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.

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^[1]

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.^[2]^[3]^[4]^[5]^[6]^[7]^[8]^[9] Acoustically, the rectangular wave has been described variously as having a narrow^[10]/thin,^[11]^[3]^[4]^[12]^[13] nasal^[11]^[3]^[4]^[10]/buzzy^[13]/biting,^[12] clear,^[2] resonant,^[2] rich,^[3]^[13] round^[3]^[13] and bright^[13] sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".^[10]

References

^ Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632
^ ^a ^b ^c Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232.
^ ^a ^b ^c ^d ^e Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135.
^ ^a ^b ^c Cann, Simon (2011). How to Make a Noise, ^{[unpaginated]}. BookBaby. ISBN 9780955495540.
^ Pejrolo, Andrea and Metcalfe, Scott B. (2017). Creating Sounds from Scratch, p.56. Oxford University Press. ISBN 9780199921881.
^ Snoman, Rick (2013). Dance Music Manual, p.11. Taylor & Francis. ISBN 9781136115745.
^ Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). Handbook of Applications of Chaos Theory, ^{[unpaginated]}. CRC Press. ISBN 9781315356549.
^ "Electronic Music Interactive: 14. Square and Rectangle Waves", UOregon.edu.
^ Hartmann, William M. (2004). Signals, Sound, and Sensation, p.109. Springer Science & Business Media. ISBN 9781563962837.
^ ^a ^b ^c Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018.
^ ^a ^b Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018.
^ ^a ^b Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089.
^ ^a ^b ^c ^d ^e Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143.