Experiment 3 - Fourier Synthesis of Periodic Waveforms
Department of Electrical Engineering and Electronics
September 2020, Ver. 5.0
Experiment specifications
Module(s) ELEC270
Experiment code 3
Semester 1
Level 2
Lab location Computer lab, third/fourth floor, check the lab timetable
Work In groups
Timetabled time 7 hrs
Subject(s) of relevance Fourier theorem
Assessment method Report template
Submission deadline 7 days after lab date
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Instructions:
• Read this script before attempting the experiment.
• The Pre-Lab Questions should be answered before the lab day. They are
available online and worth 10%.
• The script questions should be answered while carrying out the experimental
procedure.
• The report template should be completed with the graphs and answers to the
questions and submitted before the deadline.
1 Objectives
This experiment is aimed at:
• Studying quantitatively Fourier components of some basic periodic waveforms.
• Verifying that periodic waveforms can be synthesised by the superposition of sinusoids
having harmonically related frequencies.
• Studying the effect of filters on waveforms.
• Investigating the effect of phase on sound.
2 Apparatus
• A computer with Matlab
Students of the University of Liverpool can download a copy of Matlab for use in their personal
computers/laptops from the following link:
https://www.liverpool.ac.uk/csd/software/software-downloads/#matlab
3 Introduction
In the fields of communications, signal processing and in electrical engineering more generally, a
signal is any time-varying or spatial-varying quantity. A graph of the amplitude of this signal
plotted against time produces a waveform (an example is shown in Figure 1). Perhaps the most
familiar type of waveforms is the sinusoid shown in Figure 1 (b). Here, the voltage (V ) at any
time (t) is given by the equation:
Figure 1: Examples of waveforms.
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V (t) = Vosin(2πf t + φ) (1)
where Vo is the amplitude of the sinusoid, f is the frequency and φ is the phase angle measured
in radians.
It may be seen from the figure that this sinusoidal waveform is periodic with period T, i.e.
the waveform repeats itself after every T seconds, or V (t + nT) = V (t) for all t, where n is an
integer. The frequency f, measured in Hertz, and the period T are related by T = 1/f. The
phase angle determines the starting point of the sine wave at t = 0.
Analysis:
Many real-life signals are periodic but non-sinusoidal. Examples are shown in Figure 2. Fourier
analysis is based on a theorem which states that periodic waveforms may be expressed as the
sum of infinite simple sines and cosines (plus possibly a constant). If T is the period of a
periodic signal f(t), which satisfies certain conditions (normally satisfied by signals of practical
interest), then f(t) can be expressed as the sum of sine and cosine functions called a Fourier
series that is uniquely defined by constants known as Fourier coefficients. Hence:
Figure 2: Non-sinusoidal periodic waveforms.
f(t) = ao +
X∞
n=1
an cos(2πnfot) + X∞
n=1
bn sin(2πnfot) (2)
where fo = 1/T and is termed as the fundamental frequency, ao, an and bn (for n = 1, 2, 3, . . . )
are the Fourier coefficients that can be calculated from the following expressions:
f(t) sin(2πnfot)dt (5)
Equation 2 may also be written in an alternative form:
f(t) = co +
X∞
n=1
cn cos(2πnfot + φn) (6)
) and tan φn = −bn/an.
Note that f(t) has been expressed as the sum of a constant (DC) voltage co and sinusoids of
frequencies fo, 2fo, 3fo, 4fo and so on. The sinusoid at frequency fo, i.e. c1 sin(2πfot + φ1),
is called the fundamental component of f(t). The other sinusoids are called harmonic components; the component at frequency 2fo being the second harmonic; the component at 3fo being
the third harmonic and so on.
Representing a periodic function f(t) as a series of sinusoids of specific amplitudes is referred
to as Fourier Analysis.
Example: Fourier analysis of a triangular wave.
It is required to analyse a triangular wave (Figure 3) using Fourier analysis. First, we need to
establish an expression for the time domain waveform f(t). It can be concluded easily that f(t)
can be written as:
Figure 3: Triangular wave.
(7)
Now, using the definition Equations 3, 4 and 5, we can find that:
The spectrum of f(t) (frequency domain view) contains an infinite number of frequency components, hence, the bandwidth of f(t) is infinite. A practical system would restrict f(t) to a finite
bandwidth by removing all frequency components above certain upper limit. This would distort
the shape of the triangular waveform. Fortunately, the first few terms in the Fourier series of a
triangular wave are the most important since the amplitudes, an, decrease rapidly as n increases.
Synthesis:
The addition of a series of harmonically-related sinusoids in order to generate a periodic waveform is referred to as Fourier synthesis, and is the reverse process of analysis.
To synthesise a periodic wave with a good approximation, the sine and cosine waves with the
proper amplitudes (as defined by the coefficients) must be electronically generated and combined up to the highest possible (and practical) value of n. The larger the value of n for which
sine and cosine wave signals are generated the more nearly the synthesised waveform matches
the desired waveform. Figure 4 is an example of successive approximation of a sawtooth wave
by adding harmonics with amplitude inversely proportional to the harmonic number. The resultant waveform at each stage of addition is shown at right.
