Pulse wave

Last updated February 21, 2024

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.

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]

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References

↑ Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632
1 2 3 Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232.
1 2 3 4 5 Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135.
1 2 3 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.
1 2 3 Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018.
1 2 Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018.
1 2 Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089.
1 2 3 4 5 Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143.

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[1] Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632

[Holmes-2] 1 2 3 Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232.

[Souvignier-3] 1 2 3 4 5 Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135.

[Cann-4] 1 2 3 Cann, Simon (2011). How to Make a Noise , ^{[unpaginated]}. BookBaby. ISBN 9780955495540.

[5] Pejrolo, Andrea and Metcalfe, Scott B. (2017). Creating Sounds from Scratch, p.56. Oxford University Press. ISBN 9780199921881.

[6] Snoman, Rick (2013). Dance Music Manual, p.11. Taylor & Francis. ISBN 9781136115745.

[7] Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). Handbook of Applications of Chaos Theory , ^{[unpaginated]}. CRC Press. ISBN 9781315356549.

[8] "Electronic Music Interactive: 14. Square and Rectangle Waves", UOregon.edu.

[9] Hartmann, William M. (2004). Signals, Sound, and Sensation, p.109. Springer Science & Business Media. ISBN 9781563962837.

[winwood-10] 1 2 3 Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018.

[Reid-11] 1 2 Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018.

[Aikin-12] 1 2 Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089.

[Basics-13] 1 2 3 4 5 Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143.

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