If you have a bunch of audio devices in a chain (say, a CD player connected to a preamplifier connected to a power amplifier connected to a loudspeaker) then one of the simplest things that you can do to improve or optimise your audio quality is to look after the gain of the signal through the system. It’s also free – and getting a lot for free is always a good thing…
Let’s start by taking a simple view of one device – a piece of audio gear. It doesn’t matter what the gear is – it could be an MP3 player, it could be a giant mixing console. What we’ll do is just look at the output of this device as it tries to play an audio signal with a varying level.
Let’s use a very simple example of a sine wave as our audio signal; we’ll look at the output of the Audio Device as we increase the level of our sine wave from very quiet to very loud.
This screen shot shown in Figure 2, by itself, is not that interesting. Let’s zoom in to the three points on the plot to see what’s going on.
Figure 3 shows a zoomed-in view of point “A” in Figure 2. Notice that you cannot see a sine wave in that signal – it’s just noise. This is the noise that is naturally generated by the device for some reason. This may be natural noise in the analogue chain – caused by thermal movement of electrons in resistors, amplified by the device itself. It may be intentional noise like dither which is added to the signal to randomise errors in a digital audio chain. Or, it may be something else entirely…
But be careful not to jump to conclusions… Just because you can’t see a sine wave there doesn’t mean that you won’t be able to hear it. As the level of the sine wave is increased, we’ll be able to hear it along with the noise before we’ll be able to see it on the screen.
In this case, we have a very low “signal-to-noise ratio”. In other words, the level of the signal (the sine wave) divided by (because it’s a ratio) the level of the noise gives us a low number. Or, in normal English – the sine wave is “drowned out” by the noise.
Figure 4 shows a nice, clean-looking sine wave coming out of our audio device. It’s what’s going on at point “B” in Figure 2. We’ve zoomed in so much that you can’t see the increase in level over time – but trust me, it’s happening there.
The noise is still there, “riding the wave” of the sine tone. In fact, if we were to zoom in on the sine wave in that figure, we’d see the same kind of noise that we saw in Figure 3 – like little ripples on big ocean waves. Now, however, the sine wave is much louder than the noise – so we have a reasonably high “signal-to-noise ratio”. In other words, the level of the signal (the sine wave) divided by (because it’s a ratio) the level of the noise gives us a high number. Or, in normal English – the sine wave “drowns out” the noise.
Figure 5 shows what’s happening at point “C” in Figure 2. Notice that this doesn’t really look like a sine wave any more – the top and bottom has been chopped off or “clipped”. This has happened because we are trying to make our audio device have an output level that is beyond its abilities. As the sine wave increases, the audio device follows along, until its output can go no higher, so it stops and holds that output level until the sine wave comes back down.
At the point, the noise is still very much lower in level than the signal – but we have caused a problem – the input is a sine wave, but the output is not. In other words, we have distorted the shape of the audio signal.
Note that distortion of an audio signal can take an infinite number of forms. The example here is symmetrical clipping of the signal – which is what many people mean when they say “distorted” – but don’t be fooled… “Distortion” means a whole lot more than this.
So, there’s a moral to the story-thus-far: every audio device has an upper and lower limit for audio level. (Yes, even a wire has a lower limit set by thermal noise in the electrons it contains and an upper limit set by the amount of current it will pass through before melting.) That range of dynamics or dynamic range is (hopefully) big – in other words, the noise floor (the quietest sound) should be MUCH MUCH quieter than a just-clipped signal (the loudest sound). Because this difference is so big, we’ll measure it in decibels (for kind of the same reason it doesn’t make sense to measure the speed of a car in millimetres per year, or the area of Canada in square micrometres.)
We can also represent these two numbers (the level of the noise floor and the level of a just-clipped signal) as two values relative to each other. Let’s say, for the purposes of keeping the numbers pretty, that we have an audio device that just so happens to have a level of noise floor that it 100 dB below the level of a signal that just starts to clip at its output.
