# MATHEMATICAL METHODS FOR ENGINEERING

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Versione italiana
2021/2022
Teacher
MICHELE MIRANDA
Credits
6
Didactic period
Secondo Semestre
SSD
MAT/05

#### Training objectives

The aim of this course is to present in a reasonable rigorous way some mathematical tools particularly suited for the applications in many fields of engineering, in particular in signal processing. Among these, the more important will be the Z-transform and the two simplest integral transform, the Laplace transform (for causal signals) and the Fourier one (for more general signals). Time permitting, we will also briefly discuss the Hilbert transform.
The main goal is to help the students to fill the gap between elementary calculus and advanced courses.

The main acquired knowledge will be:

- definitions, properties and applications of the Z-transform
- definitions, properties and applications of the Laplace transform
- definitions, properties and applications of the Fourier transform
- basics about the theory of distributions

The main skills will be:

- computation of the integral transforms of the most common signals
- solution of ordinary differential equations by means of integral transforms
- time discretization of some simple linear differential equations and approximate solution
- theoretical construction of a low-pass filter
- use of the isolated singularities of a complex function for computing a contour integral (needed, for example, to invert a Z-transform)

#### Prerequisites

The whole usual contents of a Calculus' class, Mathematical Analysis 1A, 1B and 2

#### Course programme

FUNCTIONS OF ONE COMPLEX VARIABLE (15h)

- complex numbers
- continuous functions
- holomorphic functions
- Cauchy-Riemann equations
- conjugate harmonic functions
- Cauchy's theorem
- deformation of contour
- Cauchy's integral formula
- mean value formula for harmonic functions
- power series
- holomorphic functions are analytic
- zeros of holomorphic functions
- Liouville's theorem
- the fundamental theorem of Algebra
- isolated singularities
- residue's theorem
- bilateral series
- Laurent's theorem
- partial fraction decomposition

Z-TRANSFORM (5h)

- definition
- properties and remarkable formulas
- initial value theorem
- final value theorem (non-tangential form)
- regularity
- inversion formula
- applications of the Z-transform to finite difference equations

LEBESGUE INTEGRAL AND L^P SPACES (7,5 h)

- informal definition of Lebesgue integration
- theorems on exchanging limit and integration (Fatou, dominated convergence, monotone convergence)
- Fubini's and Tonelli's theorems
- definition of L^p spaces
- Young's, Holder's and Minkowski's inequalities
- density theorem
- continuity in norm of translations
- convolutions: definition and Young's inequalities for convolutions
- examples
- regularization by convolution
- approximation by convolution

LAPLACE TRANSFORM (10 h)

- L-transformable causal signals
- definition of Laplace transform
- abscissa of convergence
- derivative of the Laplace trasformata
- the Laplace transform is holomorphic
- Laplace transform of the derivative
- Laplace transform of the convolution
- formulas for scalings and temporal delay
- translations of the Laplace transform in the complex plane
- inversion formula for piecewise C^1 signals
- inversion of rational functions, by means of the partial fraction decomposition
- solution of Cauchy's problem for an ordinary linear differential equation, with constant coefficients
- transfer function and impulse response
- bilateral Laplace transform
- Mellin transform

FOURIER TRANSFORM (11 h)

- definition for an L^1 signal
- comparison with Laplace transform
- continuity and boundedness of the Fourier transform
- Riemann-Lebesgue lemma
- differentiability of the Fourier transform
- Fourier transform of the derivative
- link between regularity and decay
- inversion formula for piecewise C^1 signals
- duality formula
- uncertainty principle
- the Schwartz class S
- properties of the Fourier transform in S
- Parseval's and Plancherel's formulas
- band-limited signals
- example of (ideal) low-pass filters
- example of (ideal) band-pass filters
- the Shannon-Whittaker sampling formula
- the aliasing phenomenon: examples

DISTRIBUTIONS (11,5 h)

- informal introduction: derivative of the Heaviside step function
- convergence of functions in S
- definition of tempered distribution
- Dirac's delta
- slowly growing functions and regular tempered distributions generated
- series of deltas
- the tempered distribution "principal value of 1/t"
- elementari operations with tempere distributions: linear combinations, change of variable, multplication by a regular function, convolution with a function
- converging sequences of tempered distributions
- distributional derivative
- Fourier transform of a tempered distribution
- examples of Fourier transforms (Heaviside step function, sign, Dirac's delta, constants, principal value of 1/t)
- Sochocki-Plemelj formula
- Dirac's comb
- Poisson's summation formula
- Hilbert transform

#### Didactic methods

The course is organized as follows: lectures on the topics of the course, held by the teacher in charge. The lectures consist of theoretical parts, interspersed by exercises and examples which illustrate the concepts presented.

#### Learning assessment procedures

The examination consists of a colloquium lasting about 1 hour. Such a colloquium consists of at least 3 different questions, concerning at least 2 topics touched by the course. At least one of the questions will be of theoretical nature.

In order to facilitate the oral exam and in order to keep its duration within 1 hour the student is requested to solve and send to the theacher by e-mail at least the day before the exam some home exercise proposed during the lectures and divided by chapters; it is is up to the student the choice of the numbers and type of exercises to submit (at least one per chapter and in any case at least three exercises in total would be appropriate, bearing in mind that the more exercises you submit, the better you will be evaluated during the exam)

Passing the final exam is the proof that knowledge and abilities outlined in the training objectives of the course have been achieved.

#### Reference texts

- Lecture notes that will be made available on the GoogleClassRoom of the course

- C. Barozzi "Matematica per l'Ingegneria dell'Informazione" Zanichelli (for all the topics, except for the Z-transfom)

- M. Giaquinta, G. Modica "Note di metodi matematici per ingegneria informatica" Pitagora (only for the part on the Z-transform)

FURTHER TEXTS

- D. Mari "Trasformata di Laplace per l'Ingegneria" Pitagora
- L. Badia, D. Mari "MatES - Esercizi di Matematica per l'Elaborazione dei Segnali" Pitagora
- G. Gilardi "Analisi tre" Mc Graw Hill