Fourier series

The Fourier series module can compute and graph the Fourier coefficients of a periodic function $f$, and can plot partial sums of the Fourier series and the error (difference) between these partial sums and the function $f$.

For concreteness, we will write $f(x)$ for the function being considered, even though you can change the name of the independent variable from $x$ to something else like $t$, if you want.

After selecting Fourier series from the Iode main menu, the Fourier series window will open up, showing two graphs. The top graph plots a function $f(x)$ in dark blue and a partial sum of its Fourier series in red. These are plotted over two period lengths. Recall that a partial sum of the Fourier series is an expression of the form

$$\frac{A_0}{2} + \sum_{n=1}^N (A_n \cos \frac{n\pi x}{L} + B_n \sin \frac{n\pi x}{L})$$ (4.1)

where

$$A_n = \frac{1}{L} \int_{x_1}^{x_2} f(x) \cos \frac{n\pi x}{L} dx$$   and$$\qquad B_n = \frac{1}{L} \int_{x_1}^{x_2} f(x) \sin \frac{n\pi x}{L} dx$$

are the $n^{\text{th}}$ Fourier coefficients and $L$ is the half-period, $L = (x_2-x_1)/2$.

The top harmonic number $N$ tells you the highest frequency that is included in the partial sum. Try increasing or decreasing the value of the top harmonic used in your plot, by clicking on the ``arrow'' buttons in the middle of the window. Alternatively, you can type a number directly into the Current top harmonic box between the arrows.

The bottom graph in the window plots the error between the function $f(x)$ and the partial sum of its Fourier series:

   error $$(x) = f(x) - \left[ \frac{A_0}{2} + \sum_{n=1}^N (A_n \cos \frac{n\pi x}{L} + B_n \sin \frac{n\pi x}{L}) \right] .$$

Notice that the vertical scale on this error plot is generally different from the scale on the top plot, in which the function is plotted. In fact, the vertical scale on the error plot will change as you step through the partial sums (increasing or decreasing the top harmonic).

Across the top of both graphs you will find the function written out, along with the basic period interval $x_1 \leq x < x_2$. The function is extended periodically by Iode, and is shown over two periods.

Controls

• Plot partial sums and errors: plots $f$ and its partial sum in the top graph, and the error (difference) in the bottom plot.
• Plot coefficients A_n and B_n: plots the $A_n$-coefficients in the top plot, and the $B_n$-coefficients in the bottom one, for $0<n<$top harmonic. Notice $A_0$ is never plotted (because it is the least interesting Fourier coefficient, affecting only how much the graphs are translated up or down in the $y$-direction).
• Plot coefficients C_n=(A_n^2+B_n^2)^(1/2): plots the Fourier magnitudes $C_n = \sqrt{ A_n^2 + B_n^2}$ in the top graph. The bottom graph can be used to investigate the rate of decay of the $C_n$, using an arbitrary comparison function (see Options below).
• Current top harmonic: increase or decrease this value by clicking on the ``arrow'' buttons in the middle of the window. (Alternatively, you can type a number directly into the box between the arrows.) You can change the top harmonic while in any one of the three plotting modes above.

Function menu

• Enter function: You will be asked first for the left and right endpoints of the basic period interval $x_1 \leq x < x_2$ for your function, and then you enter the function using Matlab syntax as usual (e.g., exp(x) for $e^x$). Very often the interval of interest is just $-\pi \leq x < \pi$, in which case you enter -pi and pi for the left and right endpoints.
  Remark 4.1   The interval $x_1 \leq x < x_2$ is part of the definition of the function! For example, it tells us that the period of the function is $P = x_2- x_1$.

• Comparison functions... This item is only accessible when you are in the Plot coefficients C_n mode. Its purpose is to compare the rate of decay of the Fourier coefficients with a function like $\frac{1}{n}$. It has two subitems.
  • Enter comparison function: prompts you to enter a function of $n$. The idea is to enter a function that might be decaying at the same rate as the Fourier coefficients, which in most cases means you should enter something like $1/n$ or $1/n^2$ or $1/n^3$.
  • Show comparison function: plots the ratio of $C_n$ over your comparison function, in the bottom graph. You will typically need top harmonic to be 25 or so, to see a convincing pattern in this ratio graph.

    To get rid of the ratio graph, just choose Show comparison function again.

    If the ratio graph is decaying steeply overall, then the coefficients $C_n$ are decaying faster than your comparison function, so you should enter a faster-decaying comparison function (such as a larger power of $1/n$). If the ratio graph is growing steeply overall, then the coefficients $C_n$ are decaying slower than your comparison function, so you should enter a slower-decaying comparison function (such as a smaller power of $1/n$). If the ratio graph remains bounded but does not approach zero, as $n$ gets bigger, then congratulations! You have probably hit upon the correct rate of decay of the Fourier coefficients.

Remark 4.2   It is interesting to plot the values of $C_n$ and determine their rate of decay because $C_n$ equals the amplitude of the combined oscillations at the $n$th frequency level, in the Fourier series of $f$. To see this, observe that

$$A_n \cos \frac{n\pi x}L + B_n \sin \frac {n\pi x}L = C_n \cos (\frac {n \pi x}L - \alpha_n)$$ (4.2)

where the angle $\alpha_n$ is defined by requiring $\cos \alpha_n = A_n/C_n$ and $\sin \alpha_n = B_n/C_n$. (Formula (4.2) is proved just by substituting the identity $\cos(\beta-\alpha) = \cos \beta \cos \alpha + \sin \beta \sin \alpha$ on the right hand side.)

Options menu

• Change plot resolution: this item is only accessible when you are in the Plot partial sums and errors mode. You should probably increase the plot resolution if you are studying rapidly oscillating functions, or if you choose top harmonic bigger than, say, 100. The prompt, ``Number of points to be plotted'', refers to the number of plot points in the interval $x_1 \leq x < x_2$.