in OPs post image basically does not have to be windowed at all, it’s decently close to zero at the edges. Plug it into a FT and you’ll get a decently accurate FR.
Physx32 t1_j5sopan wrote
Can you explain what you mean by "time delays do not alter frequency domain". A delay in time domain causes a phase shift in frequency domain. So it does alter the Fourier transform.
SoNic67 t1_j5tfktx wrote
Only if you have infinite bandwidth.
limit the Fourier series to just a couple of terms (2-3 harmonics) and you will see that's not at all relevant.
No-Tune-9435 t1_j5t87ir wrote
Sure, but like you said, it’s a phase shift in the frequency domain. It doesn’t alter the magnitude of the FR spectrum, just the imaginary vs real distribution. For the purposes of measuring audio equipment, we can pretty safely ignore that distinction you raise, and it doesn’t negate my point that time and frequency domain are not equivalent in the OP.
chargedcapacitor t1_j5tbga0 wrote
Ha, found the non engineer. You are most certainly wrong. This is a basic math/engineering principle.
No-Tune-9435 t1_j5urhv7 wrote
Not a fan of the snide remark. Would you care to at least substantiate this “basic principle” with a link, or did you just intend to negate my position via name calling?
SoNic67 t1_j5tfsd9 wrote
Ivory tower.
SoNic67 t1_j5tfgwh wrote
I have argued with people about that. They think that theoretical considerations about Fourier transformations are all they need to understand.
They lack the understanding that real systems are not of infinite bandwidth. Especially headphones.
So that response is more important than they think, and it cannot be derived by just looking at 20Hz-30kHz response.
blutfink t1_j5v8018 wrote
> real systems are not of infinite bandwidth
I don’t see anyone making that assumption here. Could you elaborate?
> that response […] cannot be derived by just looking at [frequency] response.
Of course not from the magnitude response as it is typically plotted. But from the complex-valued frequency response, of course it can, no assumption of infinite bandwidth required.
SoNic67 t1_j5vo2ux wrote
>I don’t see anyone making that assumption here. Could you elaborate?
Every single person that has a fetish about FFT math and assumes it applies in real world 100%.
blutfink t1_j5vp0v8 wrote
Well, I’m allowed to have that fetish. Digital signal processing was literally my job for decades. Let me tell you that the math and the methods very much apply to the real world. That’s how they were conceived; to analyze the filter properties of physical systems.
Rasyad95 t1_j611nft wrote
Your comment makes me assume that you do not understand the use cases of FT. Do you?
NearlyCompressible t1_j5rhwo0 wrote
This is a side point, I agree with everything you've said, but the Ananda does not measure like that when you're getting a proper seal.
SupOrSalad t1_j5rtl56 wrote
Are you claiming that Amir doesn't take extra care to have a proper seal and make sure that variables like pad wear are accounted for? /s
audioen t1_j5sp8ch wrote
Hmm, okay. I couldn't find any other measurement of this headset, so I will leave it up to the air whether it is representative of the headset, then. Certainly the noisy/spiky character of it should be discarded as a measurement artifact, but the other curve parts going up and down in it might be real. The ideal group delay plot is just a flat line at 0 ms that gradually rises towards the bass due to inevitable high-pass filtering somewhere in the amplifier or such.
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