## Advanced Course of Higher Mathematics (Lectures & supervised practical works).

*Level*

Third–year students (CIR3 & U3) of the ISEN Engineering School.
*Detailed Content*

- Theory of functions of a complex variable.
- Recalls of basic notions from mathematical analysis: sequences of numbers, functions of a real variable, functions of several real variables, limits,
asymptotic notations (e.g. big
*O*notation), differential calculus, Taylor and Maclaurin series, integral calculus, Riemann integral, line integrals, Green's, Stokes' & Ostrogradsky–Gauss theorems. - Complex numbers: algebraic form, polar or trigonometric form, complex plane, elementary operations, complex conjugate,
modulus and argument of a complex number, trigonometric and hyperbolic functions of a complex number, Euler's formulæ, de Moivre's formula, exponential function of a complex number,
the
*n*th roots of a complex number, logarithm of a complex number (Ln and ln functions). - Functions of a complex variable: analytic functions (3 different points of view: Cauchy–Riemann conditions & harmonic functions, Cauchy's contour integral, Weiererstrass), classification of the singularities, Laurent series & Weiererstrass expansion theorem, Cauchy's integral formula and its applications to the evaluation of contour integrals, Cauchy's residue theorem and its numerous applications (Jordan's lemma, evaluation of integrals), Mittag–Leffler theorem and versatile methods for the summation of series.
- Introduction to Operational Calculus and Fourier series: Laplace, Fourier and Hartley transforms and their applications,
- Introduction to Special Functions of Mathematical Physics and to orthogonal polynomials: Gamma function and its derivatives (Digamma function and Trigamma function), Beta function, Riemann Zeta function, Bessel functions, Legendre polynomials, Hermite polynomials.
- Introduction to probability theory and statistics: random variables, probability, stochastic processes, ergodicity, elements of the estimation theory.

*Downloadable documents*

- Test for the evaluation of the initial level in mathematics (September 19, 2011).
- First intermediate examination for CIR3 students (October 12, 2011).
- First intermediate examination for U3 students (October 19, 2011).
- Second intermediate examination for CIR3 students (November 30, 2011).
- First final examination for U3 & CIR3 students (January 17, 2012).
- Third intermediate examination for U3 students (March 14, 2012).
- Third intermediate examination for CIR3 students (April 18, 2012).
- Fourth intermediate examination for CIR3 students (May 23, 2012).
- Second final examination for CIR3 students (June 26, 2012).

*Note that during the examination graphic calculators, laptops, mobile phones and tablets are*

**strictly forbidden.***Recommended Literature*

- A. M. Legendre.
*Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures. Tomes I–III*. Mme Ve Courcier, Imprimeur–Libraire pour les Mathématiques, rue du Jardinet 12, quartier Saint–André–des–Arc, 1811–1817 (there are some recent reeditions of this book). - G. Baranenkov, B. Demidovich, V. Efimenko, S. Kogan, G. Lunts, E. Porshneva, E. Sycheva, S. Frolov, R. Shostak and A. Yanpolsky.
*Problems in Mathematical Analysis*. Mir Publishers, 1964. - M. R. Spiegel.
*Theory and problems of complex variables*. McGraw–Hill, 1964. - L. I. Volkovyskii, G. L. Lunts and I. G. Aramanovich.
*A Collection of Problems on Complex Analysis*. Pergamon Press, 1965. - R. V. Churchill.
*Complex variables and applications.* - R. V. Churchill.
*Complex variables and applications. Student solutions manual.* - M. A. Evgrafov.
*Analytic Functions*. W. B. Saunders Company, 1966. - P. Louquet & A. Vogt.
*Analyse (3)*. Librairie Armand Colin, Paris, 1974. - B. A. Fuchs & B. V. Shabat,
*Functions of a Complex Variable and Some of Their Applications (in 2 vols.)*. (International Series of Monographs in Pure and Applied Mathematics). Pergamon Press, 1961, 1964. - N. W. McLachlan.
*Complex Variable & Operational Calculus with Technical Applications*. Cambridge University Press, 1942. - A. I. Markushevich.
*Theory of Functions of a Complex Variable (2nd edition)*. AMS Chelsea Publishing, American Mathematical Society, 2005. - Н. Я. Авдеев.
*Задачник по курсу ТФКП.* - И. М. Ангилейко, Р. В. Козлова.
*Задачи по ТФКП.* - М. Б. Балк, В. А. Петров, А. А. Полухин.
*Задачник-практикум по курсу ТФКП.* - М. А. Евграфов, К. А. Бежанов, Ю. В. Сидоров, М. В. Федорюк, М. И. Шабунин.
*Сборник задач по теории аналитических функций.* - Л. И. Волковыский, Г. Л. Лунц, И. Г. Араманович.
*Сборник задач по ТФКП.* - Е. С. Половинкин.
*Курс лекций по ТФКП.*