Figure 6 shows one way to represent this. The red vertical rectangle on the left shows the range of audio levels that is possible to achieve with “Device #1”. It has a noise floor of 10 µV and will clip at 1 V – therefore it has a total dynamic range of 100 dB. Since, in this example, Device #1 is the only device in our audio system, the dynamic range of the entire system is also 100 dB (shown as the ride rectangle on the right) – since the entire system consists of just one device.
What happens if we add another device in our chain? Let’s say, for example, that we put a second device in the system after Device #1. Let’s also say that Device #2 can play louder signals than Device #1 – and it has a lower noise floor, as is shown in Figure 7.
There are three things to notice in Figure 7:
- Device #2 can play louder than Device #1
- Device #2 has a lower noise floor than Device #1
- Therefore Device #2 has a wider dynamic range than Device #1
- The dynamic range of the total system is set by Device #1, since it is not limited by Device #2.
However, we should be careful here. The fact that Device #2 has a wider dynamic range than Device #1 does not automatically mean that the total system has a dynamic range that is defined by the “weakest link” (Device #1). Look at Figure 8, for example.
In Figure 8, we have not changed the devices – Device #2 still has a 120 dB dynamic range – but the Total System has a dynamic range that is reduced to 90 dB because of the alignment of levels in the system. Now, the noise floor of the system comes from Device #2 because we have not been careful about setting up the alignment of the levels of the devices.
Another way to think of this is that Device #2 is set up with the expectation that it will go much louder – but it doesn’t because of the limitations of Device #1. Because of that incorrect setup, the noise that you hear at the output of the system comes from Device #2.
An example of a system like the one shown in Figure 8 is when you connect a low-end audio device’s output (say, the headphone jack of your computer or phone) to a better device that is built to handle a much higher input level. The possible result is that the “headroom” (the amount by which the better device can handle higher level signals) is wasted (since the lower-quality device doesn’t deliver those high levels) and the total system has a degraded dynamic range.
So, the moral of the story here so far is that you should always try to ensure that your system’s dynamic range is not limited by the way it’s connected.
For example, if you have a system that has an adjustable input sensitivity, you should set it so that the input is not expecting more level than the device that’s feeding it can deliver. If your output device can only deliver 2 V RMS maximum, it my not be helpful for the thing it’s connected to to be “expecting” to see 4 V coming from it. If this is the way things are setup, then you might be “throwing away” 6 dB of dynamic range (because 4 V is 6 dB louder than 2 V).
Generally, there are two good “rules of thumb” that can help you here.
The first one is to try to align all your maximum levels as much as possible. So, as in the last example above, if your source device has a maximum output of 2.0 V RMS, set the input sensitivity of your next device to “expect” 2.0 V RMS maximum. This will make the tops of the red rectangles all align, and the dynamic range will be defined by the worst link in the chain instead of the way the devices are connected.
The second rule of thumb is to put as much gain as possible at the beginning of the chain. This is particularly true if you’re working in a recording studio. This is because every piece of gear contributes noise to the audio signal. If you put all the gain at the end of the chain, then you are making the signal louder, but you’re also making all of the noise from all of the gear “upstream” louder as well. If you put all the gain at the beginning of the chain, then you might wind up in a situation where you have to turn DOWN the signal through the chain, those reducing your signal to the correct level, and bringing the noise floor down with it. (Two obvious examples of this are using lots of gain at your mic preamp in a recording studio, or getting a RIAA preamp with a healthy output level for your turntable…) Another good example of this is the case where you have a headphone output from your phone connected to the aux input of a small stereo system. You want to turn up the phone as much as possible, and turn down the stereo volume. If you do the opposite, you’ll be using the stereo system to turn up the noise output of your phone.
One last thing: connecting devices digitally will probably help with your dynamic range, however, this is not necessarily always true. You certainly cannot make an automatic conclusion that a digital connection is better in all respects than an analogue one – or vice versa. For example, in some cases, the errors in a sampling rate converter at a digital input stage may result in a higher level of “noise” floor than the analogue noise caused by an analogue-to-digital converter on the same device. Or, it might be that these two inputs have the same measurable noise floor, but those two noises have very different characteristics. Typically analogue noise is program independent – meaning it’s unrelated to the signal – whereas poorly-implemented digital transmission and processing typically results in program-dependent errors. These can be interpreted by the listener as being part of the signal (more like distortion artefacts than noise) and therefore will be different for different signals. To make things even more confusing, different digital inputs on the same device (e.g. Optical, S/P-DIF, and HDMI) may (or may not) behave differently – so any decisions you make about one of them may (or may not) be applicable to the others…
Inspired by a conversation with Jamie Angus following my last posting, I did some digging into the probability density functions (PDF’s) of a bunch of test tracks that I use for tuning and testing loudspeakers.
