Author: geoff

Don’t believe your eyes

Sometimes, someone will use a plot to show the relative levels of different frequency bands in a signal. Even I have done this from time to time…. However, it’s important to have the skills to be able to read these plots with a little-more-knowledge-than-normal in order to not be distracted into thinking something that isn’t true.

One way to calculate the relative levels of frequency bands of a signal (whether it’s a measurement of a loudspeaker, a black box, or your favourite track on your favourite CD) is to so something called a “Fourier Transform”. This is a set of calculations that can be used to show how much energy there is in a signal, by frequency.

Typically, we do a Discrete Fourier Transform or “DFT” – although most people call it a Fast Fourier Transform or “FFT”. We will not discuss the difference between these things in this posting. I’ll just use the term “FFT” here, in order to be like everyone else…

(If you’d like to know how to do your own FFT’s by hand, this is one place to start learning…)

In order to give me something to analyse, I made a signal comprised of a sine tone with a frequency around 997 Hz. (I’ll explain at the end why it’s not exactly 997 Hz. I’ll also explain in another posting why I chose 997 Hz instead of a good-old-fashioned 1 kHz.)

I set that sine tone to have a level of -1 dB FS.

Then, I made some white noise and set its level to be exactly 80 dB below the level of the sine tone. In order to calculate this, I found the total RMS value of the noise signal, and used this to create a gain that makes it a level of -81 dB FS. (Just for the sake of being as pedantic as possible, the white noise that I created was the result of a “rand” function in Matlab, which, as you can see in this posting, has a rectangular probability density function.)

Therefore, I have an input that has a signal-to-noise ratio of 80 dB. (Note that this measurement does not use any band-limiting on the white noise… Typically a SNR measurement would apply some low pass filter to the noise.)

To keep things looking pretty on my graphs, I set the sampling rate to 65536 Hz (2^16).

 

 

Then, I pretended that this signal was coming in from some unknown device, and I do an FFT on it to find out the relative balance between the signal (the sine tone) and the noise (which I already “secretly” know is 80 dB lower)

 

If I do an FFT of 256 points on the signal (and therefore, I’m only looking at the first 256 samples of the signal – this is an important point that we’ll come back to later…), the result looks like Figure 1.

Fig 1: The magnitude response of the signal, calculated using a 256-bin FFT.

Note that the sine tone is a little higher than 997 Hz – but this is not really important. (the explanation is at the end!).

There are some things to notice here:

The first is that the plot does not extend lower than a frequency of 256 Hz. This is because the resolution of a 256-point FFT is 256 Hz – so, there is a “point” on the plot every 256 Hz – typically called a “bin” – since it contains information about the level of a collection of frequencies around its centre frequency. (If you’re new to FFT’s, don’t jump to the conclusion that the frequency resolution is equal to the length of the FFT. This is incorrect. The frequency resolution is equal to the sampling rate divided by the FFT length – 65536 Hz / 256 bins  = 256 Hz.) Limiting the length of the FFT limits its resolution, which has an obvious impact when we plot the results on a logarithmic frequency scale.

The second is that, although the SNR of the signal is 80 dB, on the magnitude response, it appears that the noise is generally lower than -100 dB. This is not that difficult to believe, since the noise is spread over a wide frequency range – so, although any one frequency may, indeed, be more than 100 dB below the signal – the sum of the energy in all of those frequency bands totals more than any of the individual contributions. (In the same way that 1000 people can shout louder than 1 person – even if all 1000 people are, individually, shouting at the same level.)

One thing that is not obvious from the plot, but that we have to keep in mind is that this shows us the level of the different frequency ranges over the entire length of the signal (all 256 samples of it). However, the noise that I created that is part of that signal is exactly that – noise. Since it is noise, there is no guarantee that all frequencies are represented at the same level at any one time – in fact, they’re not. “White noise” has the characteristic of having equal probability of having the same level at all frequencies. But if 1000 people have equal probability of winning a lottery, that doesn’t mean than 1000 of them will win. In order to ensure that you actually get the same level at all frequencies, you would have to listen to white noise forever – and I’m not willing to wait that long…

 

Fig 2: The magnitude response of the signal, calculated using a 512-bin FFT.

Figure 2 shows the same analysis done on the same signal, but with a 512-bin FFT instead. There, you can see that the the resolution of the plot is better – we have a bin or point every 128 Hz (remember 65536 Hz / 512 bins = 128 Hz). Also, the sine tone has the same level (-1 dB FS) but the noise, which we know is 80 dB lower, appears to be even lower than it does in Figure 1… Strange…

Let’s do some more FFT’s with more and more bins to see what happens…

 

Fig 3: The magnitude response of the signal, calculated using a1024-bin FFT

 

Fig 4: The magnitude response of the signal, calculated using a 2046-bin FFT.

 

 

Fig 5: The magnitude response of the signal, calculated using a 4096-bin FFT.

 

 

Fig 6: The magnitude response of the signal, calculated using a 8192-bin FFT

 

Fig 7: The magnitude response of the signal, calculated using a 16384-bin FFT

 

 

Fig 8: The magnitude response of the signal, calculated using a 32768-bin FFT

 

 

Fig 9: The magnitude response of the signal, calculated using a 65536-bin FFT.

 

So, by going from a 256-bin FFT to a 65536-bin FFT, we appear to have dropped the noise floor by more than 20 dB.

