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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.

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

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]

References

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

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

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

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

[Cann-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.

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

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

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

[Basics-13] Hurtig, Brent (1988). Synthesizer Basics, p.23. Hal Leonard. ISBN 9780881887143.

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@@ Line 1: / Line 1: @@
+{{short description|Periodic rectangular waveform}}
-{{About|a rectangular pulse wave or train|a heart beat|Pulse|a Dirac pulse train|Dirac comb|the aperiodic version|Pulse function||Pulse (disambiguation)}}
+{{about|a pulse [[waveform]]|a heart beat|Pulse|a Dirac pulse train|Sampling function|the aperiodic version|Pulse function}}
+{{Other uses|Pulse (disambiguation)}}{{Multi image
+| direction         = vertical
+| image1            = PulseTrain.png
+| image2            = PWM duty cycle with label.gif
+| footer            = A pulse wave's duty cycle D is the ratio between pulse duration 𝜏 and period T.
+| width             = 200
+}}
+A '''pulse wave''' or '''pulse train''' or '''rectangular wave''' is a [[non-sinusoidal]] [[waveform]] that is the [[Periodic function|periodic]] version of the [[rectangular function]]. It is held high a percent each cycle ([[Period of a function|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 [[Modulation|modulate]] an aspect of the pulse wave, for instance:
-{{Refimprove|date=June 2013}}
+* [[Pulse-width modulation]] (PWM) refers to methods that encode information by varying the duty cycle of a pulse wave.
-[[File:Dutycycle.svg|thumb|350px|The shape of the pulse wave is defined by its duty cycle D, which is the ratio between the pulse duration (<math>\tau</math> or "T+") and the period (T), the wave in the image has a duty cycle of 1/3 or {{repitan|33.|3}}%]]
+* [[Pulse-amplitude modulation]] (PAM) refers to methods that encode information by varying the [[amplitude]] of a pulse wave.
-[[File:PWM duty cycle with label.gif|thumb|Duty cycles]]
+==Frequency-domain representation==
-A '''pulse wave''' or '''pulse train''' is a kind of [[non-sinusoidal]] [[waveform]] that includes [[square wave]]s (duty cycle of 50%) and similarly periodic but asymmetrical waves (duty cycles other than 50%). It is a term common to [[Analog synthesizer|synthesizer]] programming, and is a typical waveform available on many synthesizers. The exact shape of the wave is determined by the [[duty cycle]] of the [[oscillator]]. In many synthesizers, the duty cycle can be modulated (sometimes called pulse-width modulation) for a more dynamic timbre.<ref>Reid, Gordon (February 2000). "[http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm  Synth Secrets: Modulation]", ''SoundOnSound.com''. Retrieved May 4, 2018.</ref>
+[[File:Pulse wave 33.33 percent Fourier series 50 harmonics.png|thumb|Fourier series of a {{repitan|33.|3}}% pulse wave, first fifty harmonics (summation in red)|200x200px]]The [[Fourier series]] expansion for a rectangular pulse wave with period <math>T</math>, amplitude <math>A</math> and pulse length <math>\tau</math> is<ref>Smith, Steven W. ''The Scientist & Engineer's Guide to Digital Signal Processing'' {{ISBN|978-0966017632}} </ref>
-The [[pulse (signal processing)|pulse]] wave is also known as the '''rectangular wave''', the [[Periodic function|periodic]] version of the [[rectangular function]].
