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2025 | Book

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Engineering Mathematics by Example

Vol. III: Special Functions and Transformations

Author: Robert Sobot

Publisher: Springer Nature Switzerland

Part of: Springer Professional "Business + Economics & Engineering + Technology" , Springer Professional "Engineering + Technology" , Springer Professional "Business + Economics"

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About this book

This textbook is a complete, self-sufficient, self-study/tutorial-type source of mathematical problems. It serves as a primary source for practicing and developing mathematical skills and techniques that will be essential in future studies and engineering practice. Rigor and mathematical formalism is drastically reduced, while the main focus is on developing practical skills and techniques for solving mathematical problems, given in forms typically found in engineering and science. These practical techniques are split into three separate books: the topics of algebra, complex algebra, and linear algebra (Vol. I), calculus of single and multiple argument functions (Vol. II), continues and discrete Convolution and Fourier integrals/sums of typical functions used in signal processing, and Laplace transform examples (Vol. III).

Table of Contents

Frontmatter

Continuous Time Domain

Frontmatter
1. Elementary Special Functions
Abstract
In the field of mathematics, aside from elementary and algebraic functions, there is a large group of special functions. In this chapter, a few of special functions that are more often used in signal processing are introduced. Namely, complex exponential, sinus cardinal, Heaviside step (a.k.a. unit step), sign, ramp, rectangular, and triangular functions are briefly studied. A bit more space is devoted to Dirac delta function (distribution) \(\delta (t)\) and its properties. For the sake of transition to mathematics for signal processing, the traditional “x” variable is replaced with “t” implying time domain progression.
Robert Sobot
2. Continuous Time Convolution
Abstract
Solution to convolution integral of the product of two continuous functions, for example, \(x(t) h(t)\), is a third function of the same variable, e.g., \(f(t)\). In this chapter, various combinations of the integrals that include special functions (distributions) are solved in detail. The “integral” implies that functions being convoluted are continuous (including quasi-continuous special functions), that is to say, non-sampled. In addition, typical problems related to energy and power of continuous functions are solved.
Robert Sobot
3. Continuous Time Fourier Transform
Abstract
Continuous time Fourier transform is defined in the form of a complex function integral, where the complex function computes product of continuous and complex exponent functions. In this chapter, classical integrals that include special functions (distributions) are solved in detail. The “integral” implies that functions being transformed are continuous (including quasi-continuous special functions), that is to say, non-sampled.
Robert Sobot
4. Fourier Series
Abstract
Continuous time Fourier transform is defined in the form of a complex function integral, where the complex function computes product of continuous and complex exponent functions. In this chapter, classical integrals that include special functions (distributions) are solved in detail. The “integral” implies that functions being transformed are continuous (including quasi-continuous special functions), that is to say, non-sampled.
Robert Sobot
5. Laplace Transform
Abstract
Bilateral version, i.e., if calculated from \(-\infty \) to \(+\infty \), of Laplace transforms may be seen as a more general case of Fourier transform. Laplace transform is calculated as complex integral of complex variable, while Fourier transform is calculated as complex integral of real variable. Arguably, most practical application of Laplace transform in the engineering field is to convert some categories of differential equations into ordinary polynomial equations. By doing so, basic algebra techniques are used to derive solutions of these differential equations.
Robert Sobot

Discrete Time Domain

Frontmatter
6. Series
Abstract
A sequence of ordered numbers (or functions) that may contain finite or infinite number of terms is referred to as a series where each term a is indexed as, for example, \(a_i\) to indicate its position. The indices i of the first element may start at any arbitrary integer value and increase by one until the last term in the series is indexed. Sum of all the elements may be finite (convergent) or infinite (divergent). The infinite sequence of additions is resolved by limit after resolving the sum of the first n elements. There is no significance in the choice of letters used to index multiple counters in the given problem: i, n, k, j, l, m... are all commonly used.
Robert Sobot
7. Discrete Time Convolution
Abstract
In general, convolution sum of discrete elements is easier to tabulate relative to the integrals. Sampled functions may be defined in analytical form or explicitly given in the form of sequence of numbers. Otherwise, the parallel between continuous and sampled functions is evident. In this chapter, some of the typical discrete time forms and techniques are illustrated.
Robert Sobot
8. Discrete Time Fourier Transform
Abstract
Discrete time Fourier transform is defined as the sum of the product of a sequence and complex exponent. In this chapter, classical sums that include fundamental functions (distributions) are solved in details. The “sum” implies that functions being transformed are sampled, that is to say, in the form of a discrete sequence.
Robert Sobot
Backmatter
Metadata
Title
Engineering Mathematics by Example
Author
Robert Sobot
Copyright Year
2025
Electronic ISBN
978-3-031-81104-3
Print ISBN
978-3-031-81103-6
DOI
https://doi.org/10.1007/978-3-031-81104-3