A pulse wave's duty cycle D is the ratio between pulse duration 𝜏 and period T.

Apulse wave orpulse train orrectangular wave is anon-sinusoidalwaveform that is theperiodic version of therectangular function. It is held high a percent each cycle (period) called theduty cycle and for the remainder of each cycle is low. A duty cycle of 50% produces asquare wave, a specific case of a rectangular wave. The average level of a rectangular wave is also given by the duty cycle.

The pulse wave is used as a basis for other waveforms thatmodulate an aspect of the pulse wave, for instance:

• Pulse-width modulation (PWM) refers to methods that encode information by varying the duty cycle of a pulse wave.
• Pulse-amplitude modulation (PAM) refers to methods that encode information by varying theamplitude of a pulse wave.

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

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

Equivalently, if duty cycle${\displaystyle d=\frac{\tau }{T}}$ is used, and${\displaystyle \omega =2\pi f}$:$${\displaystyle x(t)=Ad+\frac{2A}{\pi }\sum _{n=1}^{\mathrm{\infty }}\left(\frac{1}{n}\sin \left(\pi nd\right)\cos \left(n\omega t\right)\right)}$$

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 theSinc function, using the definition${\displaystyle \mathrm{sinc}x=\frac{\sin \pi x}{\pi x}}$, as$${\displaystyle x(t)=A\frac{\tau }{T}\left(1+2\sum _{n=1}^{\mathrm{\infty }}\left(\mathrm{sinc}\left(n\frac{\tau }{T}\right)\cos \left(2\pi nft\right)\right)\right)}$$ or with${\displaystyle d=\frac{\tau }{T}}$ as$${\displaystyle x(t)=Ad\left(1+2\sum _{n=1}^{\mathrm{\infty }}\left(\mathrm{sinc}\left(nd\right)\cos \left(2\pi nft\right)\right)\right)}$$

A pulse wave can be created by subtracting asawtooth wave from a phase-shifted version of itself. If the sawtooth waves arebandlimited, the resulting pulse wave is bandlimited, too.

Theharmonic 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 manySteve Winwood songs, such as "While You See a Chance".^[10]

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

• Waves

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