Acoustic Measurements: Understanding Time and Frequency

[amirm]

Modern computing power has put in our hands incredibly powerful measurement tools. Prime example is Room EQ Wizard (REW), a free program which is an amazing toolbox of acoustic measurements. Alas, while computing power may be free enabling us to run such analysis that used to cost a lot of money in dedicated hardware, the fundamentals of what the tool measures must still be understood. Otherwise, it is exceedingly easy to arrive at the wrong data and worse yet, wrong conclusions about the science.

The purpose of this article is to demonstrate a key element: the relationship between time and frequency resolution. This is an underlying signal processing concept that is core to functionality of everything from audio and video to compression to how we measure room acoustics. Alas, while the concept is rather simple, it is not intuitive nor is it talked about much outside of the circle of industry researchers and professionals. Simply put, time and frequency resolution are enemies of each other. If attempt to get a lot of frequency resolution, you lose time resolution and vice versa. Oh, please stay with me . This will get interesting in a hurry as we apply it to electronic correction/EQ of the sound in our rooms.

When we look at bass response of our room, often we find massive peaks and valleys. These are caused by so called room modes and we could see swings of 20 or 30 dB (every 10 dB means 2X louder perceptually). We clearly hear the frequency response variations. These room "resonances" have another manifestation. They create "ringing." Ringing is a time domain phenomena where an impulse -- an infinitely short pulse with infinite amplitude -- continues to live on in the room even after it has vanished. The lingering effect of a low frequency note after it has gone away from our source causes the rest of our music/video soundtrack to sound bloated and boomy. Bass notes are not sharp and distinct but seem to last long and overwhelm the higher frequency tones.

An assertion by some in acoustic circles is that you must correct the time response in addition to frequency response. Some will say that this is only possible using acoustic material and not electronic means. As a matter of mathematics, this can't be right. What causes ringing is the room resonance. The very same resonance will also increase the amplitude at that frequency. Fundamentals of signal processing fall apart if what these people say is true. For their proof, they present acoustic measurements that seemingly show that correcting frequency response did not improve the time domain ringing. Let's see if we can duplicate their work and demonstrate where they have may have gone wrong and in the process cover the time and frequency relationship.

For this exercise, I took a subwoofer and stuffed it in the corner. I shut off the crossover and ran a sweep up to 200 Hz. Since the topic of interest here is both frequency smoothness and time domain ringing, we invoke the "waterfall" display in REW. Here is what I got:

The graph shows at time zero the original signal and its frequency response and then in successive "slices" moving forward in time it will perform the same analysis showing that frequency response. It will keep going until the limits of the graph setting are reached which in this case is 300 milliseconds or 1/3 of a second. I suspect other than noticing the non-smooth frequency response in the original signal in the back, you are scratching your head as to what else this graph means. It sure is pretty though . But it is not giving us much insight. The problem is one of settings for that measurement and graph. Let's dig into them.

The Noise Floor
The biggest problem here is that we have not paid attention to the vertical scale. This axis shows the amplitude of the sound in the room. Lowest level is 15 dB SPL and highest is 95 dB SPL. The max is OK. The problem is the min. You can't just pick a random number here. There is a minimum below which we are just measuring the noise in the room which in the case of low frequencies can be a lot. See my article on room noise: http://www.madronadigital.com/Library/RoomDynamicRange.html.

Let's put this to practice. Here is the same REW waterfall display, this time with me turning off the output so that nothing was playing:

We see noise floor that reaches up to 45 dB, a whopping 30 dB higher than what we were measuring before! So a bunch of data in the previous graph was simply noise that is independent of what our system is doing. Let's overlay this noise on top of our sub response and see the two together:

Let's correct for that by raising the bottom level of our measurement to top of the noise floor. We do this simply by sliding the scroll bar or setting the limits in REW for our graph. While we are at it, let's perform some other changes. For one, let's re-orient our viewpoint onto the 3-d graph to make it better to see the time response. Hit "Controls" button on top right and change X to 21, Y to 122 and Z to 50. This is what we get now:

I am liking this better already! We have a much clearer picture since we got rid of the noise. We see the frequency response changes and with it, the associated time domain activity. The peaks in the amplitude of the frequency response are accompanied by time domain forward/back continuation of signal.

