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# earfluff and eyecandy

## a very irregularly-updated blog on random topics

# Category: math

# My new hero!

# Don’t believe your eyes

# Paper organ

# Noise, Averages, and Incorrect Assumptions, Part 1

## Frequency content

## Averaging

# So much for professionals…

# Early computer music

# Grebulons from Babylon???

# For your basic encryption needs…

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.

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…

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…

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.

Let’s talk about the subject of noise. To begin with, we have to agree on a definition, which might become the biggest part of the problem. For normal people, going about their daily lives, “noise” is a word that usually means “unwanted sound”, as the Grinch explained, just before hitching up his dog to a sleigh:

However, “noise” means something else to an audio professional, who is consequently not “normal” and does not have a life – daily or otherwise.

When an audio professional says “noise”, they mean something very specific – “noise” is an audio signal that is random. For example, if the noise signal is digital, there would be no way to predict the value of the next sample – even if you know everything about all previous samples.

So, “noise”, according to the professionals, is a signal made of a random sequence. However, we can be a little more specific than this. For example, in order to make the noise plotted in Figure 2, above, I asked Matlab to generate 64 random numbers between -1 and 1 using the code

rand(1,64)*2-1

So, although this will generate random numbers for me (we’ll assume for the purposes of this discussion that Matlab is able to generate truly random numbers… actually they’re “pseudorandom” – but let’s pretend for today…), there are some restrictions on those values. No matter how many random values I ask for using this code, I will never get a value greater than 1 or less than -1.

However, if I used a slightly different Matlab function – “randn” instead of “rand”, using the following code

randn(1,64)

I would get 64 random numbers, but they are not restricted as my previous bunch were, as you can seen in Figure 2a, below.

Again, it’s impossible to guess what the next value will be, but now you can see that it might be greater than 1 or less than -1 – but it will more likely to be closer to 0 than to be a “big” number.

So, Figures 2 and 2a show two different types of noise. Both are random numbers, but the *distribution* of the numbers in those two sequences are different.

To be more specific:

Let’s use the “rand(1,1000000)*2-1” function to make a sequence of 1,000,000 samples. We know that this will produce values ranging from -1 to 1. So, we’ll divide this range into 100 steps (-1.00 to -0.98, -0.98 to -0.96, -0.96 to -0.94, … and so on up to 0.98 to 1.00 – I know, I know, there’s some overlap here, but I didn’t want to use a bunch of less than or equal to symbols… The concept is the important point here… Back off!). For each of those divisions, we’ll count how many of the 1,000,000 samples fall inside that smaller range, and we’ll plot it. This is shown in Figure 3a.

As you can see in Figure 3a, there are about 10,000 samples in each “bin”. This means that about 10,000 samples have a value that fall within each of the smaller slices of the total range. Also, if I added up the values of all 100 of those numbers plotted in 3a, I would get 1,000,000, since that’s the total number of values that we analysed.

If I do the same thing to the numbers coming out of the other function: “randn(1,1000000)” it would look very different. As we already saw, the numbers can be outside the range of -1 to 1, and they’re more likely to be around 0 than to be far away from it. This can be seen in Figure 3b.

Just to clean up a loose end – there are official terms to describe these differences. The noise plotted in Figure 2 has what is called a “rectangular distribution” because a plot of the distribution of the values in it (if we measure enough sample values) eventually looks like a rectangle (a blue rectangle, in the case of Figure 3a). The noise plotted in Figure 2a has a “normal distribution”, as can be seen in the bell-like shape of the plot in Figure 3b. There are many other types of distributions (for example, rolling 2 dice will give you random numbers with a “triangular distribution”), but we’ll leave it at 2 for now.

So, we’ve seen that we can have at least two different “types” of random number sets – but let’s do a little more digging.

Here’s where things get a little interesting. If I take the 1,000,000 samples that I used to make the plot in Figure 3a and I pretend it’s an audio signal, and then calculate the frequency content (or spectrum) of the signal, it would look like the green plot in Figure 4a, below. If I did a frequency analysis of the signal used to make the plot in Figure 3b, it would look like the red plot.

What’s interesting about this is that, basically speaking, the two signals, although they’re very different, have basically the same spectra – meaning they have the same frequency content, and therefore they’ll sound the same. This is really evident if I smooth these two plots using, for example, a 1/3 octave smoothing function, as is shown below in Figure 4b.

As you can see there, the very fine peaks and dips in the response shown in Figure 4a, when averaged, smooth to the flat response shown in Figure 4b.

