Lab 1: Quantisation

ECE3141 Lab 1

Information and Networks, ECE3141
Lab 1: Quantisation

1. Introduction
Many signals of interest to us originate, and are finally reproduced, in analogue form. Sound
and image are obvious familiar examples, since they start as sound pressure levels and light
intensities at specific wavelengths respectively and are finally reproduced in a similar form
using loudspeakers or earbuds, and TV, computer or smartphone screens. In between these
ends, though, we almost always carry these signals as digital signals1
, for the many
advantages that digital gives us and which we will explore in lectures.
Conversion of these analogue signals to digital form requires sampling the signals
periodically in time (and, in the case of images and video, in space as well), and representing
each sample value as a number with finite accuracy. Here, we focus on the accuracy with
which values at each of the sample points are represented, and how we can represent the set
of all the sample values most efficiently.
Quantisation does not always involve conversion from an analogue input. It can involve any
sort of operation to represent a quantity more coarsely, as one of a discrete set of possible
values. In particular, a signal may already be in digital form but at very high resolution and
we may decide to represent it less accurately (e.g., to save bits in its representation). This, in
fact, is what we will be doing in this laboratory; we will start with signals already digitised at
high definition (large number of bits per sample) and quantise them to a lower definition
(fewer bits per sample).
In this laboratory class, you will carry out the following as you investigate the effect of
quantisation and sample representations of one-dimensional signals:
• Understand some of the decisions involved in designing a quantiser.
• Listen to the effect as you vary the number of quantisation steps used to represent a
sinusoidal signal, as well as segments of recorded sound (speech and music).
• Measure the SNR resulting from quantisation of audio signals as you increase the
accuracy of the representation, and compare the result with theory
• Develop your skills in Matlab and audio processing
Note on Matlab scripting
You should make sure you understand the Matlab processing you are doing and are
encouraged to write your own code to carry out the task required. However, you have limited
time and do not want to spend it all debugging, so example script segments are provided in
1 The most familiar exceptions are AM and FM radio, which are still analogue. All TV broadcasts have now moved to digital, and with
the emergence of DAB and DAB+, radio is moving in the same direction.
ECE3141 Lab 1
2
this lab document, so you can use these to help if you want to. You can either copy and use
them, or just get hints about how to use certain functions – whatever works for you. In any
case, you still need to stitch the pieces together and add extra code to make it all work, so
you need to make every effort to understand the processes being carried out.
2. Pre-lab
Prior to the laboratory class, you must read through this entire laboratory description and
complete the pre-lab online quiz. You may also be able to complete answers to some of the
questions below which are based on theory (i.e. that don’t depend on experimental results).
You should also consider each exercise given, and think about the Matlab processing you
need to do (that is, how will the algorithm work? What instructions will you need? Do you
know how to use them?).
If you want to use your own audio clips (e.g. a short segment of music), you need to organise
this (that is, extract a short segment from the original audio) in advance of the lab class.
However, you MUST respect copyright as described in the next section.
3. Copyright
The audio clips you will be using are short segments of copyrighted content. You may use
them for your laboratory investigation, but must then delete all copies, whether original or
processed, and must not under any circumstances publish (in any sense, including posting on
social media) any version (even if it is highly distorted from the original) of these clips.
If you are bringing and using your own audio clips, these MUST have been obtained legally
according to copyright laws. The above restrictions (deleting all copies when you’ve
finished, not publishing or distributing any copies) also, of course, apply.
4. Background theory: quantifying quantisation
Figure 1 shows a continuously varying 1-D analogue signal that is sampled periodically, and
where the value at each sample point is quantised. (Of course, the quantisation is very coarse
in this figure to allow the differences to be seen; ordinarily the quantisation noise would be
much smaller.)
Suppose the original analogue function is y=f(t) defined for all t. (This is the red curve in
Figure 1.)
ys=f(nT) is the sampled sequence, where T is the sampling interval and n is an integer. ys is
defined only for 𝑛𝑛 ∈ 𝐼𝐼 (an integer). So, for example, ys(1.7T) is not defined; it has no
representation in the sampled domain. Though ys is a sampled signal, it is not yet quantised;
ys can still have any value within some bounds. For instance, −𝑉𝑉 ≤ 𝑦𝑦𝑠𝑠 ≤ 𝑉𝑉, 𝑦𝑦𝑠𝑠 ∈ 𝑅𝑅.
