Pulse wave

This article is about a pulse waveform. For a heart beat, see Pulse. For a Dirac pulse train, see Sampling function. For the aperiodic version, see Pulse function.

For other uses, see Pulse (disambiguation).

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

The pulse wave is used as a basis for other waveforms that modulate 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 the amplitude of a pulse wave.

Frequency-domain representation

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

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

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

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

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

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

Generation

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.

Applications

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".

See also

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