Mathematics-for-Engineers-by-Georges-Fiche-and-Gerard-Hebuterne2

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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

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