Figure 4: Successive approximation of sawtooth signal.
Fourier synthesis is used in many practical applications. For example, it is used extensively
in electronic music applications to generate waveforms that mimic the sounds of familiar musical instruments. Although many musicians can create very clear-tones on their instruments,
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all instruments have their own characteristic voice. Voices are differentiated by the shape of
their associated sound waves. All instrument voices display an array of frequencies which sum
together to produce a wave of characteristic shape. Fourier synthesis is also employed in laboratory instruments known as function (signal) generators that are used to generate different
waveforms for various purposes like electronic and communication systems test.
Example: Fourier synthesis of a triangular wave with Matlab.
The triangular wave analysed above can be synthesised in Matlab by generating the right
sine/cosine components at the right harmonic frequencies and calculating a weighted sum of
them. The Matlab script below shows how to do this based on the expressions shown in
equations 8, 9 and 10:
Figure 5: Matlab code to synthesise a triangular waveforms (3 harmonics).
Figure 6 compares the original triangular wave and its synthesised version with 1, 3, 5, and
9 harmonic components (left-to-right, top-to-bottom). It can be seen that the synthesised
waveform will reproduce the original triangular signal more accurately as more harmonic components are added. However, only the first few terms in the Fourier series of a triangular wave
are the most important.
4 Experimental work
As mentioned above, Fourier synthesis is a method of electronically constructing a signal with
a specific, desired periodic waveform. It works by combining sinusoidal harmonics (signals at
multiples of the lowest or fundamental frequency) in certain proportions. In this experiment
you are going to use Matlab to synthesis arbitrary waveforms using harmonics as shown in the
example above (you can use this example as a reference).
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Figure 6: Original triangular wave and its synthesised version with 1, 3, 5, and 9 harmonic components (leftto-right, top-to-bottom).
4.1 Part A (30 Marks)
Objectives: To familiarise yourself with Fourier synthesis using Matlab and to investigate the
extent of the accompanied error.
It is required to synthesise the following function:
(12)
where Vo is the maximum applicable voltage (you can select an arbitrary value, for example
Vo = 1 volt) and ω = 2πfo.
Requirements:
• Provide the Matlab code required to synthesise the waveform in equation 12 and the resulting waveform. [5 marks]
Questions:
• Explain the features of the resulting waveform (peak-to-peak amplitude, symmetry, ripple,
etc.) [5 marks]
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• How do you think the waveform would look if an unlimited number of harmonics was
available (i.e. n goes to ∞)? To support your answer, provide a couple of figures along
with their associated code. [5 marks]
• Referring to Equation 3, what is the value of ao? [5 marks]
• Find an expression (in terms of n) for an and bn. [5 marks]
• Plot the spectrum (frequency domain view) of f(t) using cn ). Provide the
figure and the Matlab code used to obtain it (you can use the stem function from Matlab
to plot the frequency components). [5 marks]
4.2 Part B (20 Marks)
Objective: To synthesise and study a sawtooth waveform.
The periodic waveform shown in Figure 7 is called a sawtooth function, and can be represented
mathematically as follows:
f(t) = V −
2V
T
t, 0 ≤ t < T (13)
Using, the Fourier analysis equations (Equations 3, 4 and 5), the Fourier coefficients can be
shown to be ao=0, an=0 and bn=2V/πn.
Figure 7: Sawtooth waveform.
Requirements:
1. Write f(t) in Fourier synthesis form, i.e. as in Equation 2. [4 marks]
2. Calculate the first 10 sinewave coefficients (i.e. b1, b2, . . . b10). [4 marks]
3. Synthesise the first 10 harmonics of this waveform and plot the result (provide your Matlab
code as well). [4 marks]
Questions:
• Plot in the same figure the original and synthesised sawtooth waveforms (provide your
Matlab code). Compare the resulting waveform with what you expected to see and discuss
the results. [4 marks]
• If the number of harmonics is reduced to 5, comment on the changes that will be observed
practically. [4 marks]
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4.3 Part C (15 Marks)
Objective: To synthesise and study various waveforms.
Requirements:
1. Synthesise the waveforms below. Plot the resulting waveforms and provide the Matlab
code used to obtain the plots as well. [1, 1, 1 and 2 marks, respectively]
(a) f1(t) = sin ωot −
1
9
sin 3ωot +
1
25 sin 5ωot −
1
49 sin 7ωot + . . ..
(b) f2(t) = 0.1(cos ωot + cos 2ωot + cos 3ωot + cos 4ωot + . . .).
(c) f3(t) = sin ωot −
4
3π
cos 2ωot −
4
15π
cos 4ωot −
4
35π
cos 6ωot −
4
63π
cos 8ωot −
4
99π
cos 10ωot.
(d) The waveform in (c) above with the fundamental component (sin ωot) switched off.
Question:
• Comment on your results for each waveform. [3, 3, 2 and 2 marks, respectively]
4.4 Part D (15 Marks)
Objectives: To study and simulate the effect of a linear filter on a wave.
