Mathematics for Engineers 1st Edition by Georges Fiche, Gerard Hebuterne ISBN 978-1118623336 1118623336

Chapter 1: Probability Theory

• Definition and properties of events

  • The concept of an event

  • Complementary events

    • Basic properties

  • Properties of operations on events

    • Commutativity

    • Associativity

    • Distributivity

    • Difference

    • DeMorgan’s rules

• Probability

  • Definition

  • Basic theorems and results

    • Addition theorem

    • Conditional probability

    • Multiplication theorem

    • The posterior probability theorem

• Random variable

  • Definition

  • Probability functions of a random variable

    • Notations

    • Cumulative distribution function

    • Probability density function

  • Moments of a random variable

    • Moments about the origin

    • Central moments

    • Mean and variance

    • Example applications

  • Couples of random variables

    • Definition

    • Joint probability

    • Marginal probability of couples of random variables

    • Conditional probability of a couple of random variables

    • Functions of a couple of random variables

    • Sum of independent random variables

    • Moments of the sum of independent random variables

    • Practical interest

• Convolution

  • Definition

  • Properties of the convolution operation

    • The convolution is commutative

    • Convolution of exponential distributions

    • Convolution of normal (Gaussian) distributions

• Laplace transform

  • Definition

  • Properties

    • Fundamental property

    • Differentiation property

    • Integration property

    • Some common transforms

• Characteristic function, generating function, z-transform

  • Characteristic function

    • Definition

    • Inversion formula

    • The concept of event indicator and the Heaviside function

    • Calculating the inverse function and residues

    • The residue theorem

    • Asymptotic formula

    • Moments

    • Some common transforms

  • Generating functions, z-transforms

    • Definition

    • Moments

    • Some common transforms

  • Convolution

Chapter 2: Probability Laws

• Uniform (discrete) distribution

• The binomial law

• Multinomial distribution

• Geometric distribution

• Hypergeometric distribution

• The Poisson law

• Continuous uniform distribution

• Normal (Gaussian) distribution

• Chi-2 distribution

• Student distribution

• Lognormal distribution

• Exponential and related distributions

  • Exponential distribution

  • Erlang-k distribution

  • Hyperexponential distribution

  • Generalizing: Coxian distribution

  • Gamma distribution

  • Weibull distribution

• Logistic distribution

• Pareto distribution

• A summary of the main results

  • Discrete distributions

  • Continuous distributions

Chapter 3: Statistics

• Descriptive statistics

  • Data representation

  • Statistical parameters

    • Fractiles

    • Sample mean

    • Sample variance

    • Moments

    • Mode

    • Other characterizations

• Correlation and regression

  • Correlation coefficient

  • The regression curve

    • The least squares method

• Sampling and estimation techniques

• Estimation

  • Point estimation

    • Average

    • Variance

    • Estimating the mean and variance of a normal distribution

    • Example: estimating the average lifetime of equipment

  • Estimating confidence intervals

    • Example 1: estimating the mean of a normal distribution

    • Example 2: Chi-2 distribution in reliability

    • Estimating proportion

    • Estimating the parameter of a Poisson distribution

• Hypothesis testing

  • Example: testing the mean value of a normal distribution

  • Chi-2 test: uniformity of a random generator

  • Correlation test

Chapter 4: Signal Theory

• Concept of signal and signal processing

• Linear time-invariant systems and filtering

  • Linear time-invariant systems

  • Impulse response and convolution function of an LTI system

  • Filtering function

• Fourier transform and spectral representation

  • Decomposing a periodic signal using Fourier series

  • Fourier transform of an arbitrary signal

  • Dirac delta function and its Fourier transform

  • Properties of Fourier transforms

    • Time and frequency shifts

    • Convolution product and filtering

    • Product of functions and transform convolution

    • Product of functions and modulation

    • Energy conservation and Parseval’s theorem

• Sampling

  • Sampling function

  • Shannon sampling theorem

• Quantization and coding

  • Quantization noise

  • Coding power

• Discrete LTI system

• Transforms for digital signal processing

  • The z-transform

    • Definition

    • Time translation

    • Discrete convolution

    • Inversion

  • Fourier transform of a discrete signal

  • Discrete Fourier transform

    • Definition

    • Properties

  • Cosine transform

  • The fast Fourier transform (FFT)

