FourierSynthese

FourierSynthese, commonly known as Fourier synthesis, refers to the construction of signals by summing sinusoidal components with specific amplitudes and phases. The idea is rooted in Fourier’s theorem, which states that any periodic waveform can be represented as a sum of sines and cosines at harmonically related frequencies.

For periodic signals, the synthesis uses a Fourier series: x(t) = a0/2 + sum from n=1 to ∞ of [ an cos(nω0 t) + bn sin(nω0 t) ], where ω0 is the fundamental angular frequency. Non-periodic signals are treated through the Fourier transform, which expresses a signal as an integral over all frequencies: x(t) = ∫ X(ω) e^{jωt} dω. In practice, finite sums or discrete transforms are used, producing approximate representations.

In audio and music, Fourier synthesis is closely associated with additive synthesis. By selecting a set of harmonics with particular amplitudes and phases, one can shape the timbre of a sound. Common examples include constructing square, sawtooth, and triangle waves by combining a fundamental with its harmonics.

Limitations include the Gibbs phenomenon near discontinuities when a finite number of components is used, and spectral leakage or aliasing in discrete implementations. Windowing and oversampling are common techniques to mitigate these issues.

Applications span signal processing, acoustics, music synthesis, image processing (via two-dimensional Fourier transforms), and communications. FourierSynthese remains a foundational concept for understanding how complex signals arise from simple sinusoidal components.