ECE3141 Lab 1
3
Figure 1. Sampling and quantisation of a one-dimensional analogue signal.
If we are to represent the 𝑦𝑦𝑠𝑠 in a digital system, then we must use a finite number of bits to
represent each sample value. That means that we must “round” the values of 𝑦𝑦𝑠𝑠 to a subset of
allowable output values using some form of quantisation. A linear quantiser2 is shown in
Figure 2. As a transfer function which maps the input (horizontal axis) to the output (vertical
axis), whatever value the input might have, the output will be restricted to one of a set of
discrete (quantised) values.
To make sure you understand this quantisation function, consider how the output would
change as the input varies continuously, taking on every value on the horizontal axis.
Figure 2. An example of a linear (or uniform) quantiser transfer function. This is a “uniform, midtread”
quantiser.
An obvious, and important, consideration is the difference between our original signal and
the quantised, approximate representation. To characterise our quantised sequence (let’s call
it yq), we could think of it as a distorted version, or an approximation, of the unquantised
values, i.e. 𝑦𝑦𝑞𝑞 = 𝑦𝑦�𝑠𝑠. But it is often more useful to consider it, or model it, as the sum of the
2 Quantisers need not always be linear. An example is if we wanted the introduced error (or quantisation noise, as it is defined in the
coming paragraphs) to be smaller for smaller signal levels (so the ratio of error to signal is about the same for all signal magnitudes).
In this case, the quantiser steps would be smaller closer to the origin. In fact, this is exactly what is done in telephony.
ECE3141 Lab 1
4
original, exact y(nT) values, to which some noise (specifically, “quantisation noise”, nq) has
been added.
𝑦𝑦𝑞𝑞(𝑛𝑛) = 𝑦𝑦𝑠𝑠(𝑛𝑛) + 𝑛𝑛𝑞𝑞(𝑛𝑛) = 𝑦𝑦(𝑛𝑛𝑛𝑛) + 𝑛𝑛𝑞𝑞(𝑛𝑛)
Why is such a representation useful? Because, if we can characterise this signal distortion or
noise we’re calling nq, then we can calculate its power and compare it with the original signal,
to obtain a SNR (Signal to Noise Ratio)3
. Then we can select the parameters that influence the
size of the quantisation distortion, so that the SNR is kept within defined limits.
4.1. Forms of linear quantiser – mid-tread and mid-rise
Note that one of the important decisions to make when designing a quantiser is what to do at
the origin. In Figure 2, the origin (0,0) is in the middle of a horizontal “tread”. The (obvious)
alternative is to put the origin in the middle of a vertical “rise”4
, as shown in Figure 3.
Figure 3. A variation on the linear (or uniform) quantiser from Figure 2. This is a “uniform, midrise”
quantiser.
What are the advantages and disadvantages of the “midtread” and “midrise” quantisers?
(Hint: you can answer this simply by thinking about the plots in Figures 2 & 3. You do not
need to go searching for some extra theory.)
Midtread has less error for 1s, as the largest possible error occurs at 1/2ths.
Midrise has less error for 1/2ths, as the largest possible error occurs at 1s.
3 To avoid confusion with random (e.g. thermal) noise, sometimes we use the term SQNR (“Signal to Quantisation Noise Ratio”)
instead.
4 In the building industry, the terms “tread” and “riser” are used to refer to the horizontal and vertical components of stairs. The
analogy is obvious!
ECE3141 Lab 1
5
The difference between the mid-tread and mid-rise quantisers is noted for interest, but we
will concentrate in this lab on the use of the more common mid-tread quantiser. (If you have
the time and interest, you are welcome to try both, though.)
4.2.Modelling quantisation noise5
Here we obtain a quantitative measure of the SNR due to quantisation, making some
simplifying assumptions.
