| Bahman Gharesifard |
Fall 2013 |
Welcome to Signals and Systems! The main reference for this course is the book (A Mathematical Introduction to Signals and Systems, by Andrew D. Lewis). Lecture notes are provided here, mainly to guide you through the book. A major part of the materials presented in these notes are the expanded versions of the slides previously used to teach this course. Of course, at times, we might trade precision for intuition (bad practice), but hopefully these notes help answering the question: which parts of these two volumes do I need to know about?
As you will realize, our journey through this book is not chronological; we rather adapt an efficient path
which would allow us to grab enough structures from different chapters in order to be
able to study signals (and systems).
Signals (and Systems):
In this course, we are supposed to learn about signals,
and also a little bit about
systems.
But let us quickly review, with loss of precision, what we are really going to study.
We are not aiming to be mathematically precise in our description, so you will need to wait until later in the course
when, hopefully, the next paragraph makes sense to you.
Signals are really just functions,
with some physical meaning. Given this, we quickly realize
that studying them entails learning about appropriate spaces of functions. Well, we haven't defined what
we mean by space of functions,
but given your background in linear algebra
and real analysis,
you might guess, correctly, that what we need to put in place to address these
precisely are appropriate
algebraic
and analytical (well, topological) structures.
Let us describe this a bit further, in case you do not have any idea about what we mean by
algebraic and topological. One can loosely think of the algebraic structure required here
as an analog of what you have learnt in your linear algebra course as
vector spaces.
Your knowledge of vector spaces will come short, as soon as we
face infinite-dimensional spaces.
We will spend a great deal of time learning about the differences
and similarities of finite- and infinite-dimensional spaces. In
order to perform anything useful with signals, we need to equip
our algebraic structure with an additional structure (topological) which associates a notion of size to our signals,
and allows us to, roughly speaking, measure things. This relates what we are studying to your real analysis course.
After we equipped ourselves with enough structure, we finally start doing some "useful" stuff with signals,
in particular, we learn how to transform them and give a so-called
frequency-domain representations.
We hopefully agree that these transformations are useful.
You could use this paragraph at any point in the course that you ask yourself the question:
what does ... have to do with signals?
Hopefully, at least for most of you, the richness of the structures put in place to study
infinite-dimensional spaces would be the fascinating part of this course.