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.

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

The Fourier series expansion for a rectangular pulse wave with period $T$, amplitude $A$ and pulse length $\tau$ is
$x(t) = A \frac + \frac \sum_^ \left(\frac \sin\left(\pi n\frac\right) \cos\left(2\pi nft\right)\right)$
where $f = \frac$.

Equivalently, if duty cycle $d = \frac$ is used, and $\omega = 2\pi f$:
$x(t) = Ad + \frac \sum_^ \left(\frac\sin\left(\pi n d \right)\cos\left(n \omega t \right) \right)$

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 $\operatornamex = \frac$, as
$x(t) = A \frac \left(1 + 2\sum_^\infty \left(\operatorname\left(n\frac \right)\cos\left(2\pi n f t\right) \right) \right)$
or with $d = \frac$ as
$x(t) = A d \left(1 + 2\sum_^\infty \left(\operatorname\left(n d\right)\cos\left(2\pi n f t\right) \right) \right)$

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.Hartmann, William M. (2004). ''Signals, Sound, and Sensation'', p.109. Springer Science & Business Media. . Acoustically, the rectangular wave has been described variously as having a narrow/thin,Reid, Gordon (February 2000).
Synth Secrets: Modulation
, ''SoundOnSound.com''. Retrieved May 4, 2018.Souvignier, Todd (2003). ''Loops and Grooves'', p.12. Hal Leonard. .Cann, Simon (2011).
How to Make a Noise
', . BookBaby. .Aikin, Jim (2004). ''Power Tools for Synthesizer Programming'', p.55-56. Hal Leonard. .Hurtig, Brent (1988). ''Synthesizer Basics'', p.23. Hal Leonard. . nasal/buzzy/biting, clear,Holmes, Thom (2015). ''Electronic and Experimental Music'', p.230. Routledge. . 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

References

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Waves