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Teaching Signals and Systems - A First Course in Signal Processing
Conference Paper · May 2020
DOI: 10.1109/ICASSP40776.2020.9054231
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TEACHING SIGNALS AND SYSTEMS - A FIRST COURSE IN SIGNAL PROCESSING
Nikhar P. Rakhashia ?
Ankit A. Bhurane †
Vikram M. Gadre ?
?
†
Indian Institute of Technology Bombay
Indian Institute of Information Technology Nagpur
ABSTRACT
Signals and systems is a well known fundamental course in
signal processing. How this course is taught to a student can
spell the difference between whether s/he pursues a career
in this field or not. Giving due consideration to this matter,
this paper reflects on the experiences in teaching this course.
In addition, the authors share the experiences of creating and
conducting a Massive Open Online Course (MOOC) on this
subject under edX and subsequently following it up with deliberation among some students who did this course through
the platform. Further, this paper emphasizes on various active
learning techniques and modes of evaluation to ensure effective and holistic learning of the course.
Index Terms— Signals and Systems, Massive Open Online Course (MOOC), Active learning
1. INTRODUCTION
The course on signals and systems primarily not only builds
the fundamentals for advanced courses such as digital signal
processing, adaptive signal processing, wavelets and multirate signal processing, but also acts as an essential tool in
various engineering fields like electrical, mechanical, chemical, aerospace etc. A heuristic understanding of mathematics and the underlying physical interpretation helps to retain
curiosity and appreciate importance of the course. Teachinglearning methods have evolved with advancements in technology. With easy access to internet through mobile phones
and other electronic gadgets, students quickly learn from online resources. An instructor should take advantage of this
digital revolution and should structure and mould the instructions in such a way that students get attracted to learn. While
the ease of access through MOOC encourages active learning,
the classroom environment, live discussions and laboratory
sessions are essentially more important in the course. In this
paper, authors share the experiences of delivering a flipped
We hereby gratefully acknowledge the support received, from the
‘Knowledge Incubation Under TEQIP-III’ Project at IIT Bombay (TEQIPIII-KITE at IIT Bombay) for the academic work related to this paper. Further,
we gratefully acknowledge the Bharti Centre for Communication in the Department of Electrical Engineering, IIT Bombay and IEEE Signal Processing
Society for financial support to attend and present this paper.
978-1-5090-6631-5/20/$31.00 ©2020 IEEE
classroom mode of instruction [1] as well as creating and conducting MOOC on “Signals and Systems” under edX.
2. ACTIVE LEARNING APPROACHES
The two main widely accepted active learning based approaches of instructions are as follows: a modern flipped
classroom approach for a typical strength of 50 to 70 students and a MOOC based approach to reach masses through
geographical outreach. These modes of instruction largely
depend upon pre-lecture structure followed by in-class evaluation, post-lecture testing and content adaptation. Further, it
requires precisely and strategically recorded quality instructional videos and active teaching associates [2].
2.1. Modern Classroom Teaching
The classroom teaching helps to stimulate critical thinking by
bringing active and collaborative environment, thus making it
the best way to learn the course. With the advances in technology, a typical modern classroom teaching relies on a blended
approach of instructions. The course is pre-recorded in the
form of short duration video lectures that focus on core content of the course [3]. A video lecture is released prior to the
regular scheduled live lecture. The recorded lectures enhance
the curiosity for interaction, discussions and group activities
that are followed in the classroom. The lecture starts with a
recapitulation of the concepts presented in video lectures and
is supported by classroom activities. Numerical and hands-on
sessions in the classroom, together with classroom teaching
help the instructor to attend to the students in person, address
their queries, and help them in filling up subtle conceptual
gaps.
2.2. Active Learning through MOOC
To reach mass audiences, MOOC based platforms like
NPTEL, edX, and Coursera have marked their popularity
in last few years with classroom teaching traits successfully
exhibited. The MOOC lectures depict peer-to-peer learning
as well as the instructor responding to questions/doubts of the
students. During the lecture, the instructor poses challenging
questions and encourages the students to post solutions on
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• Manipulation: Sketch and perform operations on different types of signals in both the domains.
the course discussion forum. Periodically, instructor observes
and responds to the discussion forum wherein he addresses
doubts raised by the students. Thus, the feedback is obtained
about how well students are doing with the course content
which helps in adapting the flow of instruction. The lecture
notes corresponding to each video are made available in the
form of handouts. Some of the video lectures also depict the
laboratory sessions for the students to get acquainted with the
practical concepts of course.