The plots below are the results of this analysis.
Some explanations, to start…
A PDF of an audio signal is a measurement of the probability that a given level (or sample value) will happen in a given period of time. In the case of the plots below, I just counted each time every sample value the 16-bit range of possibilities (from -32768 to 32767, if you think in binary – from -1 to +1-2^-15 in steps of 2^-15 if you prefer to think in floating point decimal) occurred in an entire track (usually the full tune). That’s plotted below as the gray “curve” in a linear world.
I converted the linear levels to “instantaneous” (sample-by-sample) dB values (since they’re instantaneous, they’re not in dB FS – but let’s not get into that discussion), but kept the positive and negative polarities of the linear values separate. Those are plotted as the red (negative values, expressed in dB) and black (positive values, expressed in dB). I cheated a little here, since a linear value of “0” isn’t really -96 dB – it’s -infinity dB… but, except for that one value, everything else is plotted correctly.
When I did these analyses, I noticed lots of sample values in lots of tracks that had no probability of ever occurring. Sometimes, this is just because the track is mastered low in level – so the “upper” sample values are not used. Sometimes, there are “dead values” well inside the range. This likely points to an error in the converter and/or digital mixing and/or mastering equipment.
Finally, I made a plot of the number of “dead” sample values per 128 sample values in 512 blocks (65536/128 = 512). That’s the red line above the gray one.
Some other things are noticeable in the plots, but we’ll take those as they come, below…
Note that I will not reveal the names of the tracks I used, since it’s not my purpose here to make anyone look bad. It’s to look at the differences between different recordings, types, and even equipment… Don’t ask which recordings I used.
This plot of an orchestral recording (“orchestral8”) looks fairly normal. As Jamie pointed out in the last posting, the distribution looks to be Laplacian. There is a big spike at the “0” mark – due to the silence at the beginning and end of the track. As can be seen, the track peaks around a linear value of about +/- 0.2 or about -14 dB below full scale. So, the sample values above 0.2 (or below -0.2) are unused. This can be seen in the blue lines (comprised of a blue dot at each sample value) in the top plot, and the red line at 128 (the number of “dead values” per 128 possible values) in the bottom plot.
The “solo inst3” is very similar in behaviour, even though it’s a completely different recording. This one is of a solo stringed instrument in a fairly reverberant space. Notice that its basic characteristics are very similar to those shown in “orchestra8”.
The “voice3” recording is also similar. It’s a little interesting in all three of these recordings to note the transition between levels where there are no dead sample values (around the “0” line) and levels where there are nothing but dead values. In this area (in the “voice3” recording, for example, around +/- 0.3, there are sample values that are used – but others that are skipped. This is because the track has a reasonably large crest factor (the ratio between the peak and the RMS of the track) – in other words, it has noticeable peaks. When the levels peak positively or negatively, there will be some values skipped along the way…
Although these are very different recordings of different instruments in different spaces on different labels – they all appear to have some characteristics in common.
- They very likely had little or no compression applied
- They don’t use the entire dynamic range available. This is not necessarily a bad thing, since it could be that each of those tracks was part of a larger collection, and its level made sense in the context of the entire album.
Now let’s look at another acoustic recording that was done with four microphones, and no compression – but possibly some small processing done in the mastering.
Obviously the “orchestra6” recording has something strange going on. It almost looks as though something in the recording chain “favoured” every second sample value – hence the zig-zag pattern in the probability density function. Note that this was not a small thing – the difference in how much the sample values are “preferred” is by a factor of 10.