Time and Frequency Domain Resolution
As I mentioned, the core of this article is to talk about how these two factors are enemies of each other. We can easily demonstrate this by changing the "Window" parameter in REW. The Window sets the time window or put another way, the number of audio samples uses to determined the frequency response. If you set this to 1 second, then you get a resolution of 1 Hz. If you set it to 0.1 seconds, then the resolution jumps to 10 Hz. Let's see what happens when we set this to 40 milliseconds for a 25 Hz frequency resolution:

We see a huge change in the graph. The curves left to right have become very smooth and a lot of the ups and downs have disappeared. The reason? We have lost frequency resolution. As stated, we now are dealing with chunks of 25 Hz at a time. This is way, way too big for analyzing bass frequencies because room resonances are much smaller than this. So let's optimize for frequency response by going the other direction. Here is what it looks like with 300 milliseconds window and 3.3 Hz resolution which by the way, is the default for REW:

This brought back our frequency ups and downs but notice what it did in time domain. Look at how the curve front to back for any one resonance peak has a huge hump. Just like what we had in the other direction before. Here, we know the offending frequencies but no longer know what they are really doing in time domain. I.e. determining ringing is more difficult since time domain response is distorted.

We therefore see that we cannot have high resolution in both time and frequency domain. One comes at the expense of the other and there is no way around it.

Optimized Display and Filtering
For this test, the best settings I found were 400 milliseconds for the total "Time Range" or total display and 200 milliseconds for time window. 100 milliseconds also works well but values above and below this range just cause too much distortion. I eyeballed a frequency response peak and picked 53 Hz as the first frequency to go after. I put in a parametric EQ at that frequency to pull it down and the frequencies around it. Here is what we get with the new settings and filtered response, overlaid on the non-filtered measurement.

What do you know? The math worked! I put the cursor at 53 Hz (blue line) and you can easily see that time domain ringing has heavily subsided at that frequency. The rest remains there because I have not touched them. BTW, a key note. If you change the Time/Window setting in one graph of REW and attempt to overlay another, you have to go to the other graph and apply the same settings. Otherwise, REW will happily mix the two graphs even though they were not analyzed the same.

Let's add another filter at 140 Hz:

The math works again with the both the frequency response peak going down and time domain ringing reduced.

Oh, do not be alarmed if you see some extra bits pop up in the noise floor of the system after you make changes. This is a much more advanced topic but the analysis of the system in slices can cause errors in measurements due to truncation of samples at either end. Focus on the big picture here, pun intended .

Let's go all out and add a few more filters:

We see across the board improvement in time domain and frequency domain. In practice, we would only care about response to ~100 Hz as the rest would start to get filtered by the crossover. So that sharp drop around 100 Hz is a much smaller problem than it seems.

Subjectively the bass response became all that I said at the start. I played a string track with I could now hear every one plucked even though all I was listening to was the sub! Before one pick would result in a lot of boom and the strings would all run into each other as they were played. So clearly we had made improvements in time domain. Overall bass level however seemed low. This was fixed by a boost of the entire sub by a few db. I know had the impact but kept all that was clean about my optimization. Considering that I was just playing to create these set of scenarios, these are pretty encouraging results.

For grins, let me show you how not to do this. Here is the overlay of the 53 Hz correction over no correction but with levels wrong:

Looking at this, it is very easy to conclude that we did not correct anything in time domain. It seems like a jumbled mess before and after. But per above, this is the wrong use of the tool. Set the levels right and you too can be in good hands of science .

 

[tomelex]

nice, very nice, this is a big part of what is great here at ASR, thanks man.

 

[amirm]

My pleasure Tom. I have more articles that I need to upload....

 

[Linearphass]

nice, very nice, this is a big part of what is great here at ASR, thanks man.

Having worked for Tektronix for 31 years in engineering, sales and customer training I can agree with the notion that most people do not use the tools correctly. Most common are incorrect use of the dynamic range of the instrument, incorrect scaling, incorrect probing or in the case of acoustic measurements mic positioning, etc. in short garbage in garbage out.
REW is a wonderful tool and it gets better daily. But most people use it totally incorrectly and this is made far worse by the fact that they do not understand how to interpret the results correctly.
I appreciate you efforts here to help correct these misunderstandings.

 

[AudioJester]

Having worked for Tektronix for 31 years in engineering, sales and customer training I can agree with the notion that most people do not use the tools correctly. Most common are incorrect use of the dynamic range of the instrument, incorrect scaling, incorrect probing or in the case of acoustic measurements mic positioning, etc. in short garbage in garbage out.
REW is a wonderful tool and it gets better daily. But most people use it totally incorrectly and this is made far worse by the fact that they do not understand how to interpret the results correctly.
I appreciate you efforts here to help correct these misunderstandings.

