#101 in a series of articles about the technology behind Bang & Olufsen loudspeakers
I’ve started working with a number of my colleagues on a series of videos for internal training at Bang & Olufsen. They were kind enough to make some of these videos publicly available.
This video explains how we are able to localise the direction of and the distance to a sound source in the real world.
#100 in a series of articles about the technology behind Bang & Olufsen loudspeakers
I’ve started working with a number of my colleagues on a series of videos for internal training at Bang & Olufsen. They were kind enough to make some of these videos publicly available.
This one explains some basic concepts of human hearing in the frequency domain, including how our hearing changes with level, the reason we use “loudness” processing in loudspeakers, and psychoacoustic masking.
The June, 1968 issue of Wireless World magazine includes an article by R.T. Lovelock called “Loudness Control for a Stereo System”. This article partly addresses the issue of resistance behaviour one or more channels of a variable resistor. However, it also includes the following statement:
It is well known that the sensitivity of the ear does not vary in a linear manner over the whole of the frequency range. The difference in levels between the threshold of audibility and that of pain is much less at very low and very high frequencies than it is in the middle of the audio spectrum. If the frequency response is adjusted to sound correct when the reproduction level is high, it will sound thin and attenuated when the level is turned down to a soft effect. Since some people desire a high level, while others cannot endure it, if the response is maintained constant while the level is altered, the reproduction will be correct at only one of the many preferred levels. If quality is to be maintained at all levels it will be necessary to readjust the tone controls for each setting of the gain control
The article includes a circuit diagram that can be used to introduce a low- and high-frequency boost at lower settings of the volume control, with the following example responses:
These days, almost all audio devices include some version of this kind of variable magnitude response, dependent on volume. However, in 1968, this was a rather new idea that generated some debate.
In the following month’s issue The Letters to the Editor include a rather angry letter from John Crabbe (Editor of Hi-Fi News) where he says
Mr. Lovelock’s article in your June issue raises an old bogey which I naively thought had been buried by most British engineers many years ago. I refer, not to the author’s excellent and useful thesis on achieving an accurate gain control law, but to the notion that our hearing system’s non-linear loudness / frequency behaviour justifies an interference with response when reproducing music at various levels.
Of course, we all know about Fletcher-Munson and Robinson-Dadson, etc, and it is true that l.f. acuity declines with falling sound pressure level; though the h.f. end is different, and latest research does not support a general rise in output of the sort given by Mr. Lovelock’s circuit. However, the point is that applying the inverse of these curves to sound reproduction is completely fallacious, because the hearing mechanism works the way it does in real life, with music loud or quiet, and no one objects. If `live’ music is heard quietly from a distant seat in the concert hall the bass is subjectively less full than if heard loudly from the front row of the stalls. All a `loudness control’ does is to offer the possibility of a distant loudness coupled with a close tonal balance; no doubt an interesting experiment in psycho-acoustics, but nothing to do with realistic reproduction.
In my experience the reaction of most serious music listeners to the unnaturally thick-textured sound (for its loudness) offered at low levels by an amplifier fitted with one of these abominations is to switch it out of circuit. No doubt we must manufacture things to cater for the American market, but for goodness sake don’t let readers of Wireless World think that the Editor endorses the total fallacy on which they are based.
with Lovelock replying:
Mr. Crabbe raises a point of perennial controversy in the matter of variation of amplifier response with volume. It was because I was aware of the difference in opinion on this matter that a switch was fitted which allowed a variation of volume without adjustment of frequency characteristic. By a touch of his finger the user may select that condition which he finds most pleasing, and I still think that the question should be settled by subjective pleasure rather than by pure theory.
and
Mr. Crabbe himself admits that when no compensation is coupled to the control, it is in effect a ‘distance’ control. If the listener wishes to transpose himself from the expensive orchestra stalls to the much cheaper gallery, he is, of course, at liberty to do so. The difference in price should indicate which is the preferred choice however.
In the August edition, Crabbe replies, and an R.E. Pickvance joins the debate with a wise observation:
In his article on loudness controls in your June issue Mr. Lovelock mentions the problem of matching the loudness compensation to the actual sound levels generated. Unfortunately the situation is more complex than he suggests. Take, for example, a sound reproduction system with a record player as the signal source: if the compensation is correct for one record, another record with a different value of modulation for the same sound level in the studio will require a different setting of the loudness control in order to recreate that sound level in the listening room. For this reason the tonal balance will vary from one disc to another. Changing the loudspeakers in the system for others with different efficiencies will have the same effect.