+<math display="block">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)</math>
+where <math>f = \frac{1}{T}</math>.
+Equivalently, if duty cycle <math>d = \frac{\tau}{T}</math> is used, and <math>\omega = 2\pi f</math>:
-The average level of a rectangular wave is also given by the duty cycle, therefore by varying the on and off periods and then averaging these said periods, it is possible to represent any value between the two limiting levels. This is the basis of [[pulse width modulation]]. "One of the most sonically pleasing and sought after sounds in synth history is created by modulating the [pulse] width with an [[low-frequency oscillation|LFO]]."<ref>McGuire, Sam and Van der Rest, Nathan (2015). ''[https://books.google.com/books?id=w9BWCgAAQBAJ&pg=PT25&dq=%22pulse+wave%22+harmonics&hl=en&sa=X&ved=0ahUKEwiEutml9OvaAhVEyoMKHXnKCYAQ6AEIYDAJ#v=onepage&q=%22pulse%20wave%22%20harmonics&f=false The Musical Art of Synthesis]'', {{unpaginated}}. CRC Press. {{ISBN|9781317570523}}.</ref><ref name="Scratch"/>
+<math display="block">x(t) = Ad + \frac{2A}{\pi} \sum_{n=1}^{\infty} \left(\frac{1}{n}\sin\left(\pi n d \right)\cos\left(n \omega t \right) \right) </math>
+Note that, for symmetry, the starting time (<math>t=0</math>) in this expansion is halfway through the first pulse.
-The [[Fourier series]] expansion for a rectangular pulse wave with period {{math|''T''}} and pulse time {{math|''τ''}} is
+Alternatively, <math>x(t) </math> can be written using the [[Sinc function]], using the definition <math>\operatorname{sinc}x = \frac{\sin \pi x}{\pi x}</math>, as
-:<math>f(t) = \frac{\tau}{T} + \sum_{n=1}^{\infty} \frac{2}{n\pi} \sin\left(\frac{\pi n\tau}{T}\right) \cos\left(\frac{2\pi n}{T} t\right)</math>
+<math display="block">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 n f t\right) \right) \right) </math>
+or with <math>d = \frac{\tau}{T}</math> as
+<math display="block">x(t) = A d \left(1 + 2\sum_{n=1}^\infty \left(\operatorname{sinc}\left(n d\right)\cos\left(2\pi n f t\right) \right) \right) </math>
+==Generation==
-Note that, for symmetry, the starting time ({{math|''t'' {{=}} 0}}) in this expansion is halfway through the first pulse. The phase can be offset to match the accompanying graph by replacing {{math|''t''}} with {{math|''t'' - ''τ''/2}}.
+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==
-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. Another way to create one is with a single ramp wave (sawtooth or [[Triangle wave|triangle]]) and a [[comparator]], with the ramp wave on one input, and a variable DC{{clarify|date=May 2015}} threshold on the other. The result will be a precisely controlled pulse width, but it will not be bandlimited.
+The [[harmonic spectrum]] of a pulse wave is determined by the duty cycle.<ref name="Holmes"/><ref name="Souvignier"/><ref name="Cann"/><ref>Pejrolo, Andrea and Metcalfe, Scott B. (2017). ''Creating Sounds from Scratch'', p.56. Oxford University Press. {{ISBN|9780199921881}}.</ref><ref>Snoman, Rick (2013). ''Dance Music Manual'', p.11. Taylor & Francis. {{ISBN|9781136115745}}.</ref><ref>Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). ''[https://books.google.com/books?id=CAhEDwAAQBAJ&dq=%22rectangle+wave%22+harmonics&pg=PT440 Handbook of Applications of Chaos Theory]'', {{unpaginated}}. CRC Press. {{ISBN|9781315356549}}.</ref><ref>"[http://pages.uoregon.edu/emi/14.php Electronic Music Interactive: 14. Square and Rectangle Waves]", ''UOregon.edu''.</ref><ref>Hartmann, William M. (2004). ''Signals, Sound, and Sensation'', p.109. Springer Science & Business Media. {{ISBN|9781563962837}}.</ref> Acoustically, the rectangular wave has been described variously as having a narrow<ref name="winwood"/>/thin,<ref name="Reid">Reid, Gordon (February 2000). "[http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm Synth Secrets: Modulation]", ''SoundOnSound.com''. Retrieved May 4, 2018.