Perhaps with your knowledge and expertise you could put something together about mic positioing and measuring for the amateur hobbyist?

 

[Linearphass]

Continuing one of the most incorrectly used is the ETC feature.
The comment that time and frequency are interconnected is absolutely true. They are two different domain views of the same data.
The statement
An assertion by some in acoustic circles is that you must correct the time response in addition to frequency response. Some will say that this is only possible using acoustic material and not electronic means. As a matter of mathematics, this can't be right. What causes ringing is the room resonance.
Is confusing. While minimum phase peaks can be corrected by equalization non minimum phase dips can not. Those can only be treated by acoustic absorption. Further if an anomaly is minimum phase correcting the frequency response error will automatically correct the phase error ( time domain) associated with it.

 

[Linearphass]

Perhaps with your knowledge and expertise you could put something together about mic positioing and measuring for the amateur hobbyist?

I have considered this in the past. Indeed I had started the creation of such an article. One of the difficulties associated with doing this is the complexity of the process particularly when combined with huge variations of the acoustics of the room one is trying to measure in and the knowledge level of the users. For example I have helped at least ten people design, build and set up systems in the last two years. Initially I had to be present in person to help them be successful. Now we are largely able to do it over the phone and internet. Usually this is very time consuming. It takes around at least 100 hours to get the system close to where it should be.
Out of the ten systems only 1 of the users is able to repeat the process by himself with only occasional help and guidance. The other 9 require that I basically do the process myself and then tell them what to correct.
What I am trying to say here is a little knowledge is a dangerous thing.

Do you have specific objective you are trying to accomplish?

 

[Linearphass]

I might add that I now rely one the use of active speakers only. We are using only DSP crossovers, EQ, phase compensation, etc.
I designed passive speaker from 1969 (high school) thru 2004. Once I switched to using DSP with each driver driven directly by it own amp I could never go back. Passive crossovers are at best a significant compromise. They also take a Lot longer to optimize. DSP by comparison is child’s play.
In the last two year I have learned a lot about room acoustics that I never knew before. Much of this was learned the hard way by trial and error supplemented by lots of study. I now firmly believe that not everything can be fixed with EQ alone.

 

[DonH56]

The main problem with "EQ alone" IME is that phase is neglected and/or corrupted. Time and frequency domains are ways to show the same response, but frequency magnitude alone is not sufficient. Introducing filters often significantly changes the phase so with steady-state test signals you can get "good" amplitude response and still not have good time response. Observing phase (or perhaps group delay, which is what I tend to check) as well as magnitude is required if you want both.

 

[Linearphass]

If the response error is non minimum phase you are correct. If it is minimum phase when you correct the magnitude errors you correct the phase errors at the same time. So how do you tell? The easiest way it to look at Group delay or rather EXCESS group delay. The portions of the curve that are constant (flat) are minimum phase portions and can be corrected. However phase, and the derived group delay are VERY DIFFICUT TO MEASURE IN A LIVE ROOM. If the room is very dead and the speakers have very good controlled directivity, like some horns, you can measure phase. But with typical speakers this is tough to measure unless you go outside or stack all sorts of absorbers up on the floors and towards the walls and measure close. Phase is very sensitive to any reflections. If you use gating to remove reflections then you can not measure low frequencies which is where you generally want to correct the most.
It goes without saying to do any of this require that you must Generate minimum phase curves and Estimate IR delay for EACH trace that you analyzing before you make any GD measurements.
I suppose one way to do this is to measure with the speaker elevated in a very large room and then use gating to remove the longest reflections.
You might be able to get down below 100Hz with the earliest reflection path length 10mS or greater than the direct arrival.

 

[mikewxyz]

There are a number of folks who contend that fixing problems in the time domain fixes things in the frequency domain. They focus a lot on ETC graphs in an attempt to reduce the Hass effect. There is a member here on ASR (if he is the same person who posts on Gearslutz) who is more than competent in explaining this technique. I’ll admit that he often talks over my head although I got out my string and checked a few nearby objects.

 

[QMuse]

Phase is very sensitive to any reflections. If you use gating to remove reflections then you can not measure low frequencies which is where you generally want to correct the most.

Instead of gating one should use frequency dependent windowing which eliminates reflections at LF as well. Measuring at closer distances will give cleaner initial response less affected by reflections.

