Fourier
Coefficients: Complex with Sound -- Help |
Coefficients c_{n} in the complex Fourier series f(t) = ... + c_{-2} e^{-4 i π ν t} + c_{-1} e^{-2 i π ν t} + c_{0} + c_{1} e^{2 i π ν t} + c_{2} e^{4 i π ν t} + .... are selected using the windows at bottom. The windows at lower right record the values of |c_{n}| and Arg(c_{n}) by yellow diamonds and struts. The diamond 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 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 φ . The graphing window at the top displays the graph of Re(e^{-i φ} f(t)) . Thus if φ = 0 , the real part of f(t) is displayed, while if φ = π / 2, the imaginary part of f(t) is displayed. Red vertical lines on this window also mark t = -L and t = +L , where L = 1 / (2ν) , so that 2L is the period. A cursor rollover generates crosshairs and a readout of values of t and x . At the top, a [f(t) real] key toggles the restriction on the values of the coefficients making f(t) real valued. When [f(t) real] is selected, c_{-n} is reset to equal the complex conjugate of c_{n}. A slider at the top sets the frequency ν , 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 ν 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 |