• Analysis: Analyze signals and systems by applying appropriate transformation techniques and relate them in
time and frequency domain.
• Comparison: Compare analogous continuous-time
(CT) and discrete-time (DT) operations.
• Applications: Apply and demonstrate the techniques
for signal processing and communication systems.
2.3. Students’ perspective on modern classroom learning
• How confident do you feel about the content after you
watch the videos / animations ?
It was found that more than 80% of students watched the
videos before the class. More than 90% of students found the
pre-class recapitulations, visualizations and discussions extremely useful. Around 80% of them felt confident and were
curious to know more about the content after watching the
videos and animations. These statistics show the effectiveness of the blended approach of instructions for the course,
and at the same time emphasize the importance of an instructor in a live classroom.
r
Pe
CTFS
DTFS
ic
• Do you find the recapitulation / discussions at the beginning of class helpful?
iod
iod
• Do you watch videos regularly on time?
The MOOC on Signals and Systems offered by the Indian Institute of Technology Bombay is in two parts namely
EE 210.1x and EE 210.2x [6],[7]. EE 210.1x covers topics
such as introducing and formalising signals and systems, Linear Time Invariant (LTI) systems and the Fourier Transform
(FT). EE 210.2x covers topics such as sampling and reconstruction, Discrete Time Fourier Transform (DTFT), Laplace
Transform, Z-Transform and system analysis in generalised
transform domain. An overview of the memory map of a student’s perspective of interconnection between various tools
and domains is shown in Figure 1.
ic
Pe
r
In order to know the grasp of course instructions, a questionnaire was conducted in a class of 70 students at the Indian
Institute of Information Technology Nagpur, to know their
views about the mode of instruction. The questionnaire consisted of the following questions:
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Sampling
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on
-P
er
N
on
-
Pe
r
io
di
LPF
CTFT
di
c
DTFT
LPF
LT
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ZT
3. MOOC ON SIGNALS AND SYSTEMS
Since the target audience in any MOOC ranges from beginners to professionals, it is important to plan the MOOC accordingly to make it useful and relevant for diverse participants. In MOOC, the course is recorded in the form of video
lectures divided into different modules. An empirical study
of video production [4] states that shorter videos are more
engaging. Hence, the duration of videos in a MOOC is normally 5 to 10 minutes. The most important challenge facing
any MOOC is the high dropout rates of participants from the
midst of the course [5]. Proper care needs to be taken while
designing the course to reduce dropout rates.
The course on signals and systems is a blend of mathematical tools and engineering concepts, and thus both aspects need to be covered in the course. The prerequisites for
the course are the basic understanding of complex theory and
calculus. The course usually comprises two main parts: natural (time) domain analysis and transform (frequency) domain
analysis. By the end of the course, a student should be able to
perform the following:
DFT
Fig. 1: An overview of interconnection between various domain transformation tools.
4. EMPHASIZING FUNDAMENTAL CONCEPTS
Irrespective of the platform chosen, the course should help
students answer the following fundamental questions:
• Why is a domain transformation required?
• What is the need of complex analysis in the real world?
Why are complex exponentials used as a basis?
• Why are different transformation techniques required?
• How does the abstraction of systems help in addressing
signal processing problems? What are the applications
of LTI and non-LTI systems?
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4.1. Parallel treatment of CT and DT analysis
Students usually find it difficult to interpret and deal with the
domain transformations. For the same, it is instructed that CT
and DT signals and systems be treated in parallel. While most
of the times the representation is similar and straightforward,
it is essential to emphasize the difference between continuous and discrete frequency variables. For example, the range
of frequency variable is bounded in discrete domain as compared to continuous domain where it is infinite. Similarly,
as opposed to the CT Fourier Series, the DT Fourier Series
does not face the problem of convergence. The spectrum of
sampled signals differs for the continuous and discrete case.
All such differences are very essential to be noted and can be
emphasised by covering the CT and DT transformation tools
in parallel. This would also help the students to identify and
apply appropriate tools as per scenario given and bridge the
concepts in both the domains. It is essential to cover the concept of discretization of signals in the initial few classes.
4.2. Core Concepts
Convolution being a core concept in this course should be
presented with several analogous examples and visualizations. The mathematics behind it and functional interpretation should be derived and several examples should be
extensively practiced both graphically and analytically. The
concept of sampling introduced with respect to time domain
can be revisited in the transform domain in order to relate the
operations in both the domains. As most of the properties for
domain transformation are analogous in nature, they should
be critically compared with the properties of other transformations. In that way, the students get a broader picture of
all the transformation tools and their properties. Further, the
concept of duality property of transform helps the students to
interpret the analogy between natural and transform domain.
