Fourier Series Example - The Saw Function

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On this page, we'll look at another example of finding the Fourier Series - this time on the saw function. The saw function, f(t), is illustrated in Figure 1:

saw function

Figure 1. The Saw Function.

The saw function can be mathematically written over the fundamental period [0,T] as:

saw function f(t)
[Equation 1]

The determination of the Fourier Coefficients is fairly simple. The integral below in Equation [2] can be determined with integration by parts, or a table look up, or whatever method you choose:

calculation of Fourier Series for saw function
[Equation 2]

The above integral is simplified using the fact:

exponential complex note
[Equation 3]

Observe from Equation [2] that nth term is the complex conjugate of the -nth term, so the resultant complex Fourier sum will be real. To illustrate that the above coefficients do reproduce the saw function, the coefficients from Equation [2] are plugged into Equation [1] from the Complex Fourier Coefficients page. The n=-1, 0, 1 terms give the following function [superimposed on the original f(t) ]:

two terms in the saw expansion

Figure 2. The Saw Function with 3 Fourier Coefficients (n=-1,0,1).

Going further, here is the plot of the n=-3, -2, -1, 0, 1, 2, 3 Fourier Coefficients along with the original function:

seven terms in the saw expansion

Figure 3. The Saw Function with 7 Fourier Coefficients (n=-1,0,1).

Just in case you are still not convinced, here are the n=-20 through 20 Fourier Coefficients:

many terms in the saw expansion

Figure 4. The Saw Function with 41 Fourier Coefficients (n=-20,-19,...19,20).

...and it is pretty clear the Fourier Series is converging on the original function f(t), which is awesome.

In the next section, we'll look at a complicated function that cannot be easily evaluated analytically (if at all), and determine the Fourier Series numerically.

Next: Numerical Evaluation of Fourier Series of a Complicated Function

Previous: Fourier Series Example - The Cosine Function

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