Pulse wave

Template:Short description Template:Multi image 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.

Frequency-domain representation

File:Pulse wave 33.33 percent Fourier series 50 harmonics.png

Fourier series of a Template:Repitan% 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

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

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

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

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