ELEN0070-2 : Signal processing

Study Programmes 2015-2016

ELEN0070-2

Signal processing

Duration :

45h Th, 15h Pr, 40h Proj.

Number of credits :

Bachelor in engineering		5
Bachelor in engineering		5
Master in biomedical engineering (120 ECTS)		5
Master in electrical engineering (120 ECTS)		5

Lecturer :

Jacques Verly

Language(s) of instruction :

English language

Organisation and examination :

Teaching in the second semester

Units courses prerequisite and corequisite :

Prerequisite or corequisite units are presented within each program

Course contents :

The course provides an advanced introduction to the fundamental elements of signal processing. It mainly considers signals that are functions of time, but it emphasizes the applicability of the concepts and results to functions of other types of independent variables, such as position. Occasional references are made to the extension from one-dimensional (1D) signals to multidimensional signals, in particular 2D. The course covers both continuous-time (CT) and discrete-time (DT) signals. It maintains a maximum of parallelism between the CT and DT cases. The CT and DT convolutions are covered, and the fundamental importance of convolution is explained, especially in the context of linear, time-invariant (LTI) systems. The course covers in great detail the two most important and extraordinary types of signals of signal processing, i.e. the complex exponential and the Dirac impulse. These lead to the key notions of impulse response and frequency response. A significant part of the course is dedicated to the "spectrum" of signals, and thus to the Fourier series (FS) for periodic signals, and to the Fourier transform (FT) for arbitrary signals. Fourier analysis is motivated via the study of the "plucked string". This leads to the CTFS, and to its DT equivalent, the DTFS. (Each acronym is self-explanatory; e.g. CTFT stands for "continuous-time Fourier transform".) These transforms are then taken in a very pragmatic way to the limit, which leads to the very important CTFT and DTFT. The relation between these and the corresponding series is then examined. In the end, the CTFT and DTFT are found to be also applicable to periodic signals through the use of Dirac impulses. The concept of "signal-transform pair" is emphasized and exploited throughout. The course covers sampling theory as a simple application of Fourier transforms. Because of the emphasis on FSs and FTs, the course can be viewed both as a course in signal processing and as a course on the Fourier transform and its applications. The course is taught at an advanced level. In particular, most developments are made directly for complex-valued signals, and Dirac impulses are presented both as limits of ordinary functions and as distributions.

The main topics of the course are:
l What signals are. Why deal with complex signals from the start. Examples of signals in daily life. DT and CT. Operations on the independent variable (e.g. time) and the dependent variable (i.e. the amplitude). The not-so-trivial definition of periodicity.
l Complex exponentials and their multiple interpretations. Concept of negative frequency.
l Convolution, introduced as an accumulation of "things" over time. Interpretations. How to compute convolutions.
l CT impulses, i.e. Dirac impulses, viewed both as limits of ordinary functions and as distribution. DT impulses. Introduction to distributions. Properties of impulses. Representation of signals. Bed-of-nails functions.
l Fourier series (FS), introduced via musical instruments, and more specifically via the "plucked-string" problem.
l CT Fourier transform (CTFT). Properties and theorems. Importance of duality. Power of thinking in terms of CTFT pairs. Special case of periodic signals and relation to CTFS. Sampling and sampling theorem as direct, amazing results from properties of CTFT. Methods for computing CTFTs, with emphasis on exploiting other CTFT pairs.
l DT Fourier transform (DTFT), presented by preserving a maximum of parallelism with CTFT. Properties and theorems. Special nature of duality. Power of thinking in terms of DTFT pairs. Special case of periodic signals and relation to DTFS. DT sampling and DT sampling theorem as direct, amazing results from properties of DTFT. Methods for computing DTFTs, with emphasis on exploiting other DTFT pairs.
l Connection with scientific and engineering applications, throughout the course.

Learning outcomes of the course :

The student will understand and be able to use the key concepts in the topics listed above. He/she will understand the mathematical and theoretical underpinnings of these concepts. He will have both a precise mathematical understanding and a highly visual and intuitive view of all the concepts.

The student will understand the concepts of complex exponentials and of impulses. He will understand - and perhaps discover - why the integral often used in connection with Dirac impulses is just a notation and not a real integral. He will understand the need for treating impulses in a special way, i.e. more by what they do, than by what they are. He will develop the reflex of treating a carrier wave as a periodic complex exponential (in telecommunications), and a calcification as a Dirac impulse (in medical imaging).

The student will be able to compute key "quantities" such as convolutions and Fourier transforms (CTFT, DTFT, CTFS, and DTFS). He will juggle with all the properties and theorems of these transforms, and the relations between the latter. He will adopt the mindset of thinking in terms of FT pairs. He will not hesitate to go back and forth between the time domain and the frequency domain. He will understand the power of combining previously-derived FT pairs and FT properties and theorems to obtain new results.

The student will discover that some familiar physical devices do FTs instantly ... without worrying about whether the FTs converge!

Most importantly, the student will become friend with the convolution, all the forms of FTs, and Dirac impulses. He will no longer be afraid of them! He will have a highly visual and intuitive view of them. He will have fun manipulating these concepts. He will not hesitate to put these concepts to use in applications. He may become highly creative in using them.

The student will have a chance to revisit mathematical concepts that may have become rusty, such as the true nature of complex numbers, complex-valued functions of complex variables, analytic/holomorphic functions, and integration in the complex plane. He will (re)discover one of the most beautiful and amazing equations of all of mathematics!

Through the "laboratories" (further described below), the student will have created, processed, and visualized signals and their spectra using computers.

