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Pulse wave
Periodic rectangular waveform

A pulse wave, also known as a pulse train or rectangular wave, is a periodic, non-sinusoidal waveform formed by holding a signal high for a portion of its period, defined by the duty cycle, and low for the remainder. A 50% duty cycle creates a square wave. Pulse waves serve as the foundation for modulation techniques such as pulse-width modulation (PWM), which varies the duty cycle to encode information, and pulse-amplitude modulation (PAM), which encodes data by changing the amplitude of the pulses. This makes pulse waves integral to signal processing and communication systems.

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Frequency-domain representation

The Fourier series expansion for a rectangular pulse wave with period T {\displaystyle T} , amplitude A {\displaystyle A} and pulse length τ {\displaystyle \tau } is1

x ( t ) = A τ T + 2 A π ∑ n = 1 ∞ ( 1 n sin ⁡ ( π n τ T ) cos ⁡ ( 2 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 T {\displaystyle d={\frac {\tau }{T}}} is used, and ω = 2 π f {\displaystyle \omega =2\pi f} : x ( t ) = A d + 2 A π ∑ n = 1 ∞ ( 1 n sin ⁡ ( π n d ) cos ⁡ ( n 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 x π x {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}} , as x ( t ) = A τ T ( 1 + 2 ∑ n = 1 ∞ ( sinc ⁡ ( n τ T ) cos ⁡ ( 2 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 T {\displaystyle d={\frac {\tau }{T}}} as x ( t ) = A d ( 1 + 2 ∑ n = 1 ∞ ( sinc ⁡ ( n d ) cos ⁡ ( 2 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)}

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.23456789 Acoustically, the rectangular wave has been described variously as having a narrow10/thin,1112131415 nasal16171819/buzzy20/biting,21 clear,22 resonant,23 rich,2425 round2627 and bright28 sound. Pulse waves are used in many Steve Winwood songs, such as "While You See a Chance".29

See also

References

  1. Smith, Steven W. The Scientist & Engineer's Guide to Digital Signal Processing ISBN 978-0966017632 /wiki/ISBN_(identifier) ↩

  2. Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232. /wiki/ISBN_(identifier) ↩

  3. Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135. /wiki/ISBN_(identifier) ↩

  4. Cann, Simon (2011). How to Make a Noise, [unpaginated]. BookBaby. ISBN 9780955495540. https://books.google.com/books?id=QTBVDQAAQBAJ&dq=pulse+wave+sawtooth+wave&pg=PT20 ↩

  5. Pejrolo, Andrea and Metcalfe, Scott B. (2017). Creating Sounds from Scratch, p.56. Oxford University Press. ISBN 9780199921881. /wiki/ISBN_(identifier) ↩

  6. Snoman, Rick (2013). Dance Music Manual, p.11. Taylor & Francis. ISBN 9781136115745. /wiki/ISBN_(identifier) ↩

  7. Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). Handbook of Applications of Chaos Theory, [unpaginated]. CRC Press. ISBN 9781315356549. https://books.google.com/books?id=CAhEDwAAQBAJ&dq=%22rectangle+wave%22+harmonics&pg=PT440 ↩

  8. "Electronic Music Interactive: 14. Square and Rectangle Waves", UOregon.edu. http://pages.uoregon.edu/emi/14.php ↩

  9. Hartmann, William M. (2004). Signals, Sound, and Sensation, p.109. Springer Science & Business Media. ISBN 9781563962837. /wiki/ISBN_(identifier) ↩

  10. Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018. http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240 ↩

  11. Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018. http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm ↩

  12. Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135. /wiki/ISBN_(identifier) ↩

  13. Cann, Simon (2011). How to Make a Noise, [unpaginated]. BookBaby. ISBN 9780955495540. https://books.google.com/books?id=QTBVDQAAQBAJ&dq=pulse+wave+sawtooth+wave&pg=PT20 ↩

  14. Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089. /wiki/ISBN_(identifier) ↩

  15. Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143. /wiki/ISBN_(identifier) ↩

  16. Reid, Gordon (February 2000). "Synth Secrets: Modulation", SoundOnSound.com. Retrieved May 4, 2018. http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm ↩

  17. Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135. /wiki/ISBN_(identifier) ↩

  18. Cann, Simon (2011). How to Make a Noise, [unpaginated]. BookBaby. ISBN 9780955495540. https://books.google.com/books?id=QTBVDQAAQBAJ&dq=pulse+wave+sawtooth+wave&pg=PT20 ↩

  19. Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018. http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240 ↩

  20. Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143. /wiki/ISBN_(identifier) ↩

  21. Aikin, Jim (2004). Power Tools for Synthesizer Programming, p.55-56. Hal Leonard. ISBN 9781617745089. /wiki/ISBN_(identifier) ↩

  22. Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232. /wiki/ISBN_(identifier) ↩

  23. Holmes, Thom (2015). Electronic and Experimental Music, p.230. Routledge. ISBN 9781317410232. /wiki/ISBN_(identifier) ↩

  24. Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135. /wiki/ISBN_(identifier) ↩

  25. Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143. /wiki/ISBN_(identifier) ↩

  26. Souvignier, Todd (2003). Loops and Grooves, p.12. Hal Leonard. ISBN 9780634048135. /wiki/ISBN_(identifier) ↩

  27. Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143. /wiki/ISBN_(identifier) ↩

  28. Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143. /wiki/ISBN_(identifier) ↩

  29. Kovarsky, Jerry (Jan 15, 2015). "Synth Soloing in the Style of Steve Winwood". KeyboardMag.com. Retrieved May 4, 2018. http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240 ↩