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.