Suppose that our signal before quantisation is bounded to the range −𝑉𝑉 ≤ 𝑦𝑦𝑠𝑠 ≤ 𝑉𝑉, 𝑦𝑦𝑠𝑠 ∈ 𝑅𝑅,
and that quantisation of that signal involves setting the output 𝑦𝑦𝑞𝑞 to one of M allowable
levels. If our quantiser is uniform, then the step size between the allowable levels will be a
fixed value
∆= 2𝑉𝑉
𝑀𝑀 (1)
We further assume that M is a power of 2, because that means that we can represent each
output level with an m-bit word (𝑀𝑀 = 2𝑚𝑚), without wasting bits6
.
Now, if the step size between output levels is ∆, then a moment’s thought (after looking at
the staircase function of Figures 2 or 3) will tell us that the largest possible error (provided
the input does not exceed the limits of ±𝑉𝑉) between input analogue signal and output
quantised signal will be ∆
2
. In fact, the difference between input and output values will have
the form shown in Figure 4.
Figure 4. Error signal as a function of input.
Clearly if m (and therefore M) increases, then ∆ will become smaller and so the amplitude of
the error signal will also become smaller. We would like to know not just the amplitude of
the error, but its power, because that would allow us to obtain the power ratio of SNR.
We can estimate the noise power by using a statistical argument as follows. If we are
quantising an input signal that is non-deterministic (e.g. an audio signal that could be music
or singing), then we cannot calculate in advance what the noise signal will be, but we can
deduce something about its statistical nature. We know that the probability density function
of 𝑛𝑛𝑞𝑞 will be zero outside the range −∆
2 ≤ 𝑛𝑛𝑞𝑞 ≤ ∆
2 , but what about inside that range? A
particular input value 𝑦𝑦𝑠𝑠 could be anywhere on the horizontal axis of Figure 4. However, if
we assume that M is relatively large, there is no reason to assume that any value of 𝑛𝑛𝑞𝑞 within
5 This mathematical development of the relationship between noise due to quantisation and the number of bits m used to represent
the quantisation levels is the same as that presented in lectures.
6 If M was not a power of 2, then some of the m-bit words would never be used. We could have, for instance, M=21, which would need
m=5 bits to distinguish which of the 21 possible output levels we obtain, but there would be 11 possible 5-bit words that would
never be used (25 = 32).
ECE3141 Lab 1
6
the possible output range is any more likely than another7
. Therefore, we will model the pdf
(probability density function) as constant within this range and exactly zero outside it, as
shown in Figure 5.
Figure 5. pdf of quantisation error signal.
To calculate the average noise power, we need the mean square error, or the expected value
of the square of 𝑛𝑛𝑞𝑞. That is (substituting 𝑛𝑛 = 𝑛𝑛𝑞𝑞 to keep the maths uncluttered)
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄. 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = � 𝑃𝑃(𝑛𝑛)
∞
−∞
. 𝑛𝑛2𝑑𝑑𝑑𝑑 = 1
∆ � 𝑛𝑛2𝑑𝑑𝑑𝑑
∆
2
−∆
2
= 1
∆ �
𝑛𝑛3
3 �
∆
2
− ∆
2
= ∆2
12 (2)
We have an expression for ∆ from eq. (1) that we can substitute into eq. (2), so the noise
power is
𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄. 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = ∆2
12 = 𝑉𝑉2
3. 𝑀𝑀2 = 𝑉𝑉2
3(22𝑚𝑚) (3)
The final part of the derivation is to introduce the signal power. We do not have a value for
it, but we know that the power is the square of the standard deviation 𝜎𝜎𝑦𝑦. Therefore,
substituting the quantisation noise power from eq. (3), the SNR is
𝑆𝑆𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 3. 𝜎𝜎𝑦𝑦
2
(22𝑚𝑚)
𝑉𝑉2 = �
𝜎𝜎𝑦𝑦
𝑉𝑉 �
2
3(22𝑚𝑚)
