E1: Use the default step function. Increase the order of approximation. How does the overshooting ("ringing") at the discontinuity depend on the order?

E2: Study the spectrum and try to explain its change with order.

E3:  The spectrum is ""poor in harmonics". How could this effect the audio impression when the function drives a loudspeaker?

E4:  Insert the formula for the rectangle pulse and vary parameter c that determines the pulse width. Compare the spectra at large and at small width (where it becomes nearly "white", e.g. of constant amplitude).  How does an acoustic signal of that type − with no overtone discrimination − sound?

E5: The sawtooth coarsely reproduces the sound of a violin (why? Consider the mechanics of the bow´s horse hair action on the string). Compare the harmonics spectrum with that of step and short rectangle. Observe the ringing.

E6:  Insert the sawtooth with superimposed sine and look at the spectrum. It now contains a limited range of harmonics with high amplitude, a "formant" range that influences the tone color decisively (in musical instruments there will not be a single overtone but a mixture of harmonics and anharmonics).

E7: Investigate the triangle cases. Why do spectra like that sound "dull"?

E8: Imagine that you want to realize an electric organ with interesting sounds via Fourier synthesis. The goal would be a spectrum rich in harmonics, with distinct formants of some spread. Start with a basic function like a sawtooth with superimposed sine and manipulate the formula to get suitable spectra.

(Hint: try nonlinear functions, multiply harmonics, etc.)