## Laboratory Works in Probabilty

*Level*

Third–year students of the
University of Toulon.
*General description*

There are 3 labworks in all. Duration time of each labwork is 4 hours. At the end of each labwork,
students should leave a **handwritten report**. Final evaluation is based on the mean of these reports, as well as on the personal participations and contributions during the labworks.

*Downloadable documents*

Labwork
descriptions in PDF, as well as all necessary *.m and *.mat files (versions of 2008). For more information, please
contact me or
Bernard Xerri, who designed these labworks.## University class in Signals and System (supervised practical work in classroom)

*Level*

Third–year students of the
University of Toulon.
*Content*

Analog and digital signals, Fourier transforms, Hartley transform, Z–transform, spectrum, Parseval's theorem (signal's energy), auto– and cross–correlation
functions, spectrum density, notion of systems, linear and time–invariant (LTI) systems, analog and digital filtering, convolution product, complex analysis,
residus (Cauchy's theorem) and their applications in signal processing (especially, for the calculation of inverse Fourier/Hartley/Z/Laplace transforms),
sampling and reconstruction of signals (first Shannon's theorem).
*Downloadable documents*

- A little basic formulary.
- A mock exam on Fourier transform and Bessel's function.
- An exam on periodicity and Fourier series (real and complex).
- An exam on basic filtering, residus (Cauchy's theorem), Fourier transform, Z–transform, Fourier series.
- An exam on analog filtering, Fourier transform, complex variables, residus (Cauchy's theorem).

*Recommended Literature*

- A. V. Oppenheim.
*Signals and systems.* - A. V. Oppenheim and R. W. Schafer.
*Digital signal processing*(russian translation). - J. G. Proakis and D. G. Manolakis.
*Digital signal processing. Principles, algorithms and applications (3rd edition).* - J. G. Proakis.
*Digital communications (4th edition).* - V. K. Madisetti and D. B. Williams.
*Digital Signal Processing Handbook.* - L. R. Rabiner and B. Gold.
*Theory and applications of digital processing*(russian translation). - S. V. Vaseghi.
*Advanced digital signal processing and noise reduction (2nd edition).* - R. L. Rabiner and R. W. Schafer.
*Digital processing of speech signals*(original book in english, russian translation). - P. Brémaud.
*Signal et communications. Modulation, codage et théorie de l'information (cours de l'École Polytechnique).* - Сегеенко.
*Цифровая обработка сигналов.* - H. P. Hsu.
*Signals and systems*(Schaum's series). - M. H. Hayes.
*Digital signal processing*(Schaum's series). - H. P. Hsu.
*Analog and digital communications*(Schaum's series).

## Laboratory Works in Signals and System

*Level*

Third–year students of the
University of Toulon.
*General*

There are 5 labworks in all. Duration time of each labwork is 4 hours. At the end of each labwork,
students should leave a **handwritten report**. Final evaluation is based on the mean of these reports, as well as on the personal participations and contributions during the labworks.

*Downloadable documents*

Labwork descriptions (versions of 2008):
- labwork 0 (introduction to Matlab).
- labwork 1 (deterministic signals, discrete filters).
- labwork 2 (discrete Fourier transform).
- labwork 3 (sampling, Shannon's theorem, aliasing, signal reconstruction).
- labwork 4 (random signals).

## Laboratory Works in Image Processing

*Level*

Third–year students of the
University of Toulon.
*General description*

There are 4 labworks in all. Duration time of each labwork is 3 hours. At the end of each labwork,
students should leave a **handwritten report**. Final evaluation is based on the mean of these reports, as well as on the personal participations and contributions during the labworks.