In addition, B.S. Methven also joins in to debate the circuit design.
The debate finally peters out in the September issue.
Apart from the fun that I have reading this debate, there are two things that stick out for me that are worth highlighting:
The July 1968 issue of Wireless World Magazine contains a description of an early, but interesting analysis of the relationship between phantom image placement in a 2-channel stereo system and interchannel level differences. This is an old favourite topic of mine, originally inspired by the work of Michael Williams and his “Stereophonic Zoom”, and extending to my first AES paper in 1999.
If you, like me, are interested in this (for example, if you’re making a panning algorithm or you’re testing the veracity of headphone-based “virtual” systems), some important figures from that article are shown below.
Note that, for the plots above and below, the x-axes show the position of the image in the stereo sound stage, where 0 is the centre point between the two loudspeakers and 0.5 is a position in one of the two loudspeakers. This is 0.5 because it’s one-half of the total angular distance between the two loudspeakers. So, you can consider the loudspeaker aperture as ±0.5.
For more information similar to this, see these links as a start:
I found this document from Roland, published in 1978. The information in here is still valuable – and presented as an excellent introduction.
In Part 1, we looked at what happens when you try to record a signal whose frequency is higher than 1/2 the sampling rate (which, from now on, I’ll call the Nyquist Frequency, named after Harry Nyquist who was one of the people that first realised that this limit existed). You record a signal, but it winds up having a different frequency at the output than it had at the input. In addition, that frequency is related to the signal’s frequency and the sampling rate itself.
In order to prevent this from happening, digital recording systems use a low-pass filter that hypothetically prevents any signals above the Nyquist frequency from getting into the analogue-to-digital conversion process. This filter is called an anti-aliasing filter because it prevents any signals that would produce an alias frequency from getting into the system. (In practice, these filters aren’t perfect, and so it’s typical that some energy above the Nyquist frequency leaks into the converter.)
So, this means that if you put a signal that contains high frequency components into the analogue input of an analogue-to-digital converter (or ADC), it will be filtered. An example of this is shown in Figure 1, below. The top plot is a square wave before filtering. The bottom plot is the result of low-pass filtering the square wave, thus heavily attenuating its higher harmonics. This results in a reduction in the slope when the wave transitions between low and high states.
This means that, if I have an analogue square wave and I record it digitally, the signal that I actually record will be something like the bottom plot rather than the top one, depending on many things like the frequency of the square wave, the characteristics of the anti-aliasing filter, the sampling rate, and so on. Don’t go jumping to conclusions here. The plot above uses an aggressively exaggerated filter to make it obvious that we do something to prevent aliasing in the recorded signal. Do NOT use the plots as proof that “analogue is better than digital” because that’s a one-dimensional and therefore very silly thing to claim.
… just because we keep signals with frequency content above the Nyquist frequency out of the input of the system doesn’t mean that they can’t exist inside the system. In other words, it’s possible to create a signal that produces aliasing after the ADC. You can either do this by
Let’s take a sine wave and clip it after it’s been converted to a digital signal with a 48 kHz sampling rate, as is shown in Figure 2.
When we clip a signal, we generate high-frequency harmonics. For example, the signal in Figure 2 is a 1 kHz sine wave that I clipped at ±0.5. If I analyse the magnitude response of that, it will look something like Figure 3:
The red curve in Figure 2 is not a ‘perfect’ square wave, so the harmonics seen in Figure 3 won’t follow the pattern that you would expect for such a thing. But that’s not the only reason this plot will be weird…
Figure 3 is actually hiding something from you… I clipped a 1 kHz sine wave, which makes it square-ish. This means that I’ve generated harmonics at 3 kHz, 5 kHz, 7 kHz, and so on, up to ∞ Hz..
Notice there that I didn’t say “up to the Nyquist frequency”, which, in this example with a sampling rate of 48 kHz, would be 24 kHz.
