Mathematics (Civil Engineering -1) | 11. Fourier Transform and Properties by Abraham | Learn Smarter
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11. Fourier Transform and Properties

The chapter explores the Fourier Transform and its properties, which are essential for analyzing signals in the frequency domain. It covers definitions, computation techniques, and key applications in civil engineering, including vibration analysis and heat transfer problems. Understanding Fourier Transforms is crucial for analyzing periodic and non-periodic phenomena, making the topic foundational for engineering studies.

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Sections

  • 11

    Fourier Transform And Properties

    This section provides an overview of the Fourier Transform, its definition, properties, and its applications in civil engineering.

    Learning Practice

  • 11.1

    Definition Of Fourier Transform

    The Fourier Transform is a mathematical technique for converting time-domain signals into their frequency-domain representations.

    Learning Practice

  • 11.2

    Conditions For Existence (Dirichlet’s Conditions)

    Dirichlet's Conditions outline necessary criteria for the existence of a Fourier Transform for a function.

    Learning Practice

  • 11.3

    Properties Of The Fourier Transform

    This section discusses the essential properties of the Fourier Transform that facilitate analysis in frequency domain.

    Learning Practice

  • 11.3.1

    Linearity

    The linearity property of the Fourier Transform allows for the combination of functions in the time domain to be reflected as a linear combination in the frequency domain.

    Learning Practice

  • 11.3.2

    Time Shifting

    Time shifting relates to how shifting a function in time corresponds to a phase shift in frequency representation using Fourier Transform.

    Learning Practice

  • 11.3.3

    Frequency Shifting

    Frequency shifting involves modifying the frequency spectrum of a signal through multiplication by a complex exponential in the time domain.

    Learning Practice

  • 11.3.4

    Time Scaling

    Time Scaling relates to how a function's representation in the frequency domain is compressed or stretched based on the scaling of the time variable.

    Learning Practice

  • 11.3.5

    Differentiation In Time Domain

    This section discusses how differentiation in the time domain relates to the Fourier Transform, highlighting its mathematical formulation and significance in applications.

    Learning Practice

  • 11.3.6

    Convolution Theorem

    The Convolution Theorem states that convolution in the time domain corresponds to multiplication in the frequency domain.

    Learning Practice

  • 11.3.7

    Parseval’s Theorem

    Parseval's Theorem states that the total energy of a signal is preserved when transitioning between time and frequency domains.

    Learning Practice

  • 11.4

    Fourier Cosine And Sine Transforms

    This section focuses on the Fourier Cosine Transform (FCT) and Fourier Sine Transform (FST), which are essential for analyzing functions defined on the interval [0, ∞).

    Learning Practice

  • 11.4.1

    Fourier Cosine Transform

    The section introduces the Fourier Cosine Transform, which is used for functions defined on the interval [0, ∞) and highlights its significance in solving boundary value problems in civil engineering.

    Learning Practice

  • 11.4.2

    Inverse Fourier Cosine Transform

    The Inverse Fourier Cosine Transform is a mathematical technique that converts frequency-domain data back into time-domain functions, particularly for functions defined on the interval [0, ∞).

    Learning Practice

  • 11.4.3

    Fourier Sine Transform

    The Fourier Sine Transform (FST) is a mathematical tool that represents functions defined on [0,∞) using sine wave components.

    Learning Practice

  • 11.4.4

    Inverse Fourier Sine Transform

    The Inverse Fourier Sine Transform is a technique for recovering a function defined only on the interval [0, ∞) from its frequency-domain representation.

    Learning Practice

  • 11.5

    Applications In Civil Engineering

    Fourier Transforms are extensively used in civil engineering for analyzing vibrations, heat transfer, groundwater flow, signal processing, and seismic analysis.

    Learning Practice

  • 11.6

    Important Standard Fourier Transforms

    This section lists crucial Fourier transforms for selected functions, including deltas, exponentials, sine, and cosine functions.

    Learning Practice

  • 11.7

    Derivation Of Fourier Transform Of Common Functions

    This section discusses the derivation of Fourier transforms for common functions frequently used in civil engineering applications.

    Learning Practice

  • 11.7.1

    Fourier Transform Of Rectangular Pulse

    This section focuses on deriving the Fourier Transform of a rectangular pulse, a fundamental concept in signal analysis.

    Learning Practice

  • 11.7.2

    Fourier Transform Of Exponential Decay

    This section discusses the Fourier Transform of an exponential decay function, highlighting the process and significance of this transformation.

    Learning Practice

  • 11.8

    Example Problems

    This section presents example problems that illustrate the application of Fourier Transform concepts in practical scenarios.

    Learning Practice

  • 11.9

    Solving Differential Equations Using Fourier Transform

    This section discusses how to solve second-order linear ordinary differential equations (ODEs) using the Fourier Transform, illustrating the transformation process and its significance in engineering applications.

    Learning Practice

  • 11.10

    Fourier Transform In Two Dimensions (2d Fourier Transform)

    This section introduces the 2D Fourier Transform, essential for analyzing functions of two variables in fields such as image processing and wave propagation.

    Learning Practice

  • 11.11

    Discrete Fourier Transform (Dft)

    The Discrete Fourier Transform (DFT) analyzes sampled signals in applications like sensor data.

    Learning Practice

  • 11.12

    Use Of Fft In Civil Engineering Applications

    This section discusses the applications of Fast Fourier Transform (FFT) in various civil engineering contexts.

    Learning Practice

References

Chapter_11_Fouri.pdf

Class Notes

Memorization

What we have learnt

  • The Fourier Transform conve...
  • Existence conditions for Fo...
  • The Fourier Transform exhib...

Final Test

Revision Tests