*Downloadable documents*

Labwork
descriptions in one PDF file (version of 2007). For more information, please
contact me or
Nadège Thirion–Moreau, who designed these labworks.*Recommended Literature*

- CRC Press.
*The Image Processing Handbook (3rd edition).* - Bernd Jahne.
*Springer Digital Image Processing (5th extended edition).* - CRC Press.
*Multimedia Image and Video Processing Handbook.*

## Laboratory Works in Signal Processing

*Level*

Fourth–year students of
Minatec (formely ENSERG
Engineering School), which is a part of
the INP Grenoble.
*General description*

There are 5 labworks in all. Duration time of each labwork is 4 hours. At the end of 5 labworks, there is a
2–hour handwritten examination. Final evaluation is based on the mean of these reports.
*Objectives*

To provide necessary technical skills in analog and digital signal processing, in use of the corresponding measurement
and calibrating instruments (oscilloscope, signal generator, spectrum analyzer), and in signal processing modelling software (Matlab & Comsys).
*Content*

Analog and digital filtering, Fourier transform, spectrum analysis, analog and digital modulations, frequency conversion techniques,
sampling, Shannon's theorem, digital signal transmission.
*Downloadable documents*

- Short descriptive document
^{*}(see p. 33). - Labwork descriptions (versions of 2005): theoretical part, practical part.
- Examination, labwork 1: analog modulations AM/FM.
- Examination, labwork 6: frequency conversion, direct–conversion receiver (DCR), Single Side Band (SSB) modulation, superheterodyne receiver, FM radio design.
- For the examinations of the labworks 2–5, please contact me.

^{*}Original document comes from this site.

## Top 6 Most Stupid Questions & Remarks That I'he Heard from my Students

- “What is the cotangent?”
*– Mathieu C., 3rd–year student, ISITV*^{†}. - “What is the geometric progression?”
*– 3rd–year student, ISITV*. - After an exam on complex analysis, a 3rd–year student, who got a quite bad mark, tried to convince me that he deserved a better one by
arguing: “Logarithms of negative numbers do not exist! At least, I've never heard about this...”
*– Mohamed Amine A.–H., ISITV*. - “What is the decibel?”
*– several 4th–year students, INP Grenoble*^{‡}. - During the labworks in signal processing, two 4th–year students, having a signal generator and an oscilloscope, said that they had no idea
how to measure the transfer function gain of a filter
*(INP Grenoble)*. - One of the basic questions of the second exam in signal and systems was: “What is the impulse response?”
Despite the extreme simplicity of the question, many students did not respond to this question correctly.
The “best” answers are resumed below (original answers in french are given as images that appear when the mouse is on the text;
they were scanned from original exam sheets):
- “The impulse response is
*y(t)*or*y[n]” – Mathieu C., 3rd–year student, ISITV.* - “The impulse response is
*h[n]*in discrete”*– Rémi L., 3rd–year student, ISITV.* - “The impulse response is the equation
*h[n]*or*h[t]*depending on whatever the signal is discrete or continuous”*– Alexis T., 3rd–year student, ISITV.* - “The impulse response is the signal whose variable depends on time”
*– Mahamadou C., 3rd–year student, ISITV.* - “The impulse response is transfer function ×
*pulse δ (H×δ)*” – Again*Mathieu C., 3rd–year student, ISITV.*

- “The impulse response is

^{†}After I've heard this question, just for fun, I decided to include this question into the exam (question 2.1). The BIG (and the BAD) surprise was that from 20 students, there were 2 students

*(Hind B. & Othman K.)*who did not answered this question correctly. One of them simply did not respond to the question; the other one first wrote that it was the ratio sin/cos, and then, he replaced this ingenious formula by the another one: arcsin/arccos... see the original here...

^{‡}Also, some students were unable to convert linear ratios into decibel ones. Moreover, there were two 4th–year students, which have obtained the voltage output/input amplifier ratio equal to 1000. When I asked them to express this gain in dB, they replied me: “We need calculator for this...”

**Nota Bene:**since this section of my WEB–site is under construction, some documents may be temporary unavailable.