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Finding meaning in Fourier transforms
September 13, 2009 8:48 PM   Subscribe

I'd like to develop a better understanding of the physical and mathematical meanings of the Fourier transform. Can anyone suggest a text that describes some of its applications and elaborates on the importance of the transform to science and mathematics?

I frequently encounter it in my crystallography work and so I've only ever thought of it as the mathematical link between real and reciprocal space. I'm not really a mathematician, but I know that the transform is used in a number of other applications--I'd like to learn more about these and also improve my grasp of the mathematics involved in transformations of variables.

Are there any books (not too technical) that discuss this? I'm thinking of books like The Golden Ratio, but geared toward the scientific community. Thanks!
posted by Aanidaani to Science & Nature (11 answers total) 18 users marked this as a favorite
Have you checked out the Wikipedia article on the topic?
posted by alms at 8:54 PM on September 13, 2009

Bracewell's The Fourier Transform & Its Applications

This is still a rather technical book but it isn't a big book. It really does focus on applications and is very, very readable. Highly recommended.
posted by scalespace at 8:57 PM on September 13, 2009 [1 favorite]

Who Is Fourier?: A Mathematical Adventure
posted by neuron at 8:59 PM on September 13, 2009

While you're at it, check out the short bio of Joseph Fourier: orphan, French revolutionary, companion of Napoleon Bonaparte, governor of Lower Egypt, discoverer of the Greenhouse Effect, oh and a mathematician, too.
posted by alms at 9:04 PM on September 13, 2009

Your reading should discuss the importance of the fast Fourier transform: speed up anything by a factor of a million and people will invent reasons to use it.
posted by fantabulous timewaster at 9:12 PM on September 13, 2009

Fourier is used extensively in all areas of engineering; most commonly it is used to go from time-domain to frequency-domain, which tells you the frequency components of a time-varying signal. It's really hard to overstate how fundamental this is; for example, all of communication theory (which gives us phones and internet etc) relies on it. Of all the textbooks dealing with this topic in my undergrad electrical engineering program, I thought Signals and Systems by B.P. Lathi was the most clear and helpful, in particular for building an intuitive understanding of the frequency domain.
posted by PercussivePaul at 9:28 PM on September 13, 2009 [1 favorite]

I used this book in my applied real analysis class back when I was an undergrad. Like a lot of Dover books, it's a little old-fashioned, but it's actually very well written and cheap. It's got a solid treatment of the mathematical reasons that Fourier Series/Transforms work, plus several chapters on applications.
posted by dseaton at 10:56 PM on September 13, 2009

You can make an entire career out of this thing. Really. Pick a field, and there's an application.

My own interests are in signal processing and communications. My two favorite approachable treatments are an old Tektronix application handbook called "The FFT, Fundamentals and Concepts" from 1983 and a DSP text from Smith, "The Scientist and Engineer's Guide to Digital Signal Processing".

Neither will give you too big a headache, and the former is quaintly illustrated in a wonderful 1980's manner. Everyone has his/her favorites, and while I have a bunch of stuff on this topic, all I really care about is FFT, and these two address it accurately and specifically. I am sure there are better books, more rigorous treatments, but these may hit your mark better than more formal books.

Most folks have no idea how this revolutionized communications. If there were a Nobel prize for computer science, Cooley and Tukey would have won it.
posted by FauxScot at 2:38 AM on September 14, 2009 [1 favorite]

An old applied math guy I used to work with once told me how the Fast Fourier Transform had prevented WW3. I don't know how true this is, but it seems plausible:

During the Cuban Missile Crisis, tensions were extremely high. The US was nervously monitoring Russian nuclear missiles in Cuba with every available surveillance. This included seismic monitoring off the coast, sensitive enough to detect the launch of a missile. On one particular day, the seismic monitors jumped. Analysts were able to quickly determine that the frequencies of this activity were not from a missile launch--- rather, it was a very minor earthquake off the coast of Cuba. Without the FFT, computers of the day would not have been able to perform the frequency domain transformation fast enough, and the U.S.--- forced to assume the Russians had launched a missile--- would have launched a retaliatory strike.
posted by qxntpqbbbqxl at 8:15 AM on September 14, 2009 [1 favorite]

The Fourier Transform is immensely useful in optics, as well: the act of focusing through a lens (or propagating a sufficiently long distance) is equivalent to a Fourier Transform on the original electric field. I'd suggest Goodman's Introduction to Fourier Optics as a excellent and quite readable text on the subject.
posted by Upton O'Good at 8:38 AM on September 14, 2009

I don't know if this is helpful, but here goes.

Let phi be a (measurable) function of one real variable, x. Let f_* be the operation "integrate from negative infinity to infinity, with respect to x". Let K = e^{ixy}; it is a function of two variables. Then the Fourier transform of phi(x) is the function f_*(phi(x) K(x,y)). That is,

FM(phi(x))(y) = f_*(phi(x) K(x,y))(y) = integral phi(x) e^{ixy} dx.

In words, we started with a function phi(x), then considered it as a function of two variables (which happens not to involve y). Then we multiplied by K(x,y). And in the last step, we took the integral over x, leaving a function of a single variable y.

Let's introduce two copies of the real numbers, X and Y, where x and y live, respectively. phi is a function on X. K is a function on both X and Y. FM(phi) is a function on Y.

Again, we started with a function on X. We considered it as a function on X and Y (which happens not to involve Y). Then multiplied with K. then we used f_* to get a function on Y.

If we think of X as the x-axis and Y as the y-axis, then we started with a function on X, and extended it to a function on the plane, Then multiplied with K, another function on the plane. Then pushed it down to the y-axis by integrating along the horizontal lines parallel to the x-axis.

The three steps are 1. Pull back. 2. Multiply with the Kernel. 3. Push forward.

Abstracting the steps like this allows us to talk about Fourier transforms in other settings. For example, X and Y could be elliptic curves. There is a lot of mathematics in this direction. The buzzword is Fourier-Mukai transform.
posted by water bear at 2:19 PM on September 14, 2009

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