Al-Nahrain University
College of Engineering
Department of Electronic and
Communications Engineering
Communication Systems I
Laboratory Report
Third Year Class
First Semester
Experiment No.1
Experiment Title: Signals and
Spectra
Group No.3
Test Bench No.3
Participant(s) Name(s) Ali Hassan,
Ahmed Zeyad,Riyam Hameed and
Rafal Fawzi
Academic Year 3rd
Date of Experiment 2019/10/24
Report Due Date 2019/10/28
Report Submission Date 2020/02/6
Grading
Particulars
Maximum Mark
Mark Scored
Objectives, Apparatus
Procedure
Lab paper
Discussion
Promptness, Neatness and
Organization
Total Mark
Instructor Name and Signature
Date
Communication Systems I - Laboratory
Experiment 1
Signals and Spectra
Objective
To be aware of Fourier series and Fourier Transform techniques, and to practice
their applications in communication theory, so you will be able to understand the
next Experiments .
Theory
Procedure
Part I: Fourier Series Techniques
1. First step was opening MATLAB then went to the command window and typed
Fourier Series. This would bring up the application software that could be used to
demonstrate many concepts and properties of the Fourier series expansion. And the
one that we’ve already work on.
2. For each signal we’ve typed monitor its waveform and line spectra and sketched
them down in our report.
3. I had changed the number of harmonics to be displayed in line spectra and used
in the truncated FS expansion of the different types of signal, then reported the
observations.
4. Had note the plots for amplitude and phase spectra, And Stated what kind of
symmetry is present in each type of spectrum, also we had figured out why.
Part II: Fourier Transform Techniques
1. First step was opening MATLAB then went to the command window and typed
Fourier Transform. This would bring up the application software that could be used
to demonstrate many concepts and properties of the Fourier transform.
And the basic signal used was a rectangular unit pulse.
2. I’ve had Introduced a certain time delay in the function and noticed what
happened to the amplitude spectra,
and never forget to explain why.
3. We’ve been Introduced different scaling factors and commented on what we had
observing .
4. Had Introduce a frequency shift and observed the resultant signal waveform and
its spectra. So we’ve noticed and commented what we were observing.
Discussion
1-State the symmetry properties associated with each type of signal considered,
both in time and frequency domains.
Solution:
n displaying or specifying the Fourier transform of a real-valued time function it is
necessary to display the transform only for positive values of w.
A second important property is that of time and frequency scaling, spe- cifically that
a linear expansion (or contraction) of the time axis in the time domain has the effect
in the frequency domain of a linear contraction (expan- sion). In other words, linear
scaling in time is reflected in an inverse scaling in
frequency.
2. Give a brief explanation of the Gibbs phenomenon and the signal(s) with which it
can be observed in the experiment.
Solution: Fourier Series and Gibbs Phenomenon Overview
In this experiment you work with the Fourier series representation of periodic
continuous-time signals and learn about Gibbs phenomenon.
The Fourier series representation of a periodic signal, with period T=1/fo, is defined
by
where the complex Fourier series coefficients, also expressed in polar form,
are computed via the integral formula
where T is the fundamental period of the signal. The DC component of the signal is
equal to the first Fourier series coefficient and is simply the average value of the
signal over one period.
The sinusoidal components of the signal that occur at multiples of the fundamental
frequency are called harmonics.
In general, for well-behaved (continuous) periodic signals, a sufficiently large
number of harmonics can be used to approximate the signal reasonably well. For
periodic signals with discontinuities, however, such as a periodic square wave, even
a large number of harmonics will not be sufficient to reproduce the square wave
exactly. This effect is known as Gibbs phenomenon and it manifests itself in the
form of ripples of increasing frequency and closer to the transitions of the square
signal.
An illustration of Gibbs phenomenon is shown in the figure below. The figure
shows the result of adding one, three, five, seven, and nine harmonics. In all cases,
and regardless of the number of harmonics, it is observed that the overshoot of the
ripples has a constant magnitude (around 18%).
In this experiment we study the Fourier series representation of two periodic
signals, a triangular waveform and a square waveform. Both have periods equal to 2
seconds. It will be seen that Gibbs' effect is significantly more pronounced in the
square wave case.