What could cause this? This is difficult to say by just looking at this plot, since we have to remember that these values are complied using the entire track. So, for example, three possibilities that come immediately to mind are:
- for some strange reason, the analogue-to-digital converter, or one of the DSP blocks in the mastering console “liked” every second sample value 10 times more than the others
- for some other strange reason, the original ADC only used every second sample value – and one 10th of the track was edited together (or “spliced”) from a different take that needed extra gain.
- something else
To be honest, I think that the first or third of these is more likely than the second – but either way, it certainly looks weird…
This track of another solo instrument (hence “solo inst1”) is a little different – but not terribly so. There is a little flatter behaviour in the upper plot, which corresponds to a more convex “umbrella” shape in the lower one – but generally, this is nothing serious to raise any eyebrows in my opinion… It probably indicates some minor compression.
Let’s look at some other tracks with compression…
“testtracks4” and “pop21” both have indications of compression in the flat response of the top plots and the convex shape of the lower ones. The “pop21” track has the added indication of clipping – the spikes on the sides of the plots. This indicates that we have an unusual number of samples with values of either -1 or +1. Note, however, that we do not move smoothly into that clipping – it was what we call “hard clipping”, since the values just before the +/- 1 values show no indication of a smooth transition to the spike.
The “bass20” track is interesting, not only because of the compression, but the apparent lack of silence (notice there is no spike around the “0” line). This is because this track is from a live album that is intended for gapless playback – and I just grabbed the track. So, it starts and stops with a hard transition to and from audience sound – there is no fade in or fade out.
The “bass2” track also has the spikes on the ends, showing the clipping – but, as you can see, the plots (particularly the linear plot on the bottom) starts to slope upwards just before the spike – indicating some kind of soft clipping or a peak limiting function was used to shape the envelope.
The “bass15” track is interesting, since it has the characteristic spikes on the sides that look a little like clipping (especially if you only look at the upper plot) but, as can be seen in the bottom plot, these are not at values of -1 or +1. So, this would indicate either that something else in the recording chain clipped – and then was smoothed out and reduced in level a little later in the process – or we’re looking at some kind of interesting soft-clipping processor that keeps a little “bump” in the envelope of the signal above the “clipping” area.
Now let’s look at some really strange ones…
I have no explanation for the plot in “testtracks6”. I can’t understand what would cause that bump only in the negative portion of the signal around -70 dB FS. My guess is that this is some sort of weird watermark that is inserted – but this is really stretching my imagination… Of course, if could just be that something in the processing chain is just broken… Anyone reading this have any good ideas? Seems to me that I should do a little more digging into this track to see what’s going on around those sample values…
The “pop11” track is an example of a recording that was probably done on early digital recording gear – or an early digital mastering console. As can be seen in both plots, there are missing sample values across the entire range of possible values.
One possible explanation of this is that this is a digital recording that had gain applied to it using a processor that did not use dither. This would cause the signal to not use every second sample value (or every third or fourth – depending on the gain applied). It’s also possible that it was processed or recorded using a device that had a “stuck bit”. I’ll do some simulations to show what that would look like and publish the plots in a future posting.
Note that the small “spikes” of peak limiting (looking a little like clipping on a small scale) are visible in the bottom plot – but they’re very small…
Some more recordings with strangely dead sample values are shown below. Note that some of these are very recent recordings – so the “early digital gear” excuse doesn’t hold up for all of them…
So, something is obviously broken in all four of these examples… Please don’t ask me how to explain them. The only thing I can do is to suspect that at least one piece of gear and/or software that was used in the late stages of the process was really broken… I just hope that, whatever gear/software it was, it didn’t cost a lot of money….
An interesting pair of examples are shown below…
The “jazz1” and “jazz2” recordings both come from the same album released by a small jazz ensemble. Notice that, although they are different tracks, they have similar PDF’s, seen in the “spikes” through the upper plots. It seems that there is something weird going on in the mastering console (or software) in this case – or perhaps the final mixing console…
The “speech4” track has obvious “favouring” of alternate sample values – but this was a track that was recorded on a very early DAT machine about 35 years ago… To be honest, knowing what I know about using an identical model DAT recorded, I’m surprised this looks as good as it does…
I’ll just put in a bunch more plots without comment – just to let you see some of the variety that shows up with this kind of analysis.