The laboratory sessions in small groups will help the student to develop both technical skills (such as problem solving, capacity to apply theoretical concepts to real data, critical analysis of results, and writing-up of reports), as well as soft skills (such as team work, getting organized to meet deadlines, and dealing with Murphy's law).

The student will learn to dig into the recommended references to find complementary information and to identify the problems/exercises that relate to the material covered in class.

The students will learn - hopefully early on - that mastering the course material requires a lot of practice in solving problems.

Due to the fact that the instructor makes extensive use of the blackboard and rarely uses slides, and that the lectures are highly dynamic and interactive, the learning outcomes will be greatly increased if the student attends the classes.

The style of the course is highly influenced by the teaching practices of leading American universities, such as MIT and Stanford.

Prerequisite knowledge and skills :

Knowledge of complex numbers and basic calculus.

Planned learning activities and teaching methods :

The course has three features intended to increase the attractiveness of the course for the student:
l It makes an abundant use of graphs for signals and their spectra, and is thus highly visual. For example, a FT property or theorem and a FT pair are expressed graphically whenever possible. This offers a refreshing change from the traditional mathematics courses with few or no figures.
l It makes frequent historical references to those who made significant contributions to the field and its underlying mathematics, such as Euler, the Bernoulli's, Fourier, Dirac, Schwartz, Bracewell, and Oppenheim.
l It strives to show how the theory and principles discussed are immediately applicable to domains as diverse as speech, sound (including the Doppler effect), music, telecommunications, modulation, electrical circuits, image processing, optics (in particular Fourier optics), lasers, antennas, radars, radio-astronomy, tomographic imaging, synthetic aperture radars (SAR), and medical imaging (X-ray computerized tomography (CT)and magnetic resonance imaging (MRI)).

Mode of delivery (face-to-face ; distance-learning) :

The course consists of
l "ex-cathedra" lectures (mostly by the instructor),
l Matlab demonstrations (by teaching assistants)
l occasional review/problem sessions on specific topics (by the teaching assistants),
l laboratories (in small groups),
l possibly one or more written midterm tests/quizzes (announced ahead of time),
l a written final exam.

The lectures are given mostly by the instructor, who makes intensive use of the blackboard. While the optional course notes follow quite closely what is taught in class, these notes cannot possibly reflect precisely the "spirit" that is communicated in class, and the dynamics and interactivity of the class. It is thus highly recommended that the student attend classes. The lectures present a mix of theoretical concepts and of exercises. Therefore, except for the occasional review/problem sessions, the student should not expect the class on any given day to be clearly divided into a theoretical part and an exercise part. The instructor uses the time in a flexible way, which allows him to address the specific needs of the students.

"Laboratory" refers to an activity done by small groups of students. The students in each group can do a laboratory at any time before the corresponding deadline, possibly at home. Each laboratory consists of reading, experimenting on the computer, and writing a report (one per group). There are 3 to 4 laboratories. The laboratories are not always synchronized with the topics covered in class.

The possible written midterm tests are announced ahead of time.

Recommended or required readings :

Reading:

The primary, recommended references are
l Signals & Systems, Second edition, by Alan V. Oppenheim and Alan S. Willsky, Prentice Hall, 1997.
l Applied Digital Signal Processing, by Dimitris G. Manolakis and Vinay K. Ingle, Cambridge University Press, 2011.

The secondary, recommended reference is
l The Fourier Transform and its Applications, Third edition, by Ronald N. Bracewell, McGraw Hill, 2000.

While much of the material covered in class is found in these references, the course does not follow closely any of them (or any other book). Bracewell's book is particularly recommended for the CTFT, the Dirac impulse, and the pictorial dictionary of CTFT pairs.

With reference to the book by Manolakis & Ingle, the course pretty much covers the Chapters 1 to 6 (with the exception of Chap. 3, on the z-transform).

All books contain excellent problems/exercises. Many of these problems, or variations thereof, appear on the written tests and the final exam.

The book by Manolakis & Ingle gives Matlab examples and Matlab problems.

Course notes:

The instructor uses personal, handwritten course notes during class. Before or after each class, the instructor gives copies of most of these course notes, to a representative of the students. Each student is free to use these copies to make his own copies. The course notes are thus optional. These personal course notes are not, and cannot be, placed online.

Assessment methods and criteria :

The final grade for the course is based on the grades for the written final exam, one or more written midterm tests, and the laboratories. Between 75% and 80% of the final grade is determined by the final exam, and the balance is determined by the tests and laboratories. During the year, the instructor may offer to the students that the tests be taken into consideration only in the cases where the score on the test increase their final grades; the same rule is then applied to all students.

The (written) final exam is "open notes". This means that the student can have his own notes, as well as copies of the optional course notes provided by the instructor. No other document or textbook can be used. Thus, the laboratory reports cannot be used.

The final exam consists mainly of a set of problems to be solved. Some problems are likely to come, with or without modifications, from the recommended references. It is thus important that the student learn to find and solve as many relevant problems as possible. The instructor may provide a list of relevant problems in some of the recommended references. The final exam may include questions about the laboratories.

The tests may be "open notes" or "closed notes".

All the laboratories must be done. However, the instructor and/or the teaching assistants may decide to evaluate only one laboratory, and they may decide to take into account how many of the laboratories were turned in by each group of students. Although all the students in a group normally receive the same grade for a given laboratory, the instructor reserves the right to take into account the specific contribution of each student. It is therefore very important for each group to turn in the reports for all the laboratories, and for each member of a group to participate actively.

Work placement(s) :

Organizational remarks :

None.

Contacts :

Instructor:
Prof. Jacques G. Verly
jacques.verly@ulg.ac.be

Teaching assistant:
Clémentine François
cfrancois@ulg.ac.be