Sine wave

In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.

A sine wave represents a single frequency with no harmonics and is considered an acoustically pure tone.

Adding sine waves of different frequencies results in a different waveform.

Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical pitch played on different instruments sounds different.

Sine waves of arbitrary phase and amplitude are called sinusoids and have the general form:[1]

where: Sinusoids that exist in both position and time also have: Depending on their direction of travel, they can take the form: Since sine waves propagate without changing form in distributed linear systems,[definition needed] they are often used to analyze wave propagation.

In two or three spatial dimensions, the same equation describes a travelling plane wave if position

French mathematician Joseph Fourier discovered that sinusoidal waves can be summed as simple building blocks to approximate any periodic waveform, including square waves.

Differentiating any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle:

The gain of its frequency response increases at a rate of +20 dB per decade of frequency (for root-power quantities), the same positive slope as a 1st order high-pass filter's stopband, although a differentiator doesn't have a cutoff frequency or a flat passband.

A nth-order high-pass filter approximately applies the nth time derivative of signals whose frequency band is significantly lower than the filter's cutoff frequency.

Integrating any sinusoid with respect to time can be viewed as dividing its amplitude by its angular frequency and delaying it a quarter cycle:

will be zero if the bounds of integration is an integer multiple of the sinusoid's period.

An integrator has a pole at the origin of the complex frequency plane.

The gain of its frequency response falls off at a rate of -20 dB per decade of frequency (for root-power quantities), the same negative slope as a 1st order low-pass filter's stopband, although an integrator doesn't have a cutoff frequency or a flat passband.

A nth-order low-pass filter approximately performs the nth time integral of signals whose frequency band is significantly higher than the filter's cutoff frequency.