And, if we calculate the SNR in dB, we have
𝑆𝑆𝑆𝑆𝑆𝑆𝑑𝑑𝑑𝑑 = 10𝑙𝑙𝑙𝑙𝑙𝑙10 ��
𝜎𝜎𝑦𝑦
𝑉𝑉 �
2
3(22𝑚𝑚)� = 6.02𝑚𝑚 + 10𝑙𝑙𝑙𝑙𝑙𝑙10 �3. �
𝜎𝜎𝑦𝑦
𝑉𝑉 �
2
�
We can obtain an expression with the single variable m if we can get a relationship between
the signal standard deviation 𝜎𝜎𝑦𝑦 and the maximum amplitude V. This relationship is
obviously signal dependent, but we can obtain a result through measurement of “typical
signals”. In particular, we find that speech signals typically have a ratio of peak to standard
deviation of about 4:1. In this case,
𝑆𝑆𝑆𝑆𝑆𝑆𝑑𝑑𝑑𝑑 = 6.02𝑚𝑚 + 10𝑙𝑙𝑙𝑙𝑙𝑙10 �3. �
1
4
�
2
� = 6.02𝑚𝑚 − 7.27 𝑑𝑑𝑑𝑑 (4)
So, after all this derivation, we see that the SNR due to uniform (or linear) quantisation has
the following features:
- The SNR in dB is linearly related to m, the number of bits used to represent each sample
value
7 There are some particular circumstances when this would not be true. Can you think of an example? In any practically useful
situation, though, this is a reasonable assumption.
ECE3141 Lab 1
7
- The intercept of the linear relationship depends on the ratio of peak to standard deviation
of the signal being quantised
- The slope is independent of the signal and shows that we increase the SNR by about 6 dB
for every extra bit of accuracy of representation of sample values. For instance, if we
change from using 8-bit words to represent sample values, to using 10-bit words, then our
SNR will improve by about 12 dB.
5. Quantisation distortion with a sinusoidal test signal
a) Let us start our investigation of quantisation using a simple sinusoidal test signal. In
Matlab, create a vector that represents a time-varying sinusoid of, say, 1 kHz, that lasts
for a couple of seconds8
. Choose a suitable sampling frequency to use to represent it9
. If
you generate the corresponding time vector (that is, a vector that contains the times at
which each of the sinusoid values is calculated), you will easily be able to plot one
against the other later if you’re interested to do so. Play the vector (using the sound()
command) so you can hear (and check) it. Here is some example code in case it helps:
TestFrequency_Hz = XXXX;
SampleFrequency_Hz = YYYYY;
Sampling_Period=1/SampleFrequency_Hz;
Amplitude = 1;
Duration_sec = 2;
Num_Samples=SampleFrequency_Hz*Duration_sec;
time_vec=(0:Num_Samples-1)*Sampling_Period; % Time vector
signal_vec = Amplitude*sin(2*pi*TestFrequency_Hz*time_vec); %
Signal vector
sound(signal_vec,SampleFrequency_Hz);
With reference to the last line above, note that if the sample rate is not specified (i.e. if
you just use sound(signal_vec); ), the data will be played out at a default sample
frequency of 8192 Hz. Then 1000 Hz won’t sound like 1000 Hz! (Try it!)
b) Now let’s quantise this sinusoid to varying degrees of accuracy. Add some code to your
Matlab script to quantise (with a uniform, mid-tread quantiser) the sinusoid so it is
represented by a restricted number of levels10. The number of levels should be
determined by a parameter 𝑚𝑚, which is the number of bits needed to define which level it
is. That is, with m bits, we can represent up to 2m quantisation levels. (Hint: Since we’re
talking about rounding off numbers, so each is represented by a number from a smaller
set, then the Matlab functions round, ceil and floor would seem to be good candidates for
use.) You should develop some way to check that the quantiser is doing what you expect
it to do before proceeding.
Again, some example code is provided below. This expects an input in the range (-1,1)
and the quantised output is in the same range.
