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Estimated Time:40 minutes
Students are to find the Fourier coefficients of a piecewise periodic function. Usually, we use the same function for all groups, and assign different terms in the expansion to each group. Different functions would work for a more sophisticated class. The important point is that the function should be piecewise (most of the standard ones are), and that the function is presented as a graph so that the students have to find the equation for the graph, which, for a piecewise function, has different forms in different regions.
The class will have listened to lecture instruction on projections of functions onto sines and cosines (the harmonic basis) and in the process, will have arrived at the formula for the Fourier coefficients.
Students are now assigned to groups of about three or four to work out the Fourier coefficients of a particular function. In this course, they may still need to be reminded about how to run a group efficiently.
The most important discussion is about how to interpret the function \(f\left( t \right)\) in the expression \[a_{n}=\frac{2}{T}\int\limits_{0}^{T}{f\left( t \right)\cos \left( n\omega _{0}t \right)dt}\] for example. Students usually try to substitute the series expansion into the integral, which is circular reasoning. They need to discover that the form of \(f\left( t \right)\) is known (e.g. a truncated sine, or a sawtooth or something else.)
The fact that the function is presented in graphical form and not as a formula is another source for intense discussion. The students come to grips with the fact that they have to generate a formula. Although they have usually encountered piecewise functions before, they are remarkably resistant to the idea that such a thing is a “proper” function, because “it's not one formula”. The need to include two pieces with appropriate limits on the integrals finally becomes apparent.
Choice of origin to make the function even or odd (or neither) always comes up. Guide the students to the conclusion that any choice of origin is correct; some choices are just easier than others.
Some students are worried about the fact that an “infinite number of integrals need to be done” to find the infinite number of coefficients. The mathematically more sophisticated students help resolve this issue by pointing out that they can all be done at once by using an integer parameter $n$ to indicate the appropriate term.
The wrap up is to ask each group to report the value obtained for the Fourier coefficient and so that the class jointly generates a Fourier spectrum. This emphasizes the relative importance of different terms. If groups have chosen different origins, then there are not enough terms forthcoming from the class to generate such a spectrum, so an option is force the choice of origin. However, different origins generate different representations of the same function, and this leads to valuable discussion, too. Follow-up homework includes generating the a graph of the spectrum and/or generating an alternative series to represent the same function.
Students often simply want to discuss the process they have just gone through. This is the first time they have actually been responsible for the calculation, and they have questions about details that they did not think about before.