Mathematics for Engineers by Fiche and Hébuterne

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Contents of Mathematics for Engineers eBook

• 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
• De Morgan’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
• Mathematics for Engineers
• 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- 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
• Mathematics for Engineers
• Estimating confidence intervals
• Example : estimating the mean of a normal distribution
• Example : Chi- distribution in reliability
• Estimating proportion
• Estimating the parameter of a Poisson distribution
• Hypothesis testing
• Example: testing the mean value of a normal distribution
• Chi- 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
• Mathematics for Engineers
• 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
• JPEG
• 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
• Mathematics for Engineers
• Parallelization
• Turbo codes
• Cryptology
• Encryption
• Symmetric-key encryption
• Public-key encryption
• Digital signature
• Signature and hashing
• Chapter 6. Traffic and Queueing Theory
• Traffic concepts
• The Erlang concept
• Traffic modeling
• The concept of processes
• Arrival process
• Renewal process
• Poisson arrivals
• The use of the Poisson process
• Service process
• Exponential distribution
• Residual service time
• Erlang distribution
• Hyperexponential distribution
• General arrival and service processes
• Markov and birth/death processes
• State concept
• Markov chains
• Birth and death processes
• Queueing models
• Introduction
• A general result: the Little formula
• PASTA property (Poisson arrivals see time averages)
• The elementary queue: the M/M/ system
• Resolution of the state equations
• Using generating functions
• Waiting time distribution
• The M/M/R/R model (Erlang model)
• The M/M/R queue and the Erlang-C formula
• The M/M/∞ queue and the Poisson law
• The M(n)/M/R/R queue and the Engset formula
• Models with limited capacity
• More complex queues
• Multi-bitrate Erlang model
• The embedded Markov chain
• The number of clients in a system
• Waiting times: Pollaczek formulae
• Introduction: calculation of residual service time
• The Pollaczek-Khintchine formula
• Example : the M/M/ queue
• Example : the M/D/ queue
• Generalization: Takacs’ formula
• The Benes method: application to the M/D/ system ˘
• The G/G/ queue
• Pollaczek method
• Application to the stochastic relation of the queue to one server
• (GI/G/ queue)
• Resolution of the integral equation
• Application to the M/G/ queue
• Application to the G/M/ queue
• Other applications and extension of the Pollaczek method
• Queues with priorities
• Work conserving system
• The HoL discipline
• Using approximate methods
• Reattempts
• Peakedness factor method
• Approximate formulae for the G/G/R system
• Appendix: Pollaczek transform
• Chapter 7. Reliability Theory
• Definition of reliability
• Failure rate and bathtub curve
• Reliability functions
• System reliability
• Reliability of non-repairable systems
• Reliability of the series configuration
• Reliability of the parallel configuration
• Reliability of the series-parallel configuration
• Reliability of the parallel-series configuration
• Complex configurations
• Non-repairable redundant configurations
• Reliability and availability of repairable systems
• State equations
• Reliability of redundant repairable systems
• Imperfect structures
• Using Laplace transform
• Mathematics for Engineers
• Use of matrices
• Exact resolution by inversion
• Approximate solutions
• Software reliability
• Reliability growth model, early-life period
• Useful-life period model
• Spare parts calculation
• Definitions
• Periodical restocking
• Continuous restocking
• Chapter 8. Simulation
• Roulette simulation
• Discrete-event simulation
• Measurements and accuracy
• Measurements
• Accuracy
• Random numbers
• Generation according to a distribution
• Generating pseudo-random variables
• Appendix Mathematical Refresher
• A The function of the complex variable: definition and theorems
• A Usual z-transforms
• A Series expansions (real functions)
• A Series expansion of a function of the complex variable
• A Algebraic structures
• A Polynomials over the binary finite field
• A Matrices