什麼是『傅立葉變換』？維基百科這麼講︰
In the first row is the graph of the unit pulse function 
and its Fourier transform 
, a function of frequency 
. Translation (that is, delay) in the time domain goes over to complex phase shifts in the frequency domain. In the second row is shown 
, a delayed unit pulse, beside the real and imaginary parts of the Fourier transform. The Fourier transform decomposes a function into eigenfunctions for the group of translations.
The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, similarly to how a musical chord can be expressed as the amplitude (or loudness) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is called the frequency domain representation of the original signal. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest one can define an operation that reverses this: the inverse Fourier transformation, also called Fourier synthesis, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,[note 1] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, can be expressed relatively simply as an operation on frequencies.[note 2] After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are “simpler” in one or the other, and has deep connections to almost all areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
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由於『傅立葉變換』並不是三五篇文本可以詳說，故此介紹點網路上的資源給有興趣的讀者。據聞李家同教授曾寫過一篇簡介
李家同
暨南國際大學資訊工程系
rctlee@ncnu.edu.tw
，例釋說明這個『轉換』是什麼。
若想更深入的了解，何不上一堂 Stanford 大學的公開課︰
Stanford Engineering Everywhere EE261 – The Fourier Transform and its Applications
released under terms of: Creative Commons Attribution Non-Commercial (CC-BY-NC)
The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both.
Topics include:
- The Fourier transform as a tool for solving physical problems.
- Fourier series, the Fourier transform of continuous and discrete signals and its properties.
- The Dirac delta, distributions, and generalized transforms.
- Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysis of linear systems.
- The discrete Fourier transform and the FFT algorithm.
- Multidimensional Fourier transform and use in imaging.
- Further applications to optics, crystallography.
- Emphasis is on relating the theoretical principles to solving practical engineering and science problems.
Course Homepage: http://see.stanford.edu/see/courseinfo.aspx?coll=84d174c2-d74f-493d-92ae-c3f45c0ee091
Course features at Stanford Engineering Everywhere page:
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同時認真讀讀 Brad Osgood 教授之課堂筆記耶。
The Fourier Transform and its Applications
Prof. Brad Osgood
Electrical Engineering Department
Stanford University