Acoustic Measurements: Understanding Time and Frequency

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 .

terryj said:

thanks amir, and it's good to see some of the little traps exposed and corrected for (noise floor etc)

Is there a second part? I ask cause it somehow feels incomplete or that I have not grasped your essential point.

Or is the essential point that for the most part room treatment is not needed in the bass? Or that it is not really needed for control of ringing in the bass? (maybe based on incorrect graphing as you outlined say) Or that ringing is effectively controlled by eq?

No point in making arguments for or against any of those points-which may be completely clear to others! ie I'm the dense one around here-until it becomes clearer for me.

Click to expand...

This was not meant as a complete article on acoustic. The main point I wanted to discuss was the relationship between time and frequency. I thought that by itself would be too boring so combined it with applying electronic EQ and showing that the relationship between frequency domain and time domains holds in the case of resonances.

Answering your question, if you apply a full suite of tools, you can almost get away with no bass treatment. Example of that is shown in my bass optimization article: http://www.madronadigital.com/Library/BassOptimization.html. Here is the frequency response:



Note that not only is the response smooth but also that way for multiple seats. All without use of any acoustic material. Techniques used there are multiple subs, application of multi-sub optimization by Harman (SFM), and what I have covered here: parametric EQ. Ringing does and will go down with just EQ as my measurements show.

microstrip said:

I fail to see how you can conclude that the time information is wrong just because it has "humps", and specially how after telling that graphs taken with a 300 milliseconds window have distorted time domain response can go on working with 200 milliseconds in the next part of the article.

Click to expand...

You are correct that 200 msec still represents fair amount of error in time domain. You can go further to 100 msec as I suggest. As you do however, you will notice that your frequency response peaks start to get soft and harder to pin point what value to pick for your parametric eq. Truth to be told, I picked the filters at 100 msec when I did the test. In the process of creating these graphs, I started with 200 msec and it also showed similar results so I left it that way.

Ultimately there is no perfect solution here since you want to see the best resolution in both axis. What I have done is to put priority in frequency response because that tells you what filter values to pick.

Another view here is that you really don't need to run the waterfall. You can simply run the frequency response graph and make the corrections there. The fact that ringing/time domain improvements are there is just something that happens.

Apologies for the too basic questions, but although the concept or frequency versus time resolution seems logical, the application is not easy.

Click to expand...

Exactly. As I described above, it is a difficult path as the science is not your friend here .

And a question can arise - using higher sampling resolution we can improve the accuracy of the instrument and reduce the resolution limitation?

Click to expand...

REW won't go past 48K so it won't matter one way or the other here. I have to think about whether as a matter of theory it would help here.

I can't read any of the numbers micro . It appears that you have the low limit set to 0 db SPL. Until you fix that, it is very hard to see what is going on.

terryj said:

Well, you gotta be sure with terminology I guess. Will that frequency ring as long after eq than before? No, after all it has had it's initial amplitude reduced and hence the time to fall to a given db level (noise floor say) will be correspondingly less.

Click to expand...

Well, that "after all" part is precisely what is disputed. From Ethan's web site: "Modal ringing - an extended decay at some, but not all, bass frequencies - is just as damaging as a skewed low frequency response, and EQ cannot reduce ringing. "

Turns out there is more complexity here than meets the eye. Whether correction in frequency domain causes reduction in time domain relies on a system (room) characteristics known as "minimum phase." In a minimum phase system, the phase of the system transfer function is proportional to its amplitude. In layman terms, this means that the amplitude fully describe system behavior and hence, modification of it must by definition also modify its time response since the two define the same signal. They are two sides of the same coin. Research shows that low frequency response of a room is essentially minimum phase but not so necessarily above transition frequencies (200+ Hz). So it does not follow that in all cases you can change the the amplitude and get with it, reduced time domain ringing.

Gods of acoustics are kind to us as it seems that above transition frequencies we don't hear time domain ringing and what matters is simply frequency response variations. In low frequencies we do and there, minimum phase characteristics helps us in low frequencies where it is audible.

I feel that is a bit loose in terminology tho. 'Even after reducing the peak at that frequency, is it possible it will still take longer to decay into the noise floor than the frequency an octave above (say)?' Again, the answer is 'entirely possible'. THAT is more what is meant by ringing I feel, an uneven decay rate due to the room. It is of current interest to me personally, and we looked at this in some detail at the race gtg last weekend. My FR is as flat as any sane person could demand, and in MOST cases it sounds that way. But I have been aware for quite a while now that often it STILL sounds 'boomy' at times, prob not the best word but it will do for now.

Click to expand...

Oh, I think I get it now. You are saying you did not effect enough correction of low frequencies so you still hear the boominess over the higher frequencies? If so, that is still possible as I doubt that you have textbook response in your room. Likely there are still variations and there is always the bit about level of low frequencies to highs. And ringing as you later say.

Still, if this thread takes off and it looks like a bit of 'learnin is afoot'-always a great thing to happen-I could prob find a usb stick or sumthin.

As always, very much enjoy it when you dig in and look at things, thanks.

Click to expand...

I guess I should have focused this post entirely on acoustics rather than the signal processing concept alone . To help with that, here is a different way of presenting these measurements as performed by Dr. Toole in is book:



We see that application of EQ did indeed reduce ringing in time domain (one is a coarse EQ and the other, fine)..

microstrip said:

Curious that using Firefox in an HD monitor all numbers can be read.