5. VISUALS AND ANIMATIONS
Animations are extremely helpful in visualizing the underlying concepts. For instance, visualizing a normal exponential
could be trivial, however, visualizing a complex exponential
could be difficult and can be explained with the help of a 3D animation of a rotating vector with its projection on real
and lateral (imaginary) axis. The reconstruction of a signal
using Fouries series components, and Gibbs’ phenomenon
can be demonstrated using an illustration [8]. For another
instance, convincing students that signals can be represented
in terms of complex exponentials would be difficult without
an appropriate visualization [9]. Further, the aliasing effect
can be demonstrated by relating it to the wagon-wheel effect, whereas the importance of the phase in practice can be
demonstrated with the help of two images by retaining the
magnitude spectrum and exchanging phases [10]. An example of the illustrations is shown in Figure 2.
Fig. 2: Fouries series synthesis using scaled complex exponentials, mapping rotating vectors on real and imaginary axis,
example of convolution operation
These visuals can be generated through numerical computational software. While most of the institutions rely on
professional computing platforms like M ATLAB [11], a large
community has started to explore alternate open-source platforms like GNU Octave, Scilab, and Python [12] mainly due
to two reasons: free availability and large community support.
Further, these open-source software have several libraries that
can be readily utilised for laboratory sessions as well as advanced research projects.
6. LABORATORY SESSIONS, COURSE PROJECTS,
AND COMMUNITY NETWORKING
Laboratory is an essential component of this course and
should go in parallel with the theory. The initial laboratory
sessions of this course usually focus on the basic signal operations along independent and dependent variables. The
laboratory task here could be a simple activity of recording one’s own voice followed by plotting and hearing it. This
would help students get the physical interpretation of different
signal operations. For example, scaling along the dependent
axis refers to amplification or attenuation, and scaling along
the independent axis refers to slower or faster playback. This
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is how students can appreciate and relate the operations implemented at the back end of a music player. Further, this
would not only give a feel of dealing with real-life signals but
would also help in enhancing curiosity about the new terms
like mono / stereo channels, and sampling frequency.
It is instructed to have a significant weight for course
projects that can be executed by a group of 3 to 5 students,
preferably from varied disciplines, where they work on hardware or software platforms of their choice. The course itself
has become largely interdisciplinary, thus covering large
number of research areas. A student from computer sciences
may opt for building a music equalizer or photo editing application using cloud platforms. A student from aeronautical
sciences can take a project based on simulation, calibration
and optimization of various aerodynamic models and signaling techniques. A biomedical engineering student may
consider working on the development and analysis of systems for diagnosis of various human diseases using machine
learning tools. Collaborating the students from varied disciplines boosts the active involvement and knowledge sharing.
For example, a project to build a device that monitors vein
beats can be taken by students from branches like electronics, computer science and biomedical engineering. They
can contribute in circuit development, application interfacing, and data interpretation respectively and collaboratively.
Moreover, the students can be encouraged to collaborate with
doctoral candidates, where the latter can assign small problem
tasks to the former. This would not only help the students to
be in pace with the current research trends, but also escalate
research progress.
An additional component of research know-how can be
added in the evaluation where a group of students critically
present their understanding of a relevant research paper in the
area of signal processing. The research papers can be identified from top conferences and journals like ICASSP, ICIP,
GlobalSIP, SPCOM, etc. available on popular indexing platforms like IEEE Xplore, Elsevier Science-direct, Springer or
on social platforms like Research Gate, Academia or Mendeley. Further, an instructor may also make students aware of
various trusted online Question and Answer communities like
DSP Stack Exchange [13] and Stack Overflow [14]. The collaboration can be extended by approaching local industries,
medical institutes, and other relevant organizations so that the
students get hands-on opportunities to work on live projects
and instruments. Such collaboration helps in reducing the
industry-academia gaps.
7. EVALUATION AND BEYOND THE COURSE
A typical evaluation in this course should take into consideration holistic learning of students by appropriately reserving
points for theory, laboratory, and research awareness. Some
innovative methods are listed in [15]. Apart from a typical evaluation pattern, a provision for special points (around
10 − 20% of the total weight) gives the instructor an additional flexibility that can be utilized for encouraging students
to take up theory or laboratory challenges. The continuous
assessments not only ensure the evaluation based on recently
covered topic, but also help the instructor to keep track of the
student’s progress. This may include pre-laboratory or prelecture assessment. While summative assessment evaluates
students on the basis of examinations, the formative assessment evaluates the students on the basis of projects, assignments and academic participation [16].