% midtread quantiser
>> Num_levels=2^m-1; % Note: odd number of levels
>> Quant_vec = 2*round(signal_vec*(Num_levels-1)/2) / (Num_levels-1);
8 You could carry out this and the following steps without creating an m-file, but to save you retyping and to allow for mistakes, it
would be best to use an m-file. 9You might be tempted to sample a 1 kHz sine wave at the Nyquist rate of 2 x 1 = 2 kHz. However, if you give a little thought to which
parts of the waveform are sampled if there are exactly 2 samples per cycle (the first at t=0), you'll see why we need to sample at a
rate GREATER than the Nyquist rate.
10 Matlab provides specialised calls to do quantisation, but there is value in doing it “from first principles”, to make sure you
understand exactly what is going on.
ECE3141 Lab 1
8
c) When we quantise, we want to measure the magnitude of the error introduced, so create a
quantisation noise vector, which is the difference between the original signal vector and
the quantised version. Calculate the SNR due to quantisation, in dB11.
How will you calculate SNR in dB? (That is, what is the formula and what Matlab code
will do it?)
Signal power
𝑃𝑃𝑆𝑆 = 1
𝑁𝑁� �𝑆𝑆𝑖𝑖
2�
𝑁𝑁
1
Signal to noise ratio
𝑆𝑆𝑆𝑆𝑆𝑆 = Average signal power
Average noise power
for m=2:16
 levels = 2^m-1; %
 quantVector = 2*round(signalVector*(levels-1)/2) / (levels-1);
 quantNoiseVector = signalVector - quantVector;
 Ps = 1/length(signalVector) * sum(signalVector.^2);
 Pn = 1/length(quantNoiseVector) *
sum(quantNoiseVector.^2);
 SNR_calc = 10*log10(Ps / Pn);
 SNRVector(m-1) = SNR_calc;
end
11 In the experimental parts 5 & 6, you are NOT plotting the line 6.02m-7.27 dB. If you plot a line, it doesn't make much sense to then
consider "is it linear?'. Instead, you have generated a signal vector, quantised it and can calculate the relevant signal and noise
power. You are asked to find (or measure) the actual SNR. THEN you will consider how well it matches the theoretical model that
leads to the 6.02m-7.27 formula, what the underlying assumptions of that model are, and whether they are valid.
ECE3141 Lab 1
9
d) Now, as you increase 𝑚𝑚 from 2 up to 16, listen to the quantised sinusoid and record the
SNR you obtain.
At what SNR value (and with how many bits m) can you no longer tell the difference
between the quantised and original signal when listening to it? Does the answer change
depending on whether you’re using headphones or speakers? Why?
m = 8
Easier to hear using headphones, as the bandwidth is larger (20Hz – 20kHz).
m = 2 m = 4
m = 8 m = 16
e) We want to explore the shape of the function that describes quantisation noise SNR as a
function of 𝑚𝑚.
What does theory (from lectures, and Section 4.2 above) predict will be the relationship
between SNR and m?
𝑆𝑆𝑆𝑆𝑆𝑆𝑑𝑑𝑑𝑑 = 10𝑙𝑙𝑙𝑙𝑙𝑙10 ��
𝜎𝜎𝑦𝑦
𝑉𝑉 �
2
3(22𝑚𝑚)�
The relationship between SNR and m is exponential.
0 2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 Plot of signal vector
signal
quant
0 2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 Plot of signal vector
signal
quant
0 2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 Plot of signal vector
signal
quant
0 2 4 6 8 10 12 14 16 18 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 Plot of signal vector
signal
quant
ECE3141 Lab 1
10
f) Choose a sampling rate that is a small integer multiple of the signal frequency (e.g. 8
kHz) and plot SNR against m, for m = 2 to 16.
Copy your plot here. Does it have the form you’d expect (that you described in part e, above)?
Why? (Hint: If you’re having trouble figuring out what is going on, plot the histogram of the
quantisation noise vector. This is very straightforward; just histogram(Quant_noise);
)
Slope = 5.8028
No, it does not have the form as expected. There are jagged lines because there may be other
sources of unwanted noise, such as interference and thermal noise.
2 4 6 8 10 12 14 16
10
20
30
40
50
60
70
80
90
100
110
ECE3141 Lab 1
11
g) Change the sampling rate so it is near to that above but NOT an integer multiple of the
test frequency (e.g. 7957 Hz). Repeat your plot of SNR vs m for this case.