Those harmonics above the Nyquist frequency were generated, but then stored as their aliases. So, although there’s a new harmonic at 25 kHz, the system records it as being at 48 kHz – 25 kHz = 23 kHz, which is right on top of the harmonic just below it.
In other words, when you look at all the spikes in the graph in Figure 3, you’re actually seeing at least two spikes sitting on top of each other. One of them is the “real” harmonic, and the other is an alias (there are actually more, but we’ll get to that…). However, since I clipped a 1 kHz sine wave in a 48 kHz world, this lines up all the aliases to be sitting on top of the lower harmonics.
So, what happens if I clip a sine wave with a frequency that isn’t nicely related to the sampling rate, like 900 Hz in a 48 kHz system, for example? Then the result will look more like Figure 4, which is a LOT messier.
A 900 Hz square wave will have harmonics at odd multiples of the fundamental, therefore at 2.7 kHz, 4.5 kHz, and so on up to 22.5 kHz (900 Hz * 25).
The next harmonic is 24.3 kHz (900 Hz * 27), which will show up in the plots at 48 kHz – 24.3 kHz = 23.7 kHz. The next one will be 26.1 kHz (900 Hz * 29) which shows up in the plots at 21.9 kHz. This will continue back DOWN in frequency through the plot until you get to 900 Hz * 53 = 47.7 kHz which will show up as a 300 Hz tone, and now we’re on our way back up again… (Take a look at Figure 7, below for another way to think of this.)
The next harmonic will be 900 Hz * 55 = 49.5 kHz which will show up in the plot as a 1.5 kHz tone (49.5 kHz – 48 kHz).
Depending on the relationship between the square wave’s frequency and the sampling rate, you either get a “pretty” plot, like for the 6 kHz square wave in a 48 kHz system, as shown in Figure 5.
Or, it’s messy, like the 7 kHz square wave in a 48 kHz system in Figure 6.
There are three things to remember from this little pair of posts:
One additional, but smaller problem with all of this is that, when you look at the output of an FFT analysis of a signal (like the top plot in Figure 7, for example), there’s no way for you to know which components are “normal” harmonics, and which are aliased artefacts that are actually above the Nyquist frequency. It’s another case proving that you need to understand what to expect from the output of the FFT in order to understand what you’re actually getting.
One of the best-known things about digital audio is the fact that you cannot record a signal that has a frequency that is higher than 1/2 the sampling rate.
Now, to be fair, that statement is not true. You CAN record a signal that has a frequency that is higher than 1/2 the sampling rate. You just won’t be able to play it back properly, because what comes out of the playback will not be the original frequency, but an alias of it.
If you record a one-spoked wheel with a series of photographs (in the old days, we called this ‘a movie’), the photos (the frames of the movie) might look something like this:
As you can see there, the wheel happens to be turning at a speed that results in it rotating 45º every frame.
The equivalent of this in a digital audio world would be if we were recording a sine wave that rotated (yes…. rotated…) 45º every sample, like this:
Notice that the red lines indicating the sample values are equivalent to the height of the spoke at the wheel rim in the first figure.
If we speed up the wheel’s rotation so that it rotated 90º per frame, it looks like this:
And the audio equivalent would look like this:
Speeding up even more to 135º per frame, we get this:
and this:
Then we get to a magical speed where the wheel rotated 180º per frame. At this speed, it appears when we look at the playback of the film that the wheel has stopped, and it now has two spokes.
In the audio equivalent, it looks like the result is that we have no output, as shown below.
However, this isn’t really true. It’s just an artefact of the fact that I chose to plot a sine wave. If I were to change the phase of this to be a cosine wave (at the same frequency) instead, for example, then it would definitely have an output.
At this point, the frequency of the audio signal is 1/2 the sampling rate.
What happens if the wheel goes even faster (and audio signal’s frequency goes above this)?
Notice that the wheel is now making more than a half-turn per frame. We can still record it. However, when we play it back, it doesn’t look like what happened. It looks like the wheel is going backwards like this:
Similarly, if we record a sine wave that has a frequency that is higher than 1/2 the sampling rate like this:
Then, when we play it back, we get a lower frequency that fits the samples, like this:
There is a simple way to calculate the frequency of the signal that you get out of the system if you know the sampling rate and the frequency of the signal that you tried to record.