“Lyden av BeoLab 90 er vanskelig å forklare, den må egentlig bare oppleves. Personlig har jeg aldri hørt en mer livaktig musikkgjengivelse, og flere som har vært på besøk reagerte med å klype seg i armen eller felle en tåre når de hørte et opptak de kjente, eller rettere sagt trodde de kjente!”
“The sound of the BeoLab 90 is hard to explain, it must really be experienced. Personally I have never heard a more lifelike music reproduction, and several who have been visiting reacted by pinching their own arm or shedding a tear when they heard a recording they knew, or rather thought they knew!”
The Danish version can be read here.
Over the past year or so, I’ve had lots of discussions and interviews with lots of people (customers, installers, and journalists) about Bang & Olufsen’s loudspeakers – BeoLab 90 in particular. One of the questions that comes up when the chat gets more technical is whether our loudspeakers use native sampling rates or sampling rate conversion – and why. So, this posting is an attempt to answer this – even if you’re not as geeky as the people who asked the question in the first place.
There are advantages and disadvantages to choosing one of those two strategies – but before we get to talk about some of them, we have to back up and cover some basics about digital audio in general. Feel free to skip this first section if you already know this stuff.
A very quick primer in digital audio
An audio signal in real life is a change in air pressure over time. As the pressure increases above the current barometric pressure, the air molecules are squeezed closer together than normal. If those molecules are sitting in your ear canal, then they will push your eardrum inwards. As the pressure decreases, the molecules move further apart, and your eardrum is pulled outwards. This back-and-forth movement of your eardrum starts a chain reaction that ends with an electrical signal in your brain.
If we take your head out of the way and replace it with a microphone, then it’s the diaphragm of the mic that moves inwards and outwards instead of your eardrum. This causes a chain reaction that results in a change in electrical voltage over time that is analogous to the movement of the diaphragm. In other words, as the diaphragm moves inwards (because the air pressure is higher), the voltage goes higher. As the diaphragm moves outwards (because the air pressure is lower) the voltage goes lower. So, if you were to plot the change in voltage over time, the shape of the plot would be similar to the change in air pressure over time.
We can do different things with that changing voltage – we could send it directly to a loudspeaker (maybe with a little boosting in between) to make a P.A. system for a rock concert. We could send it to a storage device like a little wiggling needle digging a groove in a cylinder made of wax. Or we could measure it repeatedly…
That last one is where we’re headed. Basically speaking, a continuous, analogue (because it’s analogous to the pressure signal) audio signal is converted into a digital audio signal by making instantaneous measurements of it repeatedly, and very quickly. It’s a little bit like the way a movie works: you move your hand and take a movie – the camera takes a bunch of still photographs in such quick succession that, if you play back the photos quickly, it looks like movement to our slow eyes. Of course, I’m leaving out a bunch of details, but that’s the basic concept.
So, a digital audio signal is a series of measurements of an electrical voltage (which was changing over time) that are transmitted or stored as numbers somehow, somewhere.
Figure 3 shows a small slice of time from Figure 2, which is, itself a small slice of time that normally is considered to extend infinitely into the past and the future. If we want to get really pedantic about this, I can tell you the actual values represented in the plot in Figure 3. These are as follows:
The actual values that I listed there really aren’t important. What is important is the concept that this list of numbers can be used to re-construct the signal. If I take that list and plot them, it would look like Figure 4.
So, in order to transmit or store an audio signal that has been converted from an analogue signal into a digital signal, all I need to do is to transmit or store the numbers in the right order. Then, if I want to play them back (say, out of loudspeaker) I just need to convert the numbers back to voltages in the right order at the right rate (just like a movie is played back at the same frame rate that the photos were take in – otherwise you get things moving too fast or too slowly).
