sinusoids

Sinusoids are mathematical curves and signals characterized by a single harmonic component. In continuous time, a sinusoid can be written as y(t) = A sin(ωt + φ) or y(t) = A cos(ωt + φ). Here A is the amplitude, ω the angular frequency, and φ the phase. The ordinary frequency is f = ω/(2π) and the period is T = 2π/ω. Sine and cosine forms differ only by phase.

Sinusoids are the simplest periodic functions and form the basis of many physical oscillations, including simple harmonic motion, sound waves, and alternating current electrical signals. In signal processing, any periodic signal can be approximated or represented as a sum of sinusoids (Fourier series) with different amplitudes and frequencies.

Mathematically, sinusoids arise as solutions to the simple harmonic oscillator equation d^2x/dt^2 + ω^2 x = 0. Over a single period, sine and cosine are orthogonal, a property that underpins their use in Fourier analysis and filter design.

In discrete-time signals, sinusoids take the form s[n] = A sin(ωn + φ) or s[n] = A cos(ωn + φ); complex exponentials e^{jωn} are often used due to Euler's formula, enabling compact representation and manipulation of sinusoidal components.

Fundamental characteristics include constant amplitude and frequency, regardless of time, making sinusoids ideal for modeling vibrations, waves, and frequency-domain analyses across science and engineering.