Let’s use the following abbreviations to make it easy to state:
IF
F_in < Fs/2
THEN
F_out = F_in
IF
Fs > F_in > Fs/2
THEN
F_out = Fs/2 – (F_in – Fs/2) = Fs – F_in
If your sampling rate is 48 kHz, and you try to record a 25 kHz sine wave, then the signal that you will play back will be:
48000 – 25000 = 23000 Hz
If your sampling rate is 48 kHz, and you try to record a 42 kHz sine wave, then the signal that you will play back will be:
48000 – 42000 = 6000 Hz
So, as you can see there, as the input signal’s frequency goes up, the alias frequency of the signal (the one you hear at the output) will go down.
Go back and look at that last figure showing the playback signal of the sine wave. It looks like the sine wave has an inverted polarity compared to the signal that came into the system (notice that it starts on a downwards-slope whereas the input signal started on an upwards-slope). However, the polarity of the sine wave is NOT inverted. Nor has the phase shifted. The sine wave that you’re hearing at the output is going backwards in time compared to the signal at the input, just like the wheel appears to be rotating backwards when it’s actually going forwards.
In Part 2, we’ll talk about why you don’t need to worry about this in the real world, except when you REALLY need to worry about it.
Let’s say that we have to do an audio measurement of a Device Under Test (DUT) that has one input and one output, as shown below.
We don’t know anything about the DUT.
One of the first things we do in the audio world is to measure what most people call the “frequency response” but is more correctly called the “magnitude response”. (It would only be the “frequency response” if you’re also looking at the phase information.)
The standard way to do this is to use an impulse response measurement. This is a method that relies on the fact that an infinitely short, infinitely loud click contains all frequencies at equal magnitude. (Of course, in the real world, it cannot be infinitely short, and if it were infinitely loud, you would have a Big Bang on your hands… literally…)
If we measure the DUT with a single-sample impulse with a value of 1, and use an FFT to convert the impulse response to a frequency-domain magnitude response and we see this:
… then we might conclude that the DUT is as perfect as it can be, within the parameters of a digital audio system. The click comes out just like it went in, therefore the output is identical to the input.
If we measure a different DUT (we’ll call it DUT #2) and we see this:
… then we might conclude that DUT #2 is also perfect. It’s just an attenuator that drops the level by half (or -6.02 dB).
However, we’d be wrong.
I made both of those DUTs myself, and I can tell you that one of those two conclusions is definitely incorrect – but it illustrates the point I’m heading towards.
If I take DUT #1 and send in a sine tone at about 1 kHz and look at the output, I’ll see this:
As you can see there, the output is a sine wave. It looks like one on the top plot, and the bottom plot tells me that there ONLY signal at 1 kHz, which proves it.
If I send the same sine tone through DUT #2 and look at the output, I’ll see this:
As you can see there, DUT #2 clips the input signal so that it cannot exceed ±0.5. This turns the sine wave into the beginnings of a square wave, and generates lots of harmonics that can be seen in the lower half of the plot.
The point is something that is well-known by people who make audio measurements, but is too easily forgotten:
An Impulse Response measurement only shows you the linear behaviour of an audio device. If the system is non-linear, then your impulse response won’t help you. In a worst case, you’ll think that you measured the system, you’ll think that it’s behaving, and it’s not – because you need to do other measurements to find out more.
The question is “what is ‘non-linear’ behaviour in an audio device?”
This is anything that causes the device to make it impossible to know what the input was by looking at the output. Anything that distorts the signal because of clipping is a simple example (because you don’t know what happened in the input signal when the output is clipped). But other things are also non-linear. For example, dynamic processors like compressors, limiters, expanders and noise gates are all non-linear devices. Modulating delays (like in a chorus or phaser effect), or a transmission system with a drifting clock are other examples. So are psychoaoustic lossy codecs like MP3 and AAC because the signal that gets preserved by the codec changes in time with the signal’s content. Even a “loudness” function can be considered to have a kind of non-linear behaviour (since you get a different filter at different settings of the volume control).