One last piece of information that you’ll need before we move on is that, in a digital audio system, the audio signal can only contain reliable information below the frequency that is one-half of the “sampling rate” (which is the rate at which you are grabbing those measurements of the voltages – each of those measurements is called a “sample”, since it’s taking an instantaneous sample of the current state of the system). It’s just like taking a blood sample or a water sample – you use it as a measurement of one portion of one thing right now. This means that if you want to record and play back audio up to 20,000 Hz (or 20,000 cycles per second – which is what textbooks say that we can hear) you will need to be making more than 40,000 measurements per second. If we use a CD as an example, the sampling rate is 44,100 samples per second – also known as a sampling rate of 44.1 kHz. This is very fast.
Sidebar: Please don’t jump to conclusions about what I have said thus far. I am not saying that “digital audio is perfect” (or even “perfect-er or worse-er than analogue”) or that “this is all you need to understand how digital audio works”. And, if you are the type of person who worships at “The Church of Our Lady of Perpetual Jitter” or “The Temple of Inflationary Bitrates” please don’t email me with abuse. All I’m trying to do here is to set the scene for the discussion to follow. Anyone really interested in how digital audio really works should read the collected writings of Harry Nyquist, Claude Shannon, John Watkinson, Jamie Angus, Stanley Lipshitz, and John Vanderkooy and send them emails instead. (Also, if you’re one of the people that I just mentioned there, please don’t get mad at me for the deluge of spam you’re about to receive…)
Filtering an audio signal
Most loudspeakers contain a filter of some kind. Even in the simplest passive two-way loudspeaker, there is very likely a small circuit called a “crossover” that divides the analogue electrical audio signal so that the low frequencies go to the big driver (the woofer) and the higher frequencies go to the little driver (the tweeter). This circuit contains “filters” that have an input and an output – the output is a modified (or filtered) version of the input. For example, in a low-pass filter, the low frequencies are allowed to pass through it, and the higher frequencies are reduced in level (which makes it useful for sending signals to the woofer).
Once again over-simplifying, this is accomplished in an electrical circuit by playing with something called electrical impedance – a measure of how much the circuit impedes the flow of current to the next device. Some circuits will impede the flow of current if it’s alternating quickly (a high frequency) other circuits might impede the flow of current if it’s alternating slowly (a low frequency). Other circuits will do something else…
It is also possible to filter an audio signal when it is in the digital domain instead. We have the series of numbers (like the one above) and we can send these to a mathematical function which will change the values into other numbers to produce the desired characteristics (like a low-pass filter, for example).
As a simple example, if we take all of the values in the list above, and multiply them by 0.5 before converting them back to voltages, then the output level will be quieter. In other words, we’ve made a volume knob in the digital domain. Of course, it’s not a very good knob, since it’s stuck at one setting… but this isn’t a course in advanced DSP…
If you want to make something more exciting than a volume knob, the typical way of making an audio filter in the digital world is to use delays to mix the signal with delayed copies of itself. This can get very complicated, but let’s make a simple filter to illustrate…
Figure 5 shows a very simple low pass filter for digital audio. Let’s think though what will happen when we send a signal through it.
If you have a very low frequency, then the current sample goes into the input and heads in two directions. Let’s ignore the top path for now and follow the lower path. The sample goes into a delay and comes out on the other side 1 sample later (in the case of a CD, this is 1/44100-th of a second later). When that sample comes out of the delay, the one that followed it (which is now “now” at the input) meets it at the block on the right with the “+” sign in it. This adds the two samples together and spits out of the result. In other words, the filter above adds the current audio sample to the previous audio sample and delivers the result as the current output.
What will this do to the audio signal?
If the frequency of the audio signal is very low, then the two samples (the current one, and the previous one) are very similar in level, as can be seen in the plot in Figures 6 and 7, below. This means, basically, that the output of the filter will be very, very similar to the output, just twice as loud (because it’s the signal plus itself). Another way to think of this is that the current sample of the audio signal and the previous sample of the same signal are essentially “in phase” – and any two audio signals that are “in phase” and added together will give you twice the output.
