Sinusoïdales

Sinusoïdales, or sinusoidal waveforms, are a class of repeating signals that follow a sine or cosine function. They are considered the simplest non-constant periodic signals and are fundamental in physics and engineering because many complex signals can be decomposed into sums of sinusoids.

Mathematically, a real-valued sinusoid is written as y(t) = A sin(2π f t + φ) or equivalently A cos(2π f t + φ). Here A is the amplitude, f the frequency, and φ the phase. Sinusoïdales can also be represented using complex exponentials as Re{A e^{j(2π f t + φ)}}. In the frequency domain, a real sinusoid corresponds to two spectral lines at ±f with equal magnitude.

Key properties include a constant amplitude and a single frequency, which means they contain no harmonics and maintain their shape over time. Sinusoïdales are orthogonal to one another over a period when their frequencies differ, a feature that underpins Fourier analysis. They superpose linearly, so any periodic signal can be approximated by a sum of sinusoids with different frequencies.

Generation and applications abound: electronic oscillators and signal generators produce clean sinusoidal tones; in communications they act as carrier waves; in audio and control systems they serve as test signals and reference inputs. Sinusoïdales also underpin the Fourier transform and spectral analysis, enabling the study of complex signals in terms of their frequency content. In practice, real-world sinusoids are approximations due to noise and non-idealities, but they remain a central idealization in theory and engineering.