Click to expand...

I am using my laptop and is not visible in my small screen.

I have now set the low limit to -20 dB. But the main difference at 3.3 Hz resolution is a difference in decay of 5dB at 18 Hz and 15 dB at 36 Hz (cursor position) after 600ms .

Click to expand...

-20? You want to go the other way around. See what I explained regarding room/system noise. Likely you should set the floor to 30 to 40 db (positive). That will get rid of what is in the front of the graph and let us properly see the decay into the room noise. Please also apply X, Y and Z params as I suggested and then let's take a look.

Raffles said:

You could make the observation that for the first few milliseconds of a transient sound reaching the listener's ear, it is unaffected by the room response. If you were to apply simple EQ to the source signal, this transient would be EQ'd (unnecessarily and therefore wrongly) before it reached the listener's ear, but with time-domain correction it would not.

Click to expand...

The thing about a resonant system and in this case, modal ringing is that even if the signal went away, the room would continue to play it. this is why an impulse response which in theory has zero duration, still lives on in the form of ringing. Think of a pendulum that tap to start and and it keeps going. The room is storing the energy, even if excited for a very brief moment, and is giving it back in time. We could do this test with an impulse which by definition has zero time and the room would still do what it does and we will hear its effects.

The resonance and lingering energy can add to the input signal and hence modify the next sound to come. We counter this with EQ by bringing the level down, inversely proportional to how much the resonant signal amplified it and with it, remove the its effect at that range of frequencies. In this regard, we are backing out the effect of the room and not modifying the signal in some random way.

It seems to me that the 'frequency response' of a room only applies to steady test tones, and that rapidly changing signals (e.g. most music) are only subject to a fraction of the huge peaks and valleys that show up in the frequency response measurement (which is why most rooms don't sound as bad as the FR suggests); it's only when you get a long, steady note, that the real boominess or ringing shows up.

Click to expand...

Well, you can perform the same test using an impulse which as I explained has zero duration yet it will still show ringing. In modern world we don't use impulses because they need to be very loud to approximate the real thing and so are unpleasant and can damage equipment. Instead, programs like REW use a swept sine and perform an inverse transform to time domain to generate the impulse response. They are however equiv. even though it may seem like we are using signals that go on for many seconds.

I'm presuming that with bass correction, the distinction between time domain correction and frequency domain EQ is less clear cut, because you only a have a fraction of a cycle before the signal component reaches the listener's ear and the rest of the room, so the effects of frequency domain EQ and time domain correction may look very similar. But my gut instinct is that the time domain is the 'correct' way to do it.

Click to expand...

The distinction is really what I explained to Terry. The peaks act in a way that are reversible in both frequency and time domain using transformation in frequency domain.

Terry, again I am not trying to make this article about how to fix the bass. The article is to demonstrate mathematical principals that are in play. These principals are disputed by Ethan with the conclusion that only acoustic products are proper solutions. That just isn't so in this region. So no, in here and now, I am not trying to convince you of anything but the core mathematics of the problem. If you agree that by changing frequency response I also correct the time domain, then we are done. If you are not, then here is some additional data from research data I post elsewhere.

From Dr. Toole:

”Using equalization to reduce the bump also attenuates the ringing so both problems are solved simultaneously.”

Dr. Toole shows the following measurements:



He does the work with both coarse and fine grain EQ and in both cases we see a reduction in time domain ringing that accompanies the flatter frequency response on the left.

Peter. Craven and Michael Gerzon in their AES DSP conferences titled, “PRACTICAL ADAPTIVE ROOM AND LOUDSPEAKER EQUALISER FOR HI-FI USE” come to the same conclusion and back it with measurements as I have done:



Per Rubak and Lars Johansen in their AES paper draw on the above work and opine the same way:

” As pointed out by Craven & Gerzon [5], room equalizers based on Digital Signal Processors are able to reduce the reverberation time considerably, even if we only use minimum-phase equalizers. Our preliminary test results are in agreement with this important potential for DSP based equalizers. Therefore we have put focus on objective test methods concerning the improvement of the room acoustics using equalizers.”

This and the fact that time and frequency are at war with each other with respect to resolution are the central points. I think the case is made clearly here for what is right and if we agree, then Ethan's notion that such processing is not as good because it only deals with half the universe creates the wrong impression of the technology.

Beyond that, if you separate your bass and allocate it to multiple subwoofers, and optimize their operation, you can get remarkably flat response even without acoustic products aimed at bass frequencies. This is not to say you should not use acoustic products. But that you are better off starting with those techniques and then see what is left over to be corrected. If you cannot deploy DSP or multiple subs then acoustic products are your only means.

Also let's note that all of this is focused on low frequencies. Above transition frequencies of 200-400 Hz, if the room is too bare, it also needs treatment to bring its mid to late reverberation times lower. And that may very well call for acoustic products.

All I am advocating here is a deeper understanding of the science and math so that an informed decision can be made.

Hope this covers the questions you were asking.

microstrip said:

Sorry - I thought you wanted to look in the noise level. Just increased the threshold and added an overlay of the two measurements.

Click to expand...

Thanks. This is more clear. Looks like the tuned absorber is operating at a higher frequency since the noticeable improvement is at 50 Hz or so in amplitude. At 36 Hz, the amplitude has not changed. The change in ringing may not be that but rather, just lost energy. Your min level is still too high relative to what I think the noise floor of your room is. Can you run the noise test with no output as I did to make sure?