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The shape of the pulse wave is defined by its duty cycle D, which is the ratio between the pulse duration ( $\tau$ ) and the period (T)

A pulse wave or pulse train is a kind of non-sinusoidal waveform that is similar to a square wave, but does not have the symmetrical shape associated with a perfect square wave. It is a term common to 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.^[1] The pulse wave is also known as the rectangular wave, the periodic version of the rectangular function.

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.

The Fourier series expansion for a rectangular pulse wave with period $T$ and pulse time $τ$ is

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)

Note that, for symmetry, the starting time ( $t = 0$ ) in this expansion is halfway through the first pulse. The phase can be offset to match the accompanying graph by replacing $t$ with $t - τ /2$ .

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. Another way to create one is with a single ramp wave (sawtooth or triangle) and a comparator, with the ramp wave on one input, and a variable DC^{[clarification needed]} threshold on the other. The result will be a precisely controlled pulse width, but it will not be bandlimited.

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.^[2]

References

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[1] ttp://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm

[winwood-2] "Synth Soloing in the Style of Steve Winwood".

[1]

[2]

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-{{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 wave form|a heart beat|Pulse}}
+{{about|a rectangular pulse train|a Dirac pulse train|Sampling function}}
+{{for|the aperiodic version|Pulse function}}
+{{Other uses|Pulse (disambiguation)}}
 {{Refimprove|date=June 2013}}
-[[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}}%]]
+[[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>) and the period (T)]]
-[[File:PWM duty cycle with label.gif|thumb|Duty cycles]]
-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 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>
+A '''pulse wave''' or '''pulse train''' is a kind of [[non-sinusoidal]] [[waveform]] that is similar to a [[square wave]], but does not have the symmetrical shape associated with a perfect square wave.  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>http://www.soundonsound.com/sos/feb00/articles/synthsecrets.htm</ref>
-The [[pulse (signal processing)|pulse]] wave is also known as the '''rectangular wave''', the [[Periodic function|periodic]] version of the [[rectangular function]].
+The pulse wave is also known as the '''rectangular wave''', the [[Periodic function|periodic]] version of the [[rectangular function]].
-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"/>
+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]].
 The [[Fourier series]] expansion for a rectangular pulse wave with period {{math|''T''}} and pulse time {{math|''τ''}} is
@@ Line 17: / Line 19: @@
 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. 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.
+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.
+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}}</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)]]
-==Sound==
-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, more 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> "Pulse waves with different duty cycles have quite different audible characteristics. Narrow cycles (usually in the range 5 to 10 percent) are thin and nasal, and are often used to create sounds such as [[oboe]]s. As the duty cycle becomes closer to 50 percent the sound thickens considerably, but at exactly 50 percent it has a distinctively hollow character that is ideal for simulating [[clarinets]]," and similar sounding instruments.<ref name="Reid"/> "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> If the duty cycle denominator is not a whole number (the numerator being 1) then harmonics are quieted but not eliminated: "Because 28.5 percent [57:200] lies somewhere between the 1:3 [{{repitan|33.|3}}%] and 1:4 [25%] duty cycles, every third harmonic is somewhat attenuated, as is every fourth, but no harmonics are completely eliminated from the signal."<ref name="Reid"/>
 ==See also==
-*[[Gibbs phenomenon]]
+*[[Sampling function]]
-*[[Pulse shaping]]
+*[[Pulse-width modulation]]
-*[[Sinc function]]
+*[[Sine wave]]
+*[[Triangle wave]]
+*[[Square wave]]
 ==References==