 

[Pio2001]

Instead of gating one should use frequency dependent windowing which eliminates reflections at LF as well. Measuring at closer distances will give cleaner initial response less affected by reflections.

Frequency dependent windowing doesn't eliminate all reflections.

Since sound power is cumulative, the more reflections are removed, the closer you get from the amplitude vs frequency response of the speakers alone. There is a direct relationship between the amount of reflections taken into account in the measurement and the amount of deviation of your curve vs the anechoic response of the speakers. Therefore frequency dependent windowing has a meaning (although I've never used it and I don't see what I could do with it).

But what about phase vs frequency ? If only one reflection can wipe off completely any information about the anechoic phase response of the speakers, then removing nothing (no windowing) or removing 90 % of the reflections (frequency dependent windowing) won't make any difference.

 

[QMuse]

Frequency dependent windowing doesn't eliminate all reflections.

Did I say it eliminates ALL reflections? What I said was that FDW eliminates enough reflections to be able to identify phase response and correct it.

If only one reflection can wipe off completely any information about the anechoic phase response of the speakers, then removing nothing (no windowing) or removing 90 % of the reflections (frequency dependent windowing) won't make any difference.

Is that so? I don't know where did you get the idea that one reflection can wipe off phase response. Btw, you cannot measure anechoic phase response in room. Room equally affects phase and frequency in range below 300-400Hz so phase measured in room will be quite different from the phase measured anechiocally.

Let's see practical example. Can you notice the difference between these 2 phase graphs (same response taken from LP, 4m from the speakers, both with no IR gating) or there is no difference?

No FDW:

FDW of 3 cycles:

 

[Linearphass]

Did I say it eliminates ALL reflections? What I said was that FDW eliminates enough reflections to be able to identify phase response and correct it.




Is that so? I don't know where did you get the idea that one reflection can wipe off phase response. Btw, you cannot measure anechoic phase response in room. Room equally affects phase and frequency in range below 300-400Hz so phase measured in room will be quite different from the phase measured anechiocally.

Let's see practical example. Can you notice the difference between these 2 phase graphs (same response taken from LP, 4m from the speakers, both with no IR gating) or there is no difference?

No FDW:

View attachment 67089

FDW of 3 cycles:

View attachment 67090

I will have to try that. I have messed with FDW before but never for phase. In general I have tried to fix the speaker phase response anechoically before dealing with room issues. It appears that the ears and brain are able to deal with some degree of room reflections and the effected they have on magnitude and phase response. That having been said the best system I have heard to date controlled reflections by judicious use of absorption and some diffusion. Very effect was measure and the treatment designed and adjusted based on the measurements. I am not able to do this by ear alone.

 

[Pio2001]

Did I say it eliminates ALL reflections?

You didn't.

What I said was that FDW eliminates enough reflections to be able to identify phase response and correct it.

You mean the phase response of the speaker alone along the axis that goes through the speaker and the listening position ?

Why would one want to correct this ?
At the crossover frequency, phase distortion affects the spinorama. It should be dealt with amplitude correction. You can't correct separately the phase of the tweeter and the phase of the woofer anyway.
Outside the crossover frequency, phase distortion consists in minimal phase + excess phase. If you correct both, you are going to ruin the sound of the system with introduction of artificial pre-ringing.
Now if you only deal with excess phase, it should be inaudible anyway, except in room modes.

I see only drawbacks and no benefits.

Is that so? I don't know where did you get the idea that one reflection can wipe off phase response.

I don't know. It just seems logical. If a reflection comes 1 ms later than the direct sound, at 1000 Hz, it comes with a 360° phase shift. The combination of the direct signal and its reflection may result in any phase value.

In order to evaluate the effect of early reflections on phase response, we must compare the anechoic phase response of a speaker with the FDW filtered phase response of the same speaker, measured in room.

Let's see practical example. Can you notice the difference between these 2 phase graphs (same response taken from LP, 4m from the speakers, both with no IR gating) or there is no difference?
No FDW:

FDW of 3 cycles:

The distance is 4 meters. Which means that, if the speakers and listening positions are both 1 meter above the floor, the first reflection on the floor will come 1.4 ms after the direct signal.
On the second graph, since the gating is 3 cycles, it means that this reflection is removed from the analysis above 2000 Hz, and included below. But we can't tell about reflections on surfaces close to the listening position, close to the speaker, or intermediate surfaces less than 1 meter from the speaker-to-listener axis.