Weird? No. Why?

Remember that every time we double the length of the FFT, we double the number of frequency bins in its output. So, that plot in Figure 9 has more individual frequencies contributing to add together to the same noise signal, 80 dB lower than the sine tone. (If you asked 1000 people to shout as loudly as 10 people, each individual in the larger group would have to be quieter to produce the same total output.)

The “punch line” here is that we cannot make a direct conclusion about the overall Signal-to-Noise ratio of the signal by looking at any of the plots above. Of course we can say that the “signal” (the sine tone) is obviously louder than the noise – by a lot. But we can’t be much more detailed than that.

So, if someone jumps between a SNR number and a spectral plot like the ones above, in an effort to convince you of something, be very careful about being led down a garden path.

 

 

Some extra information:

We also have to remember that, although the signal that I used to make these graphs was initially the same, the actual signal that was used by each of the FFT’s was different. This is because, by default, the length of the signal used by an FFT calculation is the same as the number off bins in the FFT. So, for example, a 256-bin (or 256-point) FFT only uses 256 samples as its input. A 32768-point FFT uses 32768 samples (the first 256 of which were the ones used by the 256-point FFT).  So, for example, if you load a recording of Britney Spears singing “Toxic” into Matlab, and you type the command FFT(toxic, 256) – you’ll get a 256-bin FFT of the first 5 .8 milliseconds (256 samples) of the recording  – not a representation of the spectral content of the entire song.

 

 

Initially, I started out by saying that I would use a 997 Hz sine tone. This might look a little weird because it’s not a nice number like 1000 Hz. There’s a good reason for this, and I’ll write a posting about it some other day.

Then, I said that it’s not really 997 Hz – I moved it a little. This is because I wanted the frequency of my sine tone to land exactly on one of the bins of my FFT. So for example, in the case of the 256-bin FFT, I had a frequency resolution of 256 Hz – so my bins are at the following:

256 Hz
512 Hz
768 Hz
1024 Hz

1024 Hz is the closest value to 997 Hz that occurs in the sequence so I used that instead. If I had kept the sine tone fixed at 997 Hz, the plots would not have looked as pretty, because the information about its level would have “leaked” or “been spread out” into the adjacent bins. So, instead of a nice clean spike, we would have seen a big, round bump.

Signal levels and Dynamic Range

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.

Fig 1: An audio device with an audio signal at its output. The Signal Source might be external (say, a different device) or internal (like a CD or an MP3 file)

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.

Fig 2: A recording of the output of our Audio Device as we play a sine tone going from very, 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.

 

Fig 3: A zoomed-in view of point “A” in Figure 2. Notice that this is not a sine tone – it’s noise generated by the device that we’re testing. The sine tone is quieter than this “noise floor”, so we can’t see (or hear) it.

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.

 

Fig 4: A zoomed-in view of point “B” in Figure 2.

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.

 

Fig 5: We have pushed our audio device too far by making the sine wave louder than it can go..

 

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.

Fig 6: Another way to represent the dynamic range of an audio device or system.

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.

Fig 7: An audio system with two devices in a chain. The second device has a wider dynamic range than the first.

There are three things to notice in Figure 7:

  1. Device #2 can play louder than Device #1
  2. Device #2 has a lower noise floor than Device #1
  3. Therefore Device #2 has a wider dynamic range than Device #1
  4. 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.

Fig 8: A system with the same devices as are shown in Figure 7, but the total maximum dynamic range has dropped by 10 dB.

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…

 

 

BeoPlay H9 Headphone Reviews

I was part of the development team, and one of the two persons who decided on the final sound design (aka tonal balance) of the B&O Play H9 headphones. So, I’m happy to share some of the blame for some of the comments (at least on the sound quality) from the reviews.

Because they were sound designed primarily for portable use, the H9’s have a intentional emphasis on the lower frequency band in their tuning – although a little less than would be done on a passive pair of headphones due to the noise cancelling.

 

WhistleOut

“As you’d expect with a pair of $799 headphones, the Beoplay H9s sound amazing. In short, they offer a balanced, warm, detailed sound that isn’t quite as bass driven as what you might get from a pair of Bose or Beats. That’s not to say bass is missing, just more of a healthy medium between a more sterile studio pair of headphones and big bass alternatives. As a result, you end up with a nuanced profile that helps bring your music to life without compromising a song’s mix.”

 

headphonereview.com

“Of course, a great design and high comfort level don’t really matter if they’re let down by a bad sound quality. Thankfully, on the closed-back B&O Beoplay H9’s, the sound quality is pretty damn great.

Now, let’s make something clear before we talk about the sound — these headphones are not built for the studio. They’re consumer headphones, and as such they aren’t aimed at providing the flattes (sic.) frequency response possible, but rather at coloring the music in a way that fits most of today’s music. And that’s something they do very well.”

 

Lyd & Billede

“Særligt musik optaget i flere lag demonstrerer præcis hvor godt H9 spiller. I billigere hovedtelefoner vil denne type musik ofte resultere i et mudret lydbillede. Med et sæt af høj kvalitet som H9, kan du hele vejen igennem adskille instrumenter og stemmer fra hinanden.”

(Music recorded in multiple layers demonstrates exactly how well the H9’s play. In cheaper headphones, this type of music often result in a muddy sound. With a set with high quality like the H9’s, you can easily separate instruments and voices from each other.)