FOURIER-3 illustrates that ANY periodic waveform can be made by adding pure sinusoids that are harmonics of the fundamental frequency of the periodic waveform. Excellent for learning Fourier synthesis or analysis.

Quoting from the main tutorial in the program: "Joseph Fourier (1768-1830) was a French mathematician and physicist who discovered that practically any waveform can be represented as the sum of sinusoidal waveforms. This discovery was not readily accepted in Fourier's time, due in part to the non-intutitive nature of the result. How can waveforms with sharp corners (e.g. a triangle wave) or with discontinuities (e.g. a square wave) possibly be represented to arbitrary precision by the sum of smoothly varying sinusoids? But the surprising fact is that they can, a result that has had an impact on virtually every brance of science and engineering."

"Fourier's theorem states that any periodic waveform can be synthesized as the sum of a fundamental sinusoid and its harmonics. A waveform, w(t), is said to be periodic with period T when w(t) = w(t+T) for all t. The fundamental has the same period as the waveform, namely T, and therefore has a frequency of 1/T. The harmonics of the fundamental are defined as those sinusoidal waveforms with frequencies that are integral multiples of the frequency of the fundamental. So, if the frequency of the fundamental is 1/T, then the frequency of the second harmonic is 2/T, the frequency of the third harmonic is 3/T, and so on."

"Just as Fourier's result was non-intuitive in its time, so it is today. However, with this program, FOURIER SYNTHESIS IN ACTION, you can experience the wonder of Fourier synthesis as it unfolds before your very eyes. The main menu give you several different waveforms to choose from, and you can watch each of these waveforms emerge as the sum of sinusoids. And lest you think that we have chosen particularly well-behaved waveforms, you are invited to make your own waveform (option M from the main menu). We will compute the Fourier terms of your waveform, and synthesize it for you."

For students studying electronic communications, or vibrations, or acoustics, the concept that a non-sinusoidal, periodic waveform can be made by adding an infinite number (or closely approximated by quite a few) of sinusoidal harmonics can be difficult to believe in the first place, and later to master mathematically. Perhaps more importantly, non-sinusoidal, periodic waveforms actually contain the harmonics that Fourier analysis predicts. This can explain in part why a squarewave oscillator serving as a clock in a digital system can raise havoc with RF communications systems nearby, or why mechanical vibrations at a certain frequency can cause oscillations at frequencies that are integer multiples of that frequency.

With most of the menu choices, there are context-specific tutorials as well, which discuss Fourier concepts that apply to the waveform under view. Enjoy this fine piece of software, and watch as each harmonic added makes the waveform being approximated look closer to the ideal. Return to EET Software Home Page