Pulse wave

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.

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
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 $$\omega = 2\pi f$$: $$x(t) = Ad + \frac{2A}{\pi} \sum_{n=1}^{\infty} \left(\frac{1}{n}\sin\left(\pi n d \right)\cos\left(n \omega t \right) \right) $$

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 $$x(t) = A \frac{\tau}{T} \left(1 + 2\sum_{n=1}^\infty \left(\operatorname{sinc}\left(n\frac{\tau}{T} \right)\cos\left(2\pi n f t\right) \right) \right) $$ or with $$d = \frac{\tau}{T}$$ as $$x(t) = A d \left(1 + 2\sum_{n=1}^\infty \left(\operatorname{sinc}\left(n d\right)\cos\left(2\pi n f t\right) \right) \right) $$

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".