Now we have to clear up a couple of issues…

The first is that, as you can see in the plot above, the spectrum of both random signals (both noise generators) is flat. However, due to the way that I did the math to calculate this, this means that we have an equal amount of energy in equal-sized frequency ranges throughout the entire frequency range of the signal. So, for example, we have as much energy from 20-30 Hz as we do from 1000-1010 Hz as we do from 10,000 – 10,010 Hz. In other words, we have equal energy per Hz. This might be a little counter-intuitive, since we’re looking at a semilogarithmic plot (notice the scale of the x-axis) but trust me, it’s true. This means that, in both cases, we have what is known as a “white noise” signal – which is a colourful way of saying “noise with an equal amount of energy per Hz”.

The second thing is that I just lied to you. In order to get the pretty plots in Figures 4a and 4b, I had to take long signals and analyse how much energy we got, over a long period of time. If I had taken a shorter slice of time (say, 100 samples long) then the spectra plots would not have looked as pretty – or as flat. For example, if I just take the first 100 samples of both of those noise generator outputs, and calculate the spectra in those, I get the plots in Figure 4c, below.

Notice that these plots do not look nearly as flat – even though I’ve smoothed them with a 1/3 octave smoothing filter again. Why is this?

Well, the problem is probability. Since the two signals are random, there is no guarantee that all frequencies will be contained in them for any one little slice of time. However, there is an equal probability of getting energy at all frequencies over time. So, in a really strange universe, you might get all frequencies below 1000 Hz for one second, and then all frequencies above 1000 Hz for the next second – over two seconds, you get all frequencies, but at any one moment, you don’t.

What this means is that “white noise” doesn’t necessarily contain energy at all frequencies equally – unless you wait for a long time. If you listen for long enough, all frequencies will be represented, but the shorter the measurement, the less “flat” your response.

Just for the purposes of comparison, let’s also find out what the distribution of values is for a sine wave – since this might be interesting later… This is shown in Figure 5, below.

It may be interesting to note that “normal” sounds like music and speech have distributions that look much more like Figure 3b than like Figure 5, as is shown in the examples in Figures 6, 7, and 8. This is why you shouldn’t necessarily use a sine wave to measure a piece of audio equipment and draw a conclusion about how music will sound…

If I play the first kind of white noise (the one shown in Figure 2) and measure how loud it is with a sound pressure level meter, I’ll get a number – probably higher than 0 dB SPL (or else I can’t hear it) and hopefully lower than 120 dB SPL (or else I’ll probably be in pain…). Let’s say that we turn up the volume until the meter says 70 dB SPL – which is loud enough to be useful, but not loud enough to be uncomfortable.

Then, I switch to playing the other kind of white noise (the one shown in Figure 2a) and turn up the volume until it also measures 70 dB SPL on the same meter.

Then, I switch to playing a sine wave at 1 kHz and turn up the volume until it also reads 70 dB SPL.

If I were to compare what is coming into the loudspeaker for each of those three signals, after they have been calibrated to have the same output level, they would look like the three signals in Figure 9.

These will all sound roughly the same level, but as you can see in the plot in Figure 9, they do not have the same PEAK level – they have the same average level – averaged over time…

However, if we were to do a frequency-dependent analysis of these three signals, one of these things would not look like the others… Check out Figure 4a, which I’ve copied below…

Notice that, in a frequency-dependent analysis, the sine wave (the black line sticking up around 1000 Hz) looks much louder (35 dB louder is a lot!) than the green plot and the red plot (our two types of white noise). So, even though all three of these signals sound and measure to have roughly the same level, the distribution of energy in frequency is very different.

Another way to think about this is to say that each of the three signals has the same energy – but the sine wave has all of it at only one frequency *all the time* – so there’re more energy right there. The noise signals have the energy at that frequency only some of the time (as was explained above…).

ANOTHER way to think about this is to put a glass of water in the bottom of an empty bathtub. If the depth of the water in the glass is 10 cm – and you pour it out into the tub, the depth of the water will be less, since it’s distributed over a larger area.

As you can see in the title of this posting, this is “Part 1″… All of this was to set up for Part 2, which will (hopefully) come next week. As a preview: the topic will be the use (and mis-use) of weighting functions (like “A-weighting”) when doing measurements…

What’s impressive is that the Babylonians were more mathematically advanced than we thought. The odd thing is that the reason they developed the math was suspiciously like that used by the Grebulons…

… but be warned – apparently, this code has already been cracked…

Click the image below to take you to a download site for an Enigma machine emulator in Matlab.

For more info on the Enigma machine