FOURIER COEFFICIENTS: COMPLEX WITH SOUND

Coefficients $$c_n$$ in the complex Fourier series

$f(t) = \cdots + c_{-2} e^{-4i\pi\nu t} + c_{-1} e^{-2i\pi\nu t} + c_0 + c_1 e^{2i\pi\nu t} + c_2 e^{4i\pi\nu t} + \cdots$

are selected using the windows at bottom. The windows at lower right record the values of $$|c_n|$$ and $$\text{Arg}(c_n)$$ by yellow disks and struts. The disk and strut for the chosen value of $$n$$ are gray. A mouse click on either window at lower right sets the value of $$n$$, and also the value of either $$|c_n|$$ or $$\text{Arg}(c_n)$$. Keep the mouse button depressed and drag the cursor to reset several values of $$|c_n|$$ or $$\text{Arg}(c_n)$$. Sliders at left also control these values.

The window at lower left records the value of $$c_n$$ on the complex plane, and the value of $$c_n$$ can be chosen by a click and drag on that window.

A slider at lower right selects a value of a parameter $$\phi$$. The graphing window at the top displays the graph of $$\text{Re}(e^{-i\phi}f(t))$$. Thus if $$\phi = 0$$, the real part of $$f(t)$$ is displayed, while if $$\phi = \frac{\pi}{2}$$, the imaginary part of $$f(t)$$ is displayed. Red vertical lines on this window also mark $$t = -L$$ and $$t = +L$$, where $$L = \frac{1}{2\nu}$$, so that $$2L$$ is the period. Rolling the cursor over the graphing window produces crosshairs and a readout of the values of $$t$$ and $$x$$. Depressing the mousekey suppresses the crosshairs.

At the top, a $$[f(t) \space \text{real}]$$ key toggles the restriction on the values of the coefficients making $$f(t)$$ real valued for $$n \gt 0$$. When $$[f(t) \space \text{real}]$$ is selected, $$c_{-n}$$ is reset to equal the complex conjugate of $$c_n$$.

A slider at the top sets the frequency $$\nu$$, in kilohertz.

At top right, a [Reset] key resets all the coefficients to zero.

A [Play]/[Pause] key toggles a sound production, at the frequency $$\nu$$ kilohertz.

Be sure the volume is low when [Play] is selected. Also be aware that pure sine waves carry more energy than they sound like they do, and prolonged exposure through headphones can damage the ear.

© 2008 H. Miller and J.-M. Claus