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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

• Generation

• Applications

• See also

• References

The Fourier series expansion for a rectangular pulse wave with period T {\displaystyle T} , amplitude A {\displaystyle A} and pulse length τ τ {\displaystyle \tau } iscite-ref-1[1]

x ( t ) = A τ τ T + 2 A π π ∑ ∑ n = 1 ∞ ∞ ( 1 n sin ⁡ ⁡ ( π π n τ τ T ) cos ⁡ ⁡ ( 2 π π n f t ) ) {\displaystyle 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 = 1 T {\displaystyle f={\frac {1}{T}}} .

Equivalently, if duty cycle d = τ τ T {\displaystyle d={\frac {\tau }{T}}} is used, and ω ω = 2 π π f {\displaystyle \omega =2\pi f} : x ( t ) = A d + 2 A π π ∑ ∑ n = 1 ∞ ∞ ( 1 n sin ⁡ ⁡ ( π π n d ) cos ⁡ ⁡ ( n ω ω t ) ) {\displaystyle 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 {\displaystyle t=0} ) in this expansion is halfway through the first pulse.

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

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.

The harmonic spectrum of a pulse wave is determined by the duty cycle.cite-ref-holmes-2-0[2]cite-ref-souvignier-3-0[3]cite-ref-cann-4-0[4]cite-ref-5[5]cite-ref-6[6]cite-ref-7[7]cite-ref-8[8]cite-ref-9[9] Acoustically, the rectangular wave has been described variously as having a narrowcite-ref-winwood-10-0[10]/thin,cite-ref-reid-11-0[11]cite-ref-souvignier-3-1[3]cite-ref-cann-4-1[4]cite-ref-aikin-12-0[12]cite-ref-basics-13-0[13] nasalcite-ref-reid-11-1[11]cite-ref-souvignier-3-2[3]cite-ref-cann-4-2[4]cite-ref-winwood-10-1[10]/buzzycite-ref-basics-13-1[13]/biting,cite-ref-aikin-12-1[12] clear,cite-ref-holmes-2-1[2] resonant,cite-ref-holmes-2-2[2] rich,cite-ref-souvignier-3-3[3]cite-ref-basics-13-2[13] roundcite-ref-souvignier-3-4[3]cite-ref-basics-13-3[13] and brightcite-ref-basics-13-4[13] sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".cite-ref-winwood-10-2[10]

• Gibbs phenomenon

• Pulse shaping

• Sinc function

• Sine wave

cite-note-11. ↑ Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632

cite-note-holmes-22. ↑ Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232.

cite-note-souvignier-33. ↑ Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135.

cite-note-cann-44. ↑ Cann, Simon (2011). How to Make a Noise, [unpaginated]. BookBaby. ISBN 9780955495540.

cite-note-55. ↑ Pejrolo, Andrea and Metcalfe, Scott B. (2017). Creating Sounds from Scratch, p.56. Oxford University Press. ISBN 9780199921881.

cite-note-66. ↑ Snoman, Rick (2013). Dance Music Manual, p.11. Taylor & Francis. ISBN 9781136115745.

cite-note-77. ↑ Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). Handbook of Applications of Chaos Theory, [unpaginated]. CRC Press. ISBN 9781315356549.

cite-note-88. ↑ "Electronic Music Interactive: 14. Square and Rectangle Waves", UOregon.edu.

cite-note-99. ↑ Hartmann, William M. (2004). Signals, Sound, and Sensation, p.109. Springer Science & Business Media. ISBN 9781563962837.

cite-note-winwood-1010. ↑ citerefkovarsky2015Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018.

cite-note-reid-1111. ↑ Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018.

cite-note-aikin-1212. ↑ Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089.

cite-note-basics-1313. ↑ Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143.