Pulse wave

Periodic rectangular waveform

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

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 T {\displaystyle T} , amplitude A {\displaystyle A} and pulse length τ {\displaystyle \tau } is[1]

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.[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

References

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