**PDF Free Download | Mathematics for Engineers by Georges Fiche, and Gérard Hébuterne**

## 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