We suggest the evaluation procedure that focuses on both
summative and formative assessments through two types of
assessment scores: saturated and unsaturated. In a saturated
grading scheme, the academic participation marks are limited, whereas in an unsaturated grading scheme, the students
can earn any number of marks in the participation component
without a limit. The saturated score focuses on summative assessment and gives more weight to the examination component. On the other hand, the unsaturated score focuses more
on the formative assessment giving more weight to research
and development as well as creativity component. Accordingly, saturated and unsaturated scores are computed for every student, from which a relatively better score is awarded to
the student. This ensures the balance between the formative
and summative assessment schemes.
The course may be concluded by giving a glimpse of limitations of domain transformation tools learnt in the course.
For example, one of the limitations of Fourier Analysis that
can be emphasised is the inability to distinguish time-varying
frequency components. The students can therefore be encouraged to explore time-frequency methods such as short-time
Fourier Transform. This way, the curiosity of learning new
tools can be raised. The new principles can be demonstrated
analytically as well as through simulations. For instance,
from the concepts of sampling and band-pass filters, students
can be encouraged to design reconstruction filter banks.
8. CONCLUSION
The course on signals and systems is extremely useful irrespective of the disciplines of engineering. The concepts and
tools covered in this course find numerous applications and
act as basic building blocks for analysis and synthesis of systems found in a variety of fields. The teaching of this course
should take into consideration the holistic approach of learning by focusing on laboratory sessions, projects and research
components along with the theoretical aspects. This paper
expresses the authors’ thoughts and experience in both classroom based and MOOC based teaching. Further, the role of
abstraction, visualizations, interconnection of tools, and parallel treatment of closely related topics is emphasized. Topics
to be highlighted in the course are listed along with their impact and importance.
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9. REFERENCES
[1] Jonathan Bergmann and Aaron Sams, Flip your classroom: Reach every student in every class every day, International society for technology in education, 2012.
[2] Barry Van Veen, “Flipping signal-processing instruction
[sp education],” IEEE Signal Processing Magazine, vol.
30, no. 6, pp. 145–150, 2013.
[3] Maureen J. Lage, Glenn J. Platt, and Michael Treglia,
“Inverting the classroom: A gateway to creating an inclusive learning environment,” The Journal of Economic
Education, vol. 31, no. 1, pp. 30–43, 2000.
[4] Philip J Guo, Juho Kim, and Rob Rubin, “How video
production affects student engagement: An empirical
study of mooc videos,” in Proceedings of the first
ACM conference on Learning@ scale conference. ACM,
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[5] John Kerr, Suzy Houston, Leah Marks, and Athene
Richford, “Building and executing moocs: A practical review of glasgow’s first two moocs (massive open
online courses),” 2015.
[6] EE 210.1x, “Signals and systems, part 1,” https:
//courses.edx.org/courses/course-v1:
IITBombayX+EE210.1x+1T2018a/course/
[Accessed: 20 Oct 2019].
[11] B. L. Sturm and J. D. Gibson, “Signals and systems
using matlab: an integrated suite of applications for exploring and teaching media signal processing,” in Proceedings Frontiers in Education 35th Annual Conference, Oct 2005, pp. F2E–21.
[12] M. R. Lovejoy and M. A. Wickert, “Using the ipython
notebook as the computing platform for signals and systems courses,” in 2015 IEEE Signal Processing and
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[13] Stack Exchange,
“Signal processing stack exchange,” https://dsp.stackexchange.com/
[Accessed: 20 Oct 2019].
[14] Stack Overflow,
“Stack overflow,” https://
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[15] A. Gupta and A. Farswan, “Rethinking teaching practices for signal processing education,” in ICASSP 2019
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7878–7882.
[16] Wynne Harlen and Mary James, “Assessment and learning: Differences and relationships between formative
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[7] EE 210.2x, “Signals and systems, part 2,” https:
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[8] Ankit A. Bhurane,
“Fourier series of square
wave. demo of gibbs phenomenon with overshoot calculation,” https://in.mathworks.
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[9] Amro,
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[10] Ankit A. Bhurane, “Demonstration showing importance
of phase in images,” https://in.mathworks.
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