The effect of a linear filter on a certain waveform can be simulated by passing each Fourier
component of that waveform through the filter and observing the effect on the amplitude and
phase of each separate component.
Requirements:
Consider an ideal square wave with frequency of 1 kHz expressed as a Fourier series:
V (t) = 3
π
sin(2πfot) + 1
3
sin(2π3fot) + 1
5
sin(2π5fot) + . . .
(14)
1. Synthesise and plot this square wave (provide your Matlab code as well). Calculate the
percentage overshoot of the synthesised waveform (compared with the ideal waveform) at
the discontinuity. How does this compare with the expected limit of 17.9%? [2 marks]
2. What is the name of this overshoot? Explain it. [2 marks]
3. Give the Fourier series in each case for the resulting waveform if the above square wave is
used as an input for:
(a) A low-pass filter with gain and phase responses as given in Figures 8 and 10 respectively. [1 mark]
(b) A low-pass filter with gain and phase responses as given in Figures 8 and 11 respectively. [1 mark]
(c) A band-pass filter with gain and phase responses as given in Figures 9 and 10 respectively. [1 mark]
4. Synthesise the above waveforms and draw the obtained waveforms for filters (a), (b) and
(c), providing the Matlab code as well. [3 marks]
Question:
• What is the fundamental frequency of the output from filter (c)? Why? [5 marks]
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Figure 8: Low-pass filter gain response.
Figure 9: Band-pass filter gain response.
Figure 10: Example of a phase response.
Figure 11: Example of another phase response.
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4.5 Part E (10 Marks)
Objectives: To listen to harmonics and chord, and investigate the effect of phase on sound.
In this part, you need to listen to the sound generated by the speaker of your computer when
playing one or more harmonic components of a signal.
The example below shows how to play in Matlab a tone of 500 Hz with a duration of 3 seconds.
The amplitude can be modified to adjust the volume. The sampling frequency fs = 44.1 kHz
is commonly used by most computer sound cards.
Figure 12: Example of Matlab code to play a tone.
To plot a signal generated this way, it is convenient to clip it first. Notice that the tone
generated in the example above has a frequency of 500 Hz and its period is 2 ms. This period
is repeated 1500 times in 3 seconds, so plotting the complete signal will not be very useful. The
example below shows how to clip the signal generated in the example above to just 3 periods
before plotting it.
Figure 13: Example of Matlab code to clip a long audio signal for plotting.
Consider the following wave with a fundamental frequency fo = 500 Hz:
V (t) = 10 [sin(2πfot) + sin(2π2fot) + sin(2π3fot)] (15)
Requirements:
1. For the wave in equation 15, listen to harmonics 1, 2 and 3 individually then as a chord.
Plot the chord waveform as well. Provide the Matlab code used to listen to the harmonics/chord and plot the chord. [3 marks]
2. Alter the phase of the third harmonic in equation 15 by 90◦ and repeat the tasks in the
point above. [3 marks]
Questions:
• Discuss the effect of altering the phase of the third harmonic, both on the sound and plot
of the chord. Do the sound or plot change as you alter the phase? Why? [2 marks]
• What does the above tell you about the human ear? [2 marks]
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5 Assessment and Marking Scheme
The marking scheme for this experiment is as follows:
• Results of Part A: 30 Marks
• Results of Part B: 20 Marks
• Results of Part C: 15 Marks
• Results of Part D: 15 Marks
• Results of Part E: 10 Marks
• The pre-lab test: 10 Marks
6 Plagiarism and Collusion
Plagiarism and collusion or fabrication of data is always treated seriously, and action appropriate to the circumstances is always taken. The procedure followed by the University in
all cases where plagiarism, collusion or fabrication is suspected is detailed in the University’s
Policy for Dealing with Plagiarism, Collusion and Fabrication of Data, Code of Practice on
Assessment, Category C, available on https://www.liverpool.ac.uk/media/livacuk/tqsd/
code-of-practice-on-assessment/appendix_L_cop_assess.pdf.
Follow the following guidelines to avoid any problems:
(1) Do your work yourself.
(2) Acknowledge all your sources.
(3) Present your results as they are.
(4) Restrict access to your work.
References
[1] Manual, “Fourier Synthesiser ST2603 training system”, Scientech, 2008.
[2] W Al-Nuaimy, “Lecture notes, ELEC270, Signals and Systems”, University of Liverpool.
Version history
Name Date Version
Dr M L´opez-Ben´ıtez September 2020 Ver. 5.0
Dr M L´opez-Ben´ıtez September 2019 Ver. 4.4
Dr A Al-Ataby August 2014 Ver. 4.3
Dr A Al-Ataby October 2013 Ver. 4.2
Dr A Al-Ataby and Dr W Al-Nuaimy October 2012 Ver. 4.1
Dr A Al-Ataby and Dr W Al-Nuaimy October 2011 Ver. 4.0
Dr L Momani September 2009 Ver. 3.0
Dr K Nuttall October 2003 Ver. 2.0
Dr J Marsland August 2000 Ver. 1.0