It’s also important to keep in mind that any convolution-based processing is using the impulse response as the filter that is applied to the signal. So, if you have a convolution-based effects unit, it cannot simulate the distortion caused by vacuum tubes using ONLY convolution. This doesn’t mean that there isn’t something else in the processor that’s simulating the distortion. It just means that the distortion cannot be simulated by the convolver.*
The reason for the title: “One measurement is worse than no measurements” is that, when you do a measurement (like the impulse response measurement on DUT #2) you gain some certainty about how the device is behaving. In many cases, that single measurement can tell the truth, but only a portion of it – and the remainder of the (hidden) truth might be REALLY bad… So, your one measurement makes you THINK that you’re safe, but you’re really not… It’s not the measurement that’s bad. The problem is the certainty that results in having done it.
* Actually, one of the questions on my comprehensive exams for my Ph.D. was about compressors, with a specific sub-question asking me to explain why you can’t build a digital compressor based on convolution (which was a new-and-sexy way to do processing back then…). The simple answer is that you can’t use a linear time-invariant processor to do non-linear, time-variant processing. It would be like trying to carry water in a net: it’s simply the wrong tool for the job.
#94 in a series of articles about the technology behind Bang & Olufsen
This was an online lecture that I did for the UK section of the Audio Engineering Society.
I thought that I was finished talking about (and even thinking about) the RCA Dynagroove Dynamic Styli Correlator as well as tracking and tracing distortion… and then I got an email about the last two postings pointing out that I didn’t mention two-channel stereo vinyl, and whether there was something to think about there.
My first reaction was: “There’s nothing interesting about that. It’s just two channels with the same problem, and since (at least in a hypothetical world) the two axes of movement of the needle are orthogonal, then it doesn’t matter. It’ll be the same problem in both channels. End of discussion.”
Then I took the dog out for a walk, and, as often happens when I’m walking the dog, I re-think thoughts and come home with the opposite opinion.
So, by the time I got home, I realised that there actually is something interesting about that after all.
Starting with Emil Berliner, record discs (original lacquer, then vinyl) have been cut so that the “mono” signal (when the two channels are identical) causes the needle to move laterally instead of vertically. This was originally (ostensibly) to isolate the needle’s movement from vibrations caused by footsteps (the reality is that it was probably a clever manoeuvring around Edison’s patent).
This meant that, when records started supporting two audio channels, a lateral movement was necessary to keep things backwards-compatible.
What does THIS mean? It means that, when the two channels have the same signal (say, on the lead vocal of a pop tune, for example) when the groove of the left wall goes up, the groove of the right wall goes down by the same amount. That causes the needle to move sideways, as shown below in Figure 1.
What are the implications of this on tracing distortion? Remember from the previous posting that the error in the movement of the needle is different on a positive slope (where the needle is moving upwards) than a negative slope (downwards). This can be seen in a one-channel representation in Figure 2.
Since the two groove walls have an opposite polarity when the audio signals are the same, then the resulting movement of the two channels with the same magnitude of error will look like Figure 3.
Notice that, because the two groove walls are moving in opposite polarity (in other words, one is going up while the other is going down) this causes the two error signals to shift by 1/2 of a period.
However, Figure 3 doesn’t show the audio’s electrical signals. It shows the physical movement of the needle. In order to show the audio signals, we have to flip the polarity of one of the two channels (which, in a real pickup would be done electrically). That means that the audio signals will look like Figure 4.
Notice in Figure 4 that the original signals are identical (that’s why it looks like there’s only one sine wave) but their actual outputs are different because their error components are different.
But here’s the cool thing:
One way to think of the actual output signals is to consider each one as the sum of the original signal and the error signal. Since (for a mono signal like a lead vocal) their original signals are identical, then, if you sit in the right place with a properly configured pair of loudspeakers (or a decent pair of headphones) then you’ll hear that part of the signal as a phantom image in the middle. However, since the error signals are NOT correlated, they will not be localised in the middle with the voice. They’ll move to the sides. They’re not negatively correlated, so they won’t sound “phase-y” but they’re not correlated either, so they won’t be in the same place as the original signal.
So, although the distortion exists (albeit not NEARLY on the scale that I’ve drawn here…) it could be argued that the problem is attenuated by the fact that you’ll localise it in a different place than the signal.
Of course, if the signal is only in one channel (like Aretha Franklin’s backup singers in “Chain of Fools” for example) then this localisation difference will not help. Sorry.