Pulse wave

This article is about a pulse waveform. For a heart beat, see Pulse. For a Dirac pulse train, see Sampling function. For the aperiodic version, see Pulse function.

For other uses, see Pulse (disambiguation).

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's duty cycle D is the ratio between pulse duration 𝜏 and period T.

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

Fourier series of a 33.3% pulse wave, first fifty harmonics (summation in red)

The Fourier series expansion for a rectangular pulse wave with period ${\displaystyle T}$, amplitude ${\displaystyle A}$ and pulse length ${\displaystyle \tau }$ is^[1]

$${\displaystyle x(t)=A{\frac {\tau }{T}}+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi n{\frac {\tau }{T}}\right)\cos \left(2\pi nft\right)\right)}$$ where ${\displaystyle f={\frac {1}{T}}}$.

Equivalently, if duty cycle ${\displaystyle d={\frac {\tau }{T}}}$ is used, and ${\displaystyle \omega =2\pi f}$: $${\displaystyle x(t)=Ad+{\frac {2A}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{n}}\sin \left(\pi nd\right)\cos \left(n\omega t\right)\right)}$$

Note that, for symmetry, the starting time (${\displaystyle t=0}$) in this expansion is halfway through the first pulse.

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

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]

See also

• Gibbs phenomenon
• Pulse shaping
• Sinc function
• Sine wave

References

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