Therefore, without more detailed information, we can only guess what the graph represents. We can for example make the hypothesis that the phase variations that we see below 1000 Hz are caused by the room. However, they would only be a partial measurement. The whole true phase response at listening position is given in the first graph.

What we see in the second graph is not the phase response at listening position, nor the phase response of the speaker alone, except maybe above 1000 Hz, where it seems regular.
We also don't know if the tweeter is facing exactly the listening position. If not, maybe it is the cause of the drop above 5000 Hz.

That's all I can tell about these graphs.

 

[QMuse]

You didn't.



You mean the phase response of the speaker alone along the axis that goes through the speaker and the listening position ?

Why would one want to correct this ?
At the crossover frequency, phase distortion affects the spinorama. It should be dealt with amplitude correction. You can't correct separately the phase of the tweeter and the phase of the woofer anyway.
Outside the crossover frequency, phase distortion consists in minimal phase + excess phase. If you correct both, you are going to ruin the sound of the system with introduction of artificial pre-ringing.
Now if you only deal with excess phase, it should be inaudible anyway, except in room modes.

I see only drawbacks and no benefits.



I don't know. It just seems logical. If a reflection comes 1 ms later than the direct sound, at 1000 Hz, it comes with a 360° phase shift. The combination of the direct signal and its reflection may result in any phase value.

In order to evaluate the effect of early reflections on phase response, we must compare the anechoic phase response of a speaker with the FDW filtered phase response of the same speaker, measured in room.




The distance is 4 meters. Which means that, if the speakers and listening positions are both 1 meter above the floor, the first reflection on the floor will come 1.4 ms after the direct signal.
On the second graph, since the gating is 3 cycles, it means that this reflection is removed from the analysis above 2000 Hz, and included below. But we can't tell about reflections on surfaces close to the listening position, close to the speaker, or intermediate surfaces less than 1 meter from the speaker-to-listener axis.

Therefore, without more detailed information, we can only guess what the graph represents. We can for example make the hypothesis that the phase variations that we see below 1000 Hz are caused by the room. However, they would only be a partial measurement. The whole true phase response at listening position is given in the first graph.

What we see in the second graph is not the phase response at listening position, nor the phase response of the speaker alone, except maybe above 1000 Hz, where it seems regular.
We also don't know if the tweeter is facing exactly the listening position. If not, maybe it is the cause of the drop above 5000 Hz.

That's all I can tell about these graphs.

I'm affraid you don't get that using FDW on sine sweep measured at LP is the only way to correct phase of the loudspeaker. Very gently, of course, to avoid pre-ringing. That same algorithm is used by virtually every automated EQ software on the market as there is no other way to do it.

You can verify the results very easilly by taking measurements close to the speaker (say at 1m) where reflections would affect the result much less and you wills till get the same phase curve.

Fixing the phase shift caused by passive XO is the main target here but 2nd target is to make phase response smooth to reduce GD and to make excess phase as close to zero as possibe over the entire frequency range.

Here you also don't seem to understand that excess phase is nothing but the difference between actual phase and minimum phase response. By correcting actual phase response you can get it close to minimum phase so that excess phase is close to zero over the entire frequency range.

And finally - no, correcting the phase response doesnt' affect frequency response at all so it doesn't affect spinorama.

 

[Linearphass]

Did I say it eliminates ALL reflections? What I said was that FDW eliminates enough reflections to be able to identify phase response and correct it.




Is that so? I don't know where did you get the idea that one reflection can wipe off phase response. Btw, you cannot measure anechoic phase response in room. Room equally affects phase and frequency in range below 300-400Hz so phase measured in room will be quite different from the phase measured anechiocally.

Let's see practical example. Can you notice the difference between these 2 phase graphs (same response taken from LP, 4m from the speakers, both with no IR gating) or there is no difference?

No FDW:

View attachment 67089

FDW of 3 cycles:

View attachment 67090

On the bottom graph don't you want to use excess phase instead of phase? This assumes you are trying to build a minimum phase vs a linear phase system.

 

[QMuse]

On the bottom graph don't you want to use excess phase instead of phase? This assumes you are trying to build a minimum phase vs a linear phase system.

I wanted to show how phase changes when FDW is applied. Here is the same graph with minimum phase and excess phase shown.

From LP (same as previous):

Same thing, but measured 80cm from the speaker:

 

[Linearphass]

Very nice, can you do group delay the same way?