11.1. Definition of Fourier Transform

Let f(t) be a function defined on the entire real line, i.e., t ∈ (−∞,∞).

This statement introduces the basic idea that we are dealing with a function 'f' that exists for all values of 't', ranging from negative infinity to positive infinity. This is crucial for applying the Fourier Transform because the transform relies on the function being defined over this extended domain to fully capture its behavior.

Imagine a radio station that broadcasts a signal continuously without interruption. Just like the broadcast covers an entire area, the function f(t) captures the entire range of 't' values, allowing us to analyze its frequency content.

The Fourier Transform of f(t) is a complex-valued function F(ω) defined by:

Z ∞ F(ω)=F{f(t)}= f(t)e^(-iωt) dt −∞

The Fourier Transform takes the time-domain function f(t) and transforms it into a function F(ω) in the frequency domain. The integral shown computes the transformation by multiplying the function f(t) by a complex exponential e^(-iωt) and integrating over all time. Here, ω represents the frequencies contained in the original function, reflecting how much of each frequency contributes to forming f(t).

Consider light passing through a prism. Just as a prism separates light into its various colors (frequencies of light), the Fourier Transform separates signals into their constituent frequencies, providing a clearer understanding of their composition.

where: • ω is the angular frequency in radians per second, • i is the imaginary unit ( √(−1)).

In this context, ω refers to the angular frequency, which indicates how fast something oscillates in radians per second. The imaginary unit 'i' allows us to deal with oscillations in a mathematical form that combines both sine and cosine functions when expressed using Euler's formula. This is important for analyzing periodic signals because it simplifies many mathematical operations.

Think of 'i' like a way to model how sound waves interact. Just as sound can have both pitch (frequency) and amplitude (volume), complex numbers allow us to model both the frequency and phase of a signal efficiently.

This transform converts the time-domain signal f(t) into its frequency-domain representation F(ω).

The key aim of the Fourier Transform is to convert a signal that varies over time (f(t)) into a format (F(ω)) that represents how much of each frequency is present in that signal. This transition allows engineers and scientists to analyze properties of signals that can be cumbersome in time-domain, such as frequency components and their intensities.

Consider baking a cake. The ingredients list (time-domain) tells you what goes into the cake, while the final product (frequency-domain) shows how those ingredients came together in terms of taste and texture. The Fourier Transform 'bakes' the time-domain signal into its frequency components to reveal their 'flavors'.

To recover f(t) from its Fourier Transform F(ω), we use the Inverse Fourier Transform:

1 Z ∞ f(t)=F^{-1}{F(ω)}= F(ω)e^(iωt) dω 2π −∞

The Inverse Fourier Transform provides a way to revert back from frequency representation F(ω) to the original time-domain function f(t). This process involves integrating the product of F(ω) and another exponential function e^(iωt), indicating the waveform's contributions at each frequency. The division by 2π normalizes the result.

Imagine a DJ remixing a song. The DJ can take the various tracks (frequencies) one by one and blend them back together to create the original sound. The Inverse Fourier Transform is like that process—taking the separated frequencies and recombining them to get back to the original song.

This two-way transformation is central to signal processing, vibrations, and analysis of heat conduction in civil engineering systems.

The ability to transform functions back and forth between time and frequency domains is fundamental in many fields, including civil engineering. Analysis of structures (vibrations) and heat transfer scenarios often requires understanding the frequency characteristics of signals, making the Fourier Transform essential for engineers.

Think of this two-way transformation like translating a book from one language to another and then back again. Just as you can gain insights into the text's meaning in both languages, the Fourier Transform allows engineers to analyze signals in either the time or frequency domain to understand different aspects of their behavior.

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