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I know there are lots of tutorial on different webpages.. but I am not an engineering student... those look quite complex to me..

Could anybody explain what is "Fourier transform" in the very simple sentences so non-engineer background people can actually understand?


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A transform, used by Joseph Fourier. – Ethan Dec 22 '12 at 2:32
@Ethan It is a pity I cannot down vote comments like that. If you have something that might be useful, then say it otherwise don't. – AD. Dec 25 '12 at 8:17
up vote 6 down vote accepted

The Fourier transform tells you how to express something like an acoustic signal as a combination of something like notes. A trained musician's ear performs something like a Fourier transform when she tells you what notes make up a chord you play on a piano. More generally, your ears perform something like a Fourier transform when you do things like talk to a friend while listening to a television at the same time; you have the capacity to distinguish these different parts of the acoustic signal your ears are getting so you can pay attention to one or the other. This is something like Ohm's acoustic law.

Given the above, it should not be surprising that the Fourier transform is fundamental in signal processing. The Fourier transform is also fundamental in both pure and applied mathematics, as it is very useful for solving certain kinds of partial differential equations. Since acoustic signals are governed by such a differential equation, namely the wave equation, this should not be so surprising either.

This is a vague description. If you want to understand precisely what the Fourier transform is and does, you will have to learn some math. You will need to at least have taken a course on calculus, and having taken courses in differential equations and linear algebra will also be useful. Depending on how thoroughly you want to understand the Fourier transform, it will also be useful to have taken a oourse on real analysis. Once you have these prerequisites there are various sources you can learn from; I like Stein and Shakarchi. If you have some musical background or are otherwise interested in the relationship to music, I like Benson's Music: a Mathematical Offering.

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@Quiaochu's answer is spot-on for temporal signals. In optics, however, there is also a spatial interpretation as follows. Consider some wavefront $f(x)$ that may be propagated along the $x$ axis. For example, the wavefront may be a focused, circular wave, or a diverging wave, or even some aberrated wavefront. The Fourier transform allows us to express a complex wavefront in terms of a sum over plane waves that propagate at a range of angles with respect to the $x$ axis. In this case, the Fourier transform $F(v)$ represents the amplitude of the plane wave whose angle $\theta$ of propagation with respect to the $x$ axis is given by $\frac{2 \pi}{\lambda} \cos{\theta}=v$, where $\lambda$ is the wavelength of the light. This is known as the angular spectrum representation of a wavefront, and this representation allows for beautiful analysis of wave propagation in complex optical systems.

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