Lecture

Fourier and Laplace Transforms: Concepts and Applications

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In course

MATH-207(c): Analysis IV (for EL, GM, MX)

This course serves as an introduction to the theory of complex analysis, Fourier series and Fourier transforms, the Laplace transform, with applications to the theory of ordinary and partial differential equations. These tools play an integral role in most ...

Description

This lecture covers the fundamental concepts of Fourier and Laplace transforms, focusing on their definitions, properties, and applications in analyzing signals. The instructor begins with a recap of the Fourier transform, explaining its definition for piecewise continuous functions and the conditions for its well-defined nature. The inverse Fourier transform is also introduced, emphasizing its role in recovering the original function. The lecture then transitions to the Laplace transform, detailing its definition for signals defined on the positive real line and its relationship to differential equations. The instructor highlights the similarities and differences between the two transforms, particularly in their applications to signal processing and system analysis. Key properties such as linearity, time shifting, and frequency shifting are discussed, along with theorems that underpin these concepts. The lecture concludes with practical examples illustrating how these transforms are utilized in engineering and physics, providing students with a comprehensive understanding of these essential mathematical tools.

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Instructor

Georgios Moschidis

Teaching & PhD PhD Students Abraham Gabriel Dorsaz Courses Analysis IV (for EL, GM, MX) MATH-207(c) This course serves as an introduction to the theory of complex analysis, Fourier series and Fourier transforms, the Laplace transform, with applications to the theory of ordinary and partial differential equations. These tools play an integral role in most branches of science and engineering. Differential geometry I - curves and surfaces MATH-213 This course serves as an intr

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Ontological neighbourhood

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

Frequency domain

In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal.

Time–frequency analysis

In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations. Rather than viewing a 1-dimensional signal (a function, real or complex-valued, whose domain is the real line) and some transform (another function whose domain is the real line, obtained from the original via some transform), time–frequency analysis studies a two-dimensional signal – a function whose domain is the two-dimensional real plane, obtained from the signal via a time–frequency transform.

Integral transform

In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral transform is any transform of the following form: The input of this transform is a function , and the output is another function .

Inverse Laplace transform

In mathematics, the inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t) which has the property: where denotes the Laplace transform. It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem.

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

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