Pulse wave

The Fourier series expansion for a rectangular pulse wave with period ${\displaystyle T}$ , amplitude ${\displaystyle A}$ and pulse length ${\displaystyle \tau }$ is^[1]

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

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

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

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

${\displaystyle d={\frac {\tau }{T}}}$

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

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.

The harmonic spectrum of a pulse wave is determined by the duty cycle.^{[2][3][4][5][6][7][8][9]} Acoustically, the rectangular wave has been described variously as having a narrow^[10]/thin,^{[11][3][4][12][13]} nasal^{[11][3][4][10]}/buzzy^[13]/biting,^[12] clear,^[2] resonant,^[2] rich,^[3][13] round^[3][13] and bright^[13] sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".^[10]

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