Pulse wave

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

A pulse wave or pulse train or rectangular wave is a non-sinusoidal waveform that is the periodic version of the rectangular function. It is held high a percent each cycle (period) called the duty cycle and for the remainder of each cycle is low. A duty cycle of 50% produces a square wave, a specific case of a rectangular wave. The average level of a rectangular wave is also given by the duty cycle.

A pulse wave is used as a basis for other waveforms that modulate an aspect of the pulse wave. In pulse-width modulation (PWM) information is encoded by varying the duty cycle of a pulse wave. Pulse-amplitude modulation (PAM) encodes information by varying the amplitude.

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

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

The harmonic spectrum of a pulse wave is determined by the duty cycle. Acoustically, the rectangular wave has been described variously as having a narrow/thin, nasal/buzzy/biting, clear, resonant, rich, round and bright sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".

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