Let’s make a DUT with a simple distortion problem: It clips the signal symmetrically at 0.5 of the peak value of the signal, so if I send in a sine wave at 1 kHz, it looks like this:
Figure 1: An example of a symmetrically-clipped sine wave with a fundamental frequency of 1 kHz.
Now, to be fair, what I’m REALLY doing here is to look for the peak value of the measurement signal coming into the DUT, and then clipping it. This would be equivalent to doing a measurement of the DUT and adjusting the input gain so that it looks like a peak level of – 6 dB relative to maximum is coming in.
Also, because what I’m about to do through this series is going to have radical effects on the level after processing, I’m normalising the levels. So, some things won’t look right from a perspective of how-loud-it-appears-to-be.
If I measure that DUT using the three methods, the results look like this:
Figure 2: The impulse response of a clipping DUT, measured with an impulse (plot has been normalised for level).
Figure 3: The impulse response of a clipping DUT, measured with an MLS sequence (plot has been normalised for level).
Figure 4: The impulse response of a clipping DUT, measured with a swept sine (plot has been normalised for level).
As can easily be seen above, the three systems show very different responses. So, unlike what I claimed this post (which is admittedly over-simplified, although intentionally so to make a point…), the fact that they are measuring the impulse response does not mean that we can’t see the effects of the non-linear response. We can obviously see artefacts in the linear response that are caused by the distortion, but those artefacts don’t look like distortion, and they don’t really show us the real linear response.
In the last posting, I made a big assumption: that it’s normal to measure the magnitude response of a device via an impulse response measurement.
In order to illustrate the fact that an impulse response measurement shows you only the linear response of a system (and not distortion effects such as clipping), I did an impulse response measurement using an impulse. However, it only took about 24 hours for someone to email me to point out that it’s NOT typical to use an impulse to do an impulse response measurement.
These days, that is true. In the old days, it was pretty normal to do an impulse response measurement of a room by firing a gun or popping a balloon. However, unless your impulse is really loud, this method suffers from a low signal-to-noise ratio.
So, these days, mainly to get a better signal-to-noise ratio, we typically use another kind of signal that can be turned into an impulse response using a little clever math. One method is to send a Maximum Length Sequence (or MLS) through the device. The other method uses a sine wave with a smoothly swept-frequency.
There are other ways to do it, but these two are the most common for reasons that I won’t get into.
In both the MLS and the swept-sine cases, you take the incoming signal from the DUT, and do some math that compares the outgoing signal to the incoming signal and converts the result into an impulse response. You can then use that to do your analyses of the linear response of the DUT.
If your DUT is behaving perfectly linearly, then this will work fine. However, if your DUT has some kind of non-linear distortion, then the effects of the distortion on the measurement signal will result in some kinds of artefacts that show up in the impulse response in a potentially non-intuitive way.
This series of postings is going to be a set of illustrations of these artefacts for different types of distortion. For the most part, I’m not going to try to explain why the artefacts look the way they do. It’s just a bunch of illustrations that might help you to recognise the artefacts in the future and to help you make better choices about how you’re doing the measurements in the first place.
:To start, let’s take a “perfect” DUT and
measure its impulse response using the three methods (impulse, MLS, and swept sine)
for the MLS and swept sine methods, convert the incoming signal to an impulse response and plot it
find the magnitude response of the impulse response via an FFT and plot that
The results of these three measurement methods are shown below:
Method 1: Impulse
Method 2: MLS
Method 3: Swept Sine
If you believe in conspiracy theories, then you might be suspicious that I actually just put up the same plot three times and changed the caption, but you’ll have to trust me. I didn’t do that. I actually ran the measurement three times.
If you’re familiar with the MLS and/or swept sine techniques, then you’ll be interested in a little more information:
The sampling rate is 48 kHz
Calculating in a floating point world with lots of resolution (I’m doing this all in Matlab and I’m not quantising anything… yet…)
The MLS signal is 2^16 samples long
I’m using one MLS sequence (for now)
I am not averaging the MLS measurement. I just ran it once.
The swept sine starts at 1 Hz and runs for 10 seconds.
For both the MLS and the sine sweep, I’m applying a pre-emphasis filter to the signal sent to the DUT and a reciprocal de-emphasis filter to the signal coming from it. This puts a bass-heavy tilt on the signal to be more like the spectrum of music. However, it’s not a “pinking” filter, which would cause a loss of SNR due to the frequency-domain slope starting at too low a frequency.
My DUT isn’t really a device. It’s just code that I’m applying to the signal, so there’s no input or output via some transmission system like analogue cabling, for example…
Most of that will be true for the other parts of the rest of the series. When it’s not true, I’ll mention 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.
What’s the point?
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.*
P.S.
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.
We attended a concert last week at the Danish Guitar Camp, where Carlo Marchione played a collection of music written for various instruments in the 1700s. One of those pieces was an arrangement of the Adagio from Mozart’s Klaviersonate in B-Major K. 570, which I had never heard before.
As soon as he started, I thought…. waitaminute…. this tune sounds awfully familiar… as a Canadian…
wait… what?
Have a listen to the first 10 seconds of the Mozart, then listen to the first 10 seconds of the Canadian National Anthem. I suspect that Calixa Lavallée might have had the Mozart tune in the back of his head when he sat down to work on Théodore Robitaille’s commission.
When working on the last series of posts, I stumbled on a signal that caused an FFT analysis to look a little strange to me. This post is about that strangeness.
If I make a sine wave (in a floating point world) that sits perfectly on an FFT bin, and I do an FFT of it, the noise floor that I see is the result a lack of precision of the calculations that were used to make the signal. An example of this is shown in Figure 1.
Figure 1. Two plots of the same analysis. The top plot has a logarithmic frequency scale and a range of 20 Hz to 20 kHz. The bottom plot is on a linear frequency scale, and has a range of 0 Hz to 2 kHz.
As can be seen there, the noise floor in the fit is typically at least 300 dB down from the signal level. This means that if the signal has a peak amplitude of 1, then each bin in my FFT has a peak amplitude of less than 0.000 000 000 000 001, which is very, very, low.
If I dither and quantise the sine wave with, say, a 24-bit LPCM precision, then the result would be different, as shown in Figure 2.
Figure 2. Two plots of the same analysis. The top plot has a logarithmic frequency scale and a range of 20 Hz to 20 kHz. The bottom plot is on a linear frequency scale, and has a range of 0 Hz to 2 kHz.
Now the noise floor seen in the FFT analysis is the noise that is intentionally generated as dither to randomise the quantisation error when converting to LPCM.
However, what happens if the signal is quantised but not dithered? Then the result looks like Figure 3.
Figure 3.
This is interesting because, starting the first bin, every second bin has nothing in it, so on a decibel scale, the value is -∞ dB. Why does this happen?
The short answer is symmetry. By quantising the sine wave, I made it perfectly symmetrical.
This removes the DC content, since the positive-going portion of the waveform is identical to the negative-going portion. Therefore there’s nothing at 0 Hz (which is DC) or any of its “harmonics” (at least in the world of FFT bins…).
There is content of some kind in the other bins because our sine wave is not perfectly sinusoidal. All those steps that I put in it are an error that generates information at frequency centres other than the sine tone’s itself.
So, if you do an FFT on a sinusoidal signal and you see a result where half of the bins have nothing in them, one possible reason is that you’re dealing with a perfectly symmetrical signal.