Consider again the plot of SNR vs m. What do you find, and can you explain it?
Slope = 6.2024
2 4 6 8 10 12 14 16
0
10
20
30
40
50
60
70
80
90
100
ECE3141 Lab 1
12
h) Now change the sampling rate so it is a large integer multiple of the signal frequency
(e.g. 40 kHz). Repeat your plot of SNR vs m for this case.
Consider again the plot of SNR vs m. What do you find, and can you explain it?
Slope = 6.1743
Better slope than small integer multiple.
We need to sample at rate at least twice the bandwidth, not the highest frequency of
signal.
i) What have you learned about the relationship between the signal frequency and sampling
rate if our model (from section 4.2) is to be reliable?
2 4 6 8 10 12 14 16
0
10
20
30
40
50
60
70
80
90
100
ECE3141 Lab 1
13
It cannot be a small integer multiple of the test frequency, unless a large integer multiple is
used.
j) Before we get on to investigate it, is the observation you just made also going to occur
when we quantise real world signals (e.g. audio signals rather than just sine waves)?
Why?
Yes, because audio signals have varying levels compared to sine waves that are almost
always -A or A.
6. Quantisation distortion with real audio signals
a) Now that you’ve succeeded with a test signal, let’s repeat the exercise with an audio
sample. Modify your script to read in an audio file (use both speech and music). The
audio files ShortClip_Shine On.wav and 12secSpeechSample - Speeches That Changed
The World - Intro.wav are provided for you on Moodle or, as noted earlier, you may have
brought your own clips. (Note the copyright information at the start of this laboratory
description, and make sure you respect the request to delete all copies when you are
finished.) The provided files are digitised to 16-bit accuracy and sampled at 44,100 Hz
(you can obtain this number when you read the audio file with audioread). Modify your
Matlab code so that your signal vector is read from a file, and you also read the correct
sampling rate. Some example code follows:
FilenameString=['ShortClip_Shine On.wav'];
[signal_vec,SampleFrequency_Hz]=audioread(FilenameString);
[Num_Samples,x]=size(signal_vec);
if x>1
 disp('Warning: This is not a mono recording')
end
Duration_sec = Num_Samples/SampleFrequency_Hz;
Display_String=[num2str(Duration_sec),' second audio file, sampled at
',num2str(SampleFrequency_Hz),' Hz'];
disp(Display_String);
ECE3141 Lab 1
14
b) Again, measure SNR as a function of m, and listen to the quantised data using sound.
Is the slope of the SNR vs m function the same as in the case of the sinusoidal test signal? Up
to how many bits of quantisation can you hear the quantisation noise? Is it easier to hear with
speech or music?
Music
Slope = 6.23
Speech
Slope = 6.01
Up to 6 bits we can still hear the quantisation noise.
It is easier to hear with music, than speech.
Were you correct in your speculation at the end of Section 5 above? That is, do you need to
be so careful about the choice of sample rate? (Make sure you can explain why.)
Yes
7. Non-uniform Quantisation (Optional)
(“Optional” does not mean “I stop here”! In terms of marks, you will not be penalised if you don’t do this
section, but you may be able to make up for some earlier lost marks if you do. More importantly, if you work
through the steps below, you will get a more solid understanding of how and why non-uniform quantisation is
used in many important applications – such as every telephone call you make! The section is only marked
“optional” because if things don’t all go smoothly earlier, you may struggle to get through it in the 2 hours lab
time. If you don’t/can’t do it during the lab time, it would be a good idea to complete it in your own time
afterwards.)
With non-uniform quantisation, the step size Δ is not the same for all input values, but instead
the step sizes are small for small input values and larger for large input values. In this section,
we will explore the value of non-uniform quantisation.
2 4 6 8 10 12 14 16
0
10
20
30
40
50
60
70
80
90
100
2 4 6 8 10 12 14 16
0
10
20
30
40
50
60
70
80
90
100
ECE3141 Lab 1
15
a) Repeat your quantisation test, but this time reduce the signal amplitude by a suitable
factor (divide all the values of the signal vector by, say, 8 or 10) before quantisation.
