Friday, December 27, 2013

Fourier Series

One of the most important mathematical techniques for a physicist is the Fourier series. I discussed Joseph Fourier, the inventor of this method, previously in this blog. In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Fourier series in Sections 11.4 and 11.5.

The classic example of a Fourier series is the representation of a periodic square wave: y(t) = 1 for t between 0 and T/2, and y(t) = -1 for t between T/2 and T, where T is the period. The Fourier series represents this function as a sum of sines and cosines, with frequencies of k/T, where k is an integer, k = 0, 1, 2, …. The square wave function y(t) is odd, so the contributions of the cosine functions vanish. The sine functions contribute for half the frequencies, those with odd values of k. The amplitude of each non-zero frequency is 4/πk (Eq. 11.34 in IPMB), so the very high frequency terms (large k) don’t contribute much.

Being able to calculate the Fourier series is nice, but much more important is being able to visualize it. When I teach my Medical Physics class (PHY 326), based on the last half of IPMB, I stress that students should “think before you calculate.” One ought to be able to predict qualitatively the Fourier coefficients by inspection. Being able to understand a mathematical calculation in pictures and in physical terms is crucially important for a physicist. The Wikipedia article about a square wave has a nice animation of the square wave being built up by adding more and more frequencies to the series. I always insist that students draw figures showing better and better approximations to a function as more terms are added, at least for the first three non-zero Fourier components. You can also find a nice discussion of the square wave at the Wolfram website. However, the best visualization of the Fourier series that I have seen was brought to my attention by one of the PHY 326 students, Melvin Kucway. He found this lovely site, which shows the different Fourier components as little spinning wheels attached to wheels attached to wheels, each with the correct radius and spinning frequency so that their sum traces out the square wave. Watch this animation carefully. Notice how the larger wheels rotate at a lower frequency, while the smaller wheels spin around at higher frequencies. This picture reminds me of the pre-Copernican view of the rotation of planets based on epicycles proposed by Ptolemy.

What is unique about the development of Fourier series in IPMB? Our approach, which I rarely, if ever, see elsewhere, is to derive the Fourier coefficients using a least-squares approach. This may not be the simplest or most elegant route to the coefficients, but in my opinion it is the most intuitive. Also, we emphasize the Fourier series written in terms of sines and cosines, rather than complex exponentials. Why? Understanding Fourier series on an intuitive level is hard enough with trigonometric functions; it becomes harder still when you add in complex numbers. I admit, the math appears in a more compact expression using complex exponentials, but for me it is more difficult to visualize.

If you want a nice introduction to Fourier series, click here or here (in the second site, scroll down to the bottom on the left). If you prefer listening to reading, click here for an MIT Open Courseware lecture about the Fourier series. The two subsequent lectures are also useful: see here and here. The last of these lectures examines the square wave specifically.

One of the fascinating things about the Fourier representation of the square wave is the Gibbs phenomenon. But, I have discussed that in the blog before, so I won’t repeat myself.

What is the Fourier series used for? In IPMB, the main application is in medical imaging. In particular, computed tomography (Chapter 12) and magnetic resonance imaging (Chapter 18) are both difficult to understand quantitatively without using Fourier methods.

As a new year’s resolution, I suggest you master the Fourier series, with a focus on understanding it on a graphical and intuitive level. What is my new year’s resolution for 2014? It is for Russ and I to finish and submit the 5th edition of IPMB to our publishers. With luck, you will be able to purchase a copy before the end of 2015.

1 comment:

  1. Brad Osgood gives a terrific, free, online course on the Fourier Transform. You can see lecture 1 and link to others here:

    Happy New Year Brad!