However, as the frequency of the audio signal gets higher, the relative levels of those two adjacent samples becomes more and more different (because the sampling rate doesn’t change). One of them will be closer to “0” than the other – and increasingly so as the frequency increases . So, the higher the audio frequency, the lower the level of the output (since it will not go higher than the signal plus itself, as we saw in the low frequency example…) If we’re thinking of this in terms of phase, the higher the frequency of the audio signal, the greater the phase difference between the adjacent samples that are summed, so the lower the output…
That output level keeps dropping as the audio frequency goes up until we hit a frequency where the audio signal’s frequency is exactly one half of the sampling rate. At that “magic point”, the two samples are so far apart (in terms of the audio signal’s waveform) that they have opposite polarity values (because they’re 180 degrees out-of-phase with each other). So, if you add those two samples together, you get no output – because they are equal, but opposite.
Let’s zoom in on the plot in Figure 6 to see the individual samples. We’ll take a slice of time around the 500-sample mark. This is shown below in Figure 7.
As you can see in Figure 7, any two adjacent samples for a low frequency (the red plot) are almost identical. The middle frequency (the blue plot) shows that two adjacent samples are more different than they are for this low frequency. For the “magic frequency” of “sampling rate divided by 2” (in this case, 22050 Hz) two adjacent samples are equal and opposite in polarity.
Now that you know this, we are able to “connect the dots” and plot the output levels for the filter in Figure 5 for a range of frequencies from very low to very high. This is shown in Figures 8 and 9, below.
Now we have to step up the coefficient of geekiness…
So far, we have been thinking with a fixed sampling rate of 44.1 kHz – just like that which is used for a CD. However, audio recordings that you can buy online are available at different sampling rates – not just 44.1 kHz. So, how does this affect our understanding so far?
Well, things don’t change that much – we just have to change gears a little by making our frequency scales vary with sampling rate.
So, without using actual examples or numbers, we already know that an audio signal with a low frequency going through the filter above will come out louder than the input. We also know that the higher the frequency, the lower the output until we get to the point where the audio signal’s frequency is one half the sampling rate, where we get no output. This is true, regardless of the sampling rate – the only change is that, by changing the sampling rate, we change the actual frequencies that we’re talking about in the audio signal.
So, if the sampling rate is 44.1 kHz, we get no output at 22050 Hz. However, if the sampling rate were 96 kHz, we wouldn’t reach our “no output” frequency until the audio signal gets to 48 kHz (half of 96 kHz). If the sampling rate were 176.4 kHz, we would get something out of our filter up to 88.2 kHz.
So, the filter generally behaves the same – we’re just moving the frequency scale around.
So, instead of plotting the magnitude response of our filter with respect to the actual frequency of the audio signal out here in the real world, we can plot it with respect to the sampling rate, where we can get all the way up to 0.5 (half of the sampling rate) since this is as high as we’re allowed to go. So, I’ve re-plotted Figures 8 and 9 below using what is called a “normalised frequency” scale – where the frequency of the audio signal is expressed as a fraction of the sampling rate.
These last sentences are VERY IMPORTANT! So, if you didn’t understand everything I said in the previous 6 paragraphs, go back and read them again. If you still don’t understand it, please email me or put a comment in below, because it means that I didn’t explain it well enough. ..
Note that there are two conventions for “normalised frequency” just to confuse everyone. Some people say that it’s “audio frequency relative to the sampling rate” (like I’ve done here). Some other people say that it’s “audio frequency relative to half of the sampling rate”. Now you’ve been warned.
Designing an audio filter
In the example above, I made a basic audio filter, and then we looked at its output. Of course, if we’re making a loudspeaker with digital signal processing, we do the opposite. We have a target response that we want at the output, and we build a filter that delivers that result.
For example, let’s say that I wanted to make a filter that has a similar response to the one shown above, but I want it to roll off less in the high frequencies. How could I do this? One option is shown below in Figure 10:
Notice that I added a multiplier on the output of the delay block. This means that if the frequency is low, I’ll add the current sample to half the value of the previous one, so I’ll get a maximum output of 1.5 times the input (instead of 2 like we had before). When we get to one half the sampling rate, we won’t cancel completely, so the high end won’t drop off as much. The resulting magnitude response is shown in Figures 11 and 12, below.