After you’ve quantised the signal and measured the SNR, scale all the values up again
before playing the audio out, so that the volume is the same as for the original signal. Use
m=8 for this test.
Is the quantisation noise more, or less, noticeable for a smaller amplitude signal? Why?
Measure the SNR, compare it with the result without the signal attenuation and try to
explain the quantitative difference. (Hint: make sure you calculate SNR using the scaled
down version of the signal, before you scale it up again to listen to; and make sure you
understand why.)
Explain the relevance of this observation to the nonlinear (A-law or µ-law) quantisation
that is used in digital telephony.
b) OK, if non-uniform quantisation is so good, let’s test it. Matlab provides a function for 8-
bit µ-law quantisation12. It is therefore easy to quantise a signal using this function, with
the code:
mu=lin2mu(signal_vec);
Quant_vec = mu2lin(mu);
Repeat the test in part (a) above, quantising an attenuated version of the input signal, but
this time use µ-law quantisation. Again, measure the SNR and listen to the result. (Note
that you are only doing comparisons using 8-bit quantisation, because that’s the only
accuracy that the Matlab lin2mu and mu2lin functions operate with.)
12 It is sometimes referred to as “mu-law” to avoid the need to use the Greek character “µ”.
ECE3141 Lab 1
16
What do you notice about the subjective quality of the quantised signal, compared with
what you heard in part (a) above? What is the SNR, and how does it compare with the
result from part (a)?
Explain these observations.
Explore the quantised representation. First, inspect the array “mu” (or whatever variable
name you used in the code shown above) and see the range of numbers present, to convince
yourself that this really is doing 8-bit quantisation. See if you can determine how many of
the possible 8-bit numbers (i.e. 0-255) are actually used in the mu vector to represent the
attenuated signal13. Compare this with the range of values used in the linear case, and also
with the range of possible input values, and use this to explain the advantage of nonuniform quantisation.
8. Conclusion
Quantisation is an absolutely fundamental and unavoidable component of any system that
involves processing real-world analogue signals in digital form (including all image, video or
audio capture systems, but also almost any other quantity we measure today: temperature,
pressure, weight, speed,…..). By completing this laboratory, you now have a feel for what
the effects of quantisation are, and how many bits (or what SNR) are needed to avoid
perceptible quantisation noise in the case of audio. You will also now understand some of the
important design considerations for quantisers (midtread or midrise, uniform or nonuniform).
13 Because negative numbers are stored in ones complement, this is not just a matter of using min() and max() functions. One way is
to use histograms and find how many numbers in the range 0-255 are unrepresented in mu. Or you may have a more elegant
solution?
ECE3141 Lab 1
17
Before finishing, could each student please click on the “Feedback” icon for this laboratory
on Moodle (all inputs are anonymous). Record the SNR needed before you felt that the
quantisation noise was no longer audible in your audio tests on the sinusoidal test tone, the
speech sample, and music (5(d) and 6(b) above). This will allow us to obtain an overall
average, with which you can compare your result. There is also an opportunity to provide
some brief feedback on this laboratory exercise.
Finally, please submit this completed report via Moodle by the stated deadline. In so doing,
please be aware of the following:
• Even if you have had a mark assigned to you during the lab session, this mark will
not be registered unless you have also submitted the report.
• Your mark may also not be accepted or may be modified if your report is incomplete
or is identical to that of another student.
• By uploading the report, you are agreeing with the following student statement on
collusion and plagiarism, and must be aware of the possible consequences.
Student statement:
I have read the University’s statement on cheating and plagiarism, as described in the Student
Resource Guide. This work is original and has not previously been submitted as part of
another unit/subject/course. I have taken proper care safeguarding this work and made all
reasonable effort to make sure it could not be copied. I understand the consequences for
engaging in plagiarism as described in Statute 4.1 Part III – Academic Misconduct. I certify
that I have not plagiarised the work of others or engaged in collusion when preparing
this submission.
– END –