</ref><ref name="Souvignier">Souvignier, Todd (2003). ''Loops and Grooves'', p.12. Hal Leonard. {{ISBN|9780634048135}}.</ref><ref name="Cann">Cann, Simon (2011). ''[https://books.google.com/books?id=QTBVDQAAQBAJ&dq=pulse+wave+sawtooth+wave&pg=PT20 How to Make a Noise]'', {{unpaginated}}. BookBaby. {{ISBN|9780955495540}}.</ref><ref name="Aikin">Aikin, Jim (2004). ''Power Tools for Synthesizer Programming'', p.55-56. Hal Leonard. {{ISBN|9781617745089}}.</ref><ref name="Basics">Hurtig, Brent (1988). ''Synthesizer Basics'', p.23. Hal Leonard. {{ISBN|9780881887143}}.</ref> nasal<ref name="Reid"/><ref name="Souvignier"/><ref name="Cann"/><ref name="winwood"/>/buzzy<ref name="Basics"/>/biting,<ref name="Aikin"/> clear,<ref name="Holmes">Holmes, Thom (2015). ''Electronic and Experimental Music'', p.230. Routledge. {{ISBN|9781317410232}}.</ref> resonant,<ref name="Holmes"/> rich,<ref name="Souvignier"/><ref name="Basics"/> round<ref name="Souvignier"/><ref name="Basics"/> and bright<ref name="Basics"/> [[sound]]. Pulse waves are used in many [[Steve Winwood]] songs, such as "[[While You See a Chance]]".<ref name="winwood">{{cite web|url=http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240|title=Synth Soloing in the Style of Steve Winwood |last=Kovarsky|first=Jerry|date=Jan 15, 2015|website=KeyboardMag.com|access-date=May 4, 2018}}</ref>
-[[File:Spectrum square oscillation.jpg|thumb|350px|right|Odd harmonics of a 1000 Hz pulse wave with a 1/2 (50%) duty cycle (square wave)]]
-{{Listen|filename=Additive_220Hz_Square_Wave.wav|title=Additive square demo|description=220&nbsp;Hz square wave created by harmonics added every second over sine wave|format=[[Ogg]]}}
-[[File:Pulse wave 33.33 percent Fourier series 50 harmonics.png|thumb|Fourier series of a {{repitan|33.|3}}% pulse wave, first fifty harmonics (summation in red)]]
-Acoustically, the rectangular wave has been described as having a more "narrow and nasal" sound than a perfect square wave, and its characteristic sound features prominently in many [[Steve Winwood]] songs.<ref name="winwood">{{cite web|url=http://www.keyboardmag.com/lessons/1251/synth-soloing-in-the-style-of-steve-winwood/50240|title=Synth Soloing in the Style of Steve Winwood |last=Kovarsky|first=Jerry|date=Jan 15, 2015|website=KeyboardMag.com|publisher=|quote=For example, "[[While You See a Chance]]".|access-date=May 4, 2018}}</ref> "Pulse waves with short positive [duty] cycles (10% to 20%) have more harmonics and take on more of a thin, nasal character; longer positive [duty] cycles (30% to 40%) sound richer and rounder."<ref>Souvignier, Todd (2003). ''Loops and Grooves'', p.12. Hal Leonard. {{ISBN|9780634048135}}. "Pulse waves are often used to create bass sounds...Pulse width modulation has become a familiar, slightly clichéd sound within electronic dance music."</ref> "The tone varies according to the width of the pulse, giving a range of tones going from sounding similar to square wave, through becoming increasingly thinner and more nasal, to ending with noise."<ref>Cann, Simon (2011). ''[https://books.google.com/books?id=QTBVDQAAQBAJ&pg=PT20&dq=pulse+wave+sawtooth+wave&hl=en&sa=X&ved=0ahUKEwiJws-y3vDaAhVG5oMKHcQtCFAQ6AEITDAG#v=onepage&q=pulse%20wave%20sawtooth%20wave&f=false How to Make a Noise]'', {{unpaginated}}. BookBaby. {{ISBN|9780955495540}}.</ref> "The shift away from the symmetrical square wave [to the asymmetrical pulse wave] adds variation to the harmonic content, most notably as a [[comb filter]] in the higher harmonics."<ref name="Scratch">Pejrolo, Andrea and Metcalfe, Scott B. (2017). ''Creating Sounds from Scratch'', p.56. Oxford University Press. {{ISBN|9780199921881}}. "A pulse-wave shape is colorful on its own but can really stand out when the duty cycle percentage is shifted over time by an envelope generator or LFO using what is called ''pulse-width modulation''."