Pulse wave

The Fourier series expansion for a rectangular pulse wave with period ${\displaystyle T}$, amplitude ${\displaystyle A}$ and pulse length ${\displaystyle \tau }$ is[2]

$${\displaystyle x(t)=A{\frac {\tau }{T}}+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)}$$

where ${\displaystyle f={\frac {1}{T}}}$.

Equivalently, if duty cycle ${\displaystyle d={\frac {\tau }{T}}}$ is used, and ${\displaystyle \omega =2\pi f}$:

$${\displaystyle x(t)=Ad+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi nd\right)\cos \left(n\omega t\right)\right)}$$

Note that, for symmetry, the starting time (${\displaystyle t=0}$) in this expansion is halfway through the first pulse.

Alternatively, ${\displaystyle x(t)}$ can be written using the Sinc function, using the definition ${\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}}$, as

$${\displaystyle x(t)=A{\frac {\tau }{T}}\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)\right)}$$

or with ${\displaystyle d={\frac {\tau }{T}}}$ as

$${\displaystyle x(t)=Ad\left(1+2\sum _{n=1}^{\infty }\left(\operatorname {sinc} \left(nd\right)\cos \left(2\pi nft\right)\right)\right)}$$

A pulse wave can be created by subtracting a sawtooth wave from a phase-shifted version of itself. If the sawtooth waves are bandlimited, the resulting pulse wave is bandlimited, too. A single ramp wave (sawtooth or triangle) applied to an input of a comparator produces a pulse wave that is not bandlimited. A voltage applied to the other input of the comparator determines the pulse width.

Fourier series of a 33.3% pulse wave, first fifty harmonics (summation in red)

The harmonic spectrum of a pulse wave is determined by the duty cycle.[3][4][5][6][7][8][9][10] Acoustically, the rectangular wave has been described variously as having a narrow[11]/thin,[1][4][5][12][13] nasal[1][4][5][11]/buzzy[13]/biting,[12] clear,[3] resonant,[3] rich,[4][13] round[4][13] and bright[13] sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".[11]

In digital electronics, a digital signal is a pulse train (a pulse amplitude modulated signal), a sequence of fixed-width square wave electrical pulses or light pulses, each occupying one of two discrete levels of amplitude.[14][15] These electronic pulse trains are typically generated by metal–oxide–semiconductor field-effect transistor (MOSFET) devices due to their rapid on–off electronic switching behavior, in contrast to BJT transistors which slowly generate signals more closely resembling sine waves.[16]

• Gibbs phenomenon
• Pulse shaping
• Sinc function
• Sine wave