    • Cooley-Tukey FFT algorithm

• Filter design and synthesis

  • Definitions and principles

    • Principle

    • Causality and stability

  • Finite impulse response (FIR) filters

    • Design methodology

    • FIR filter synthesis

    • Low-pass, high-pass and band-pass filters

  • Infinite impulse response (IIR) filters

    • Filter design from models of the s plane

    • Butterworth model

    • Chebychev model

    • Synthesis of IIR filters

    • Low-pass, high-pass and band-pass filters

  • Non-linear filtering

    • Median filtering at rank N

  • Filter banks and multirate systems

    • Sub- and up-sampling

    • Multirate filtering and polyphase bank

    • Signal decomposition and reconstruction

    • Half-band filters

    • Quadrature mirror filters

• Spectral analysis and random signals

  • Statistical characterization of random signals

    • Average

    • Autocorrelation function

    • Power spectral density

  • Filtering a random signal

    • Spectral density of a filtered signal

    • Filtering white noise

  • Sampled random signals

    • Autocorrelation function

    • Cross-correlation function

    • Power spectral density

  • Filtering a sampled random signal

    • Spectral density of the filtered signal

    • Correlation between input and output signals (cross-correlation)

    • Colored noise

  • Spectral estimation

    • Estimating the autocorrelation function

    • Non-parametric spectral estimation with periodogram

    • Parametric spectral estimation: ARMA, AR, MA

    • Toeplitz matrix and Levinson algorithm

• Linear prediction, coding and speech synthesis, adaptive filtering

  • Linear prediction

    • Variance minimization

    • Example of speech processing: coding and synthesis

  • Adaptive filtering

    • The least squares method with exponential forgetting

    • The stochastic gradient descent algorithm

    • Example: echo cancelation

Chapter 5: Information and Coding Theory

• Information theory

  • The basic diagram of a telecommunication system

• Information measurement

  • Algebraic definition of information

  • Probabilistic definition of information

  • Self-information

  • Unit of information

  • Conditional information

  • Mutual information

• Entropy

  • Entropy of a memoryless source

  • Entropy of a binary source

  • Maximum entropy

  • Joint entropy of random variables

  • Average conditional entropy

  • Additivity and joint entropy

  • Average mutual information

  • Conditional average mutual information

  • Extension to the continuous case

• Source modeling

  • Concept of sources with and without memory

  • Discrete Markov source in discrete time

  • The main types of source

    • The binary source

    • Text, or alphabetic source

  • Information rate of the source

• Source coding

  • Code efficiency

  • Redundancy of a code

  • Instantaneous and uniquely decipherable codes

• Shannon’s first theorem

  • Optimal coding

  • Shannon’s first theorem

• Coding and data compression

  • Huffman coding algorithm

  • Retrieval algorithms and decision trees

  • (Reversible) data compression

    • The Huffman method

    • Fano method

    • Shannon’s method

    • Arithmetic coding

    • Adaptive and dictionary methods

  • Image compression

    • Describing an image: luminance and chrominance

    • Image redundancy

    • The discrete cosine transform (DCT)

    • Quantization and coding

    • Recent methods: wavelets

    • JPEG2000

• Channel modeling

  • Definition of the channel

  • Channel capacity

  • Binary symmetric channel

• Shannon’s second theorem

  • The noisy-channel coding theorem (Shannon’s second theorem)

• Error-detecting and error-correcting codes

  • Algebraic coding

    • Principles

    • Hamming distance

    • Detection and correction capability

    • Additional definitions and properties

    • Linear block codes, group codes

    • Cyclic codes

  • Convolutional codes

    • D-transform

    • Graphical representation, graphs and trellis

    • Viterbi’s algorithm

  • Combined codes and turbo codes

    • Interleaving

    • Product codes

    • Concatenation

    • Parallelization

    • Turbo codes

• Cryptology

  • Encryption

    • Symmetric-key encryption

    • Public-key encryption

  • Digital signature

  • Signature and hashing

Chapter 6: Traffic and Queueing Theory

• Traffic concepts