</ref> "Pulse waves have a clear, resonant sound."<ref name="Holmes">Holmes, Thom (2015). ''Electronic and Experimental Music'', p.230. Routledge. {{ISBN|9781317410232}}. "The harmonic content of the pulse wave is determined by the duty cycle...The harmonic content of a pulse wave can be changed dramatically merely by altering its duty cycle."</ref> "As the pulse becomes narrower..., the wave acquires a thinner, ore biting character. A thin pulse wave is good for synthesizing [[Clavinet]] sounds."<ref>Aikin, Jim (2004). ''Power Tools for Synthesizer Programming'', p.55-56. Hal Leonard. {{ISBN|9781617745089}}.</ref> "In general, pulse waves are bright and buzzy, almost reed-like. The narrower the width, the thinner the sound. The wider the width, the rounder and richer the sound."<ref name="Basics"/> [[Double reed instrument]]s, such as the oboe, may approximate an almost square pulse wave.<ref>Johnston, Ian (2009). ''Measured Tones: The Interplay of Physics and Music'', p.203 CRC Press. 3rd edition. {{ISBN|9781439894675}}. "A double reed would probably have the gap open for longer than it is closed."</ref> The duty cycle determines the [[spectrum]] or [[timbre]] of a pulse wave,<ref name="Holmes"/><ref>Snoman, Rick (2013). ''Dance Music Manual'', p.11. Taylor & Francis. {{ISBN|9781136115745}}. "A pulse wave allows the width of the high and low states to be adjusted, therby varying the harmonic content of the sound."</ref> suppressing or "leaving out" ([[null (mathematics)|nullifying]]) the [[harmonic series (music)|harmonics]] which are divisible by the inverse of the duty cycle. Thus for a ratio of 50% (1/2) then all even harmonics (those divisible by 2/1) are suppressed, leaving only odd harmonics; for {{repitan|33.|33}}% (1/3), then every third harmonic is suppressed (those divisible by 3/1); and for 25% (1/4) then every fourth harmonic is suppressed (those divisible by 4/1), and so on.<ref name="Basics">Hurtig, Brent (1988). ''Synthesizer Basics'', p.23. Hal Leonard. {{ISBN|9780881887143}}. "There is a general rule of thumb that applies [to pulse wave harmonic spectra]: For every pulse wave with a width of <math>1/n</math>, every <math>n</math>th harmonic will be absent or weak in proportion to the other harmonics.</ref><ref>Skiadas, Christos H. and Skiadas, Charilaos; eds. (2017). ''[https://books.google.com/books?id=CAhEDwAAQBAJ&pg=PT440&dq=%22rectangle+wave%22+harmonics&hl=en&sa=X&ved=0ahUKEwjtw-ye7evaAhVkwYMKHZBSD7oQ6AEILzAB#v=onepage&q=%22rectangle%20wave%22%20harmonics&f=false Handbook of Applications of Chaos Theory]'', {{unpaginated}}. CRC Press. {{ISBN|9781315356549}}. "For example, if a rectangle wave has a duty cycle of 25%, or 1/4, every fourth harmonic is missing. If the duty cycle is 20%, or 1/5, every fifth harmonic would be missing."</ref><ref>"[http://pages.uoregon.edu/emi/14.php Electronic Music Interactive: 14. Square and Rectangle Waves]", ''UOregon.edu''. A pulse wave's, "harmonic spectrum is related to its duty cycle. For example, if a rectangle wave has a duty cycle of 25%, or 1/4, every fourth harmonic is missing. If the duty cycle is 20%, or 1/5, every fifth harmonic would be missing. Given a duty cycle of 12.5%, or 1/8, then every eighth harmonic would be missing."</ref><ref>Hartmann, William M. (2004). ''Signals, Sound, and Sensation'', p.109. Springer Science & Business Media. {{ISBN|9781563962837}}. "Duty factors of the precise form <math>1/n</math>, where <math>n</math> is an integer, can be achieved by nulling the <math>n</math>-th harmonic."</ref>
 ==See also==
@@ Line 29: / Line 40: @@
 *[[Pulse shaping]]
 *[[Sinc function]]
+*[[Sine wave]]
 ==References==
@@ Line 37: / Line 49: @@
 {{DEFAULTSORT:Pulse Wave}}
 [[Category:Waves]]
-{{Electronics-stub}}