So, we can decide on a response for the filter, and we can design a filter that delivers that response. Part of the design of that filter is the values of the “coefficients” inside it. In the case of digital filters, “coefficient” is a fancy term meaning “number” – in the case of the filter in Figure 10, it has one coefficient – the “0.5” that is multiplied by the output of the delay. For example, if we wanted less of a roll-off in the high end, we could set that coefficient to 0.1, and we would get less cancellation at half the sampling rate (and less output in the low end….)
Putting some pieces together
So, now we see that a digital filter’s magnitude response (and phase response, and other things) is dependent on three things:
- its design (e.g. how many delays and additions)
- the sampling rate
- the coefficients inside it
If we change one of these three things, the magnitude response of the filter will change. This means that, if we want to change one of these things and keep the magnitude response, we’ll have to change at least one of the other things.
For example, if we want to change the sampling rate, but keep the design of the filter, in order to get the same sampling rate, we’re going to have to change the coefficients.
Again, those last two paragraphs were important… Read’em again if you didn’t get it.
Let’s now take this information into the real world.
In order for BeoLab 90 to work, we had to put a LOT of digital filters into it – and some of those filters contain thousands of coefficients… For example, when you’re changing from “Narrow” mode to “Wide” mode, you have to change a very large filter for each of loudspeaker driver (that’s thousands of coefficients times 18 drivers) – among other things… This has at least four implications:
- there has to be enough computing power inside the BeoLab 90 to make all those multiplications at the sampling rate (which, we’ve already seen above, is very fast)
- there has to be enough computer memory to handle all of the delays that are necessary to build the filters
- there has to be enough memory to store all of those coefficients (remember that they’re not numbers like 1 or 17 – they’re very precise numbers with a lot of digits like 0.010383285748578423049 (in case you’re wondering, that’s not an actual coefficient from one of the filters in a loudspeaker – I just randomly tapped on a bunch of keys on my keyboard until I got a long number… I’m just trying to make an intuitive point here…))
- You have to be able to move those coefficients from the memory where they’re stored into the calculator (the DSP) quickly because people don’t want to wait for a long time when you’re changing modes
This is why (for now, at least) when you switch between “Narrow” and “Wide” mode, there is a small “break” in the audio signal to give the processor time to load all the coefficients and get the signal going again.
One sneaky thing in the design of the system is that, internally, the processor is always running at the same sampling rate. So, if you have a source that is playing back audio files from your hard drive, one of them ripped from a CD (and therefore at 44.1 kHz) and the next one from www.theflybynighthighresaudiodownloadcompany.com (at 192 kHz), internally at the DSP, the BeoLab 90 will not change.
Why not? Well, if it did, we would have to load a whole new set of coefficients for all of the filters every time your player changes sampling rates, which, in a worst case, is happening for every song, but which you probably don’t even realise is happening – nor should you have to worry about it…
So, instead of storing a complete set of coefficients for each possible sampling rate – and loading a new set into the processor every time you switch to the next track (which, if you’re like my kids, happens after listening to the current song for no more than 5 seconds…) we keep the internal sampling rate constant.
There is a price to pay for this – we have to ensure that the conversion from the sampling rate of the source to the internal sampling rate of the BeoLab 90 is NOT the weakest link in the chain. This is why we chose a particular component (specifically the Texas Instruments SRC4392 – which is a chip that you can buy at your local sampling rate converter store) to do the job. It has very good specifications with respect to passband ripple, signal-to-noise ratio, and distortion to ensure that the conversion is clean. One cost of this choice was that its highest input sampling rate is 216 kHz – which is not as high as the “DXD” standard for audio (which runs at 384 kHz).
So, in the development meetings for BeoLab 90, we decided three things that are linked to each other.
- we would maintain a fixed internal sampling rate for the DSP, ADC’s and DAC’s.
- This meant that we would need a very good sampling rate converter for the digital inputs.
- The choice of component for #2 meant that BeoLab 90’s hardware does not support DXD at its digital inputs.
One of the added benefits to using a good sampling rate converter is that it also helps to attenuate artefacts caused by jitter originating at the audio source – but that discussion is outside the scope of this posting (since I’ve already said far too much…) However, if you’re curious about this, I can recommend a bunch of good reading material that has the added benefit of curing insomnia… Not unlike everything I’ve written here…