Exercise: Sawtooth Wave Fourier Transform

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Problem Statement

Find the Fourier Tranform of the sawtooth wave given by the equation




Solution

As shown in class, the general equation for the Fourier Transform for a periodic function with period is given by



where



For the sawtooth function given, we note that , and an obvious choice for is 0 since this allows us to reduce the equation to . It remains, then, only to find the expression for and . We proceed first to find .



which is solved easiest with integration by parts, letting




so





Now, for we must consider the case when separately.



For , we have



which again is best solved using integration by parts, this time with




so






Therefore, the Fourier Transform representation of the sawtooth wave given is:



The figures below graph the first few iterations of this solution. The first graph shows the solution truncated after the first 100 terms of the infinite sum, as well as each of the contributing sine waves with offset. The second figure shows the function truncated after 1, 3, 5, 10, 50, and 100 terms. The last figure shows the Error between the Fourier Series truncated after the first 100 terms and the function itself. These figures were constructed using the following matlab code: SawToothFourier.


First 100 Terms.jpg


First n Terms.jpg


Error.jpg

Author

John Hawkins

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