Editing Sawtooth wave

{{Short description|Non-sinusoidal waveform}}
{{Refimprove|date=August 2008}}
{{Infobox mathematical function
| name = Sawtooth wave
| image = sawtooth-td and fd.png
| imageclass = skin-invert-image
| imagesize = 400px
| imagealt = A bandlimited sawtooth wave pictured in the time domain and frequency domain.
| caption = A [[Bandlimiting|bandlimited]] sawtooth wave<ref name="bandlimited-synthesis">{{cite conference |title=LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms |last1=Kraft |first1=Sebastian |last2=Zölzer |first2=Udo |date=5 September 2017 |book-title=Proceedings of the 20th [[International Conference on Digital Audio Effects]] (DAFx-17) |pages=255–259 |location=Edinburgh |conference=20th International Conference on Digital Audio Effects (DAFx-17) |conference-url=http://www.dafx17.eca.ed.ac.uk/}}</ref> pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220&nbsp;Hz (A<sub>3</sub>).
| general_definition = <math>x(t) = 2 \left (t - \left\lfloor t + \tfrac{1}{2} \right\rfloor \right), t - \tfrac{1}{2} \notin \Z</math>
| fields_of_application = Electronics, synthesizers
| domain = <math>\R \setminus \left\{ n - \tfrac{1}{2} \right\}, n \in \Z</math>
| codomain = <math>\left( -1,1 \right)</math>
| parity = Odd
| period = 1
| root = <math>\Z</math>
| fourier_series = <math>x(t) = -\frac{2}{\pi}\sum_{k=1}^{\infty} \frac{{\left( -1 \right)}^{k}}{k} \sin \left(2 \pi k t \right)</math>
}}

{{Listen|filename=220 Hz anti-aliased sawtooth wave.ogg|title=Sawtooth wave|description=5 seconds of 220&nbsp;Hz sawtooth wave}}

The '''sawtooth wave''' (or '''saw wave''') is a kind of [[non-sinusoidal waveform]]. It is so named based on its resemblance to the teeth of a plain-toothed [[saw]] with a zero [[Rake angle|rake angle]].  A single sawtooth, or an intermittently triggered sawtooth, is called a '''ramp waveform'''.

The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse (or inverse) sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the extreme case of an asymmetric [[triangle wave]].<ref>{{cite web | url=http://mathworld.wolfram.com/FourierSeriesTriangleWave.html |title=Fourier Series-Triangle Wave - from Wolfram MathWorld | website=Mathworld.wolfram.com |date=2012-07-02 |access-date=2012-07-11}}</ref>

The equivalent [[piecewise linear function]]s
<math display="block">x(t) = t - \lfloor t \rfloor</math>
<math display="block">x(t) = t \bmod 1</math>
based on the [[floor function]] of time ''t'' is an example of a sawtooth wave with [[Wave period|period]] 1.

A more general form, in the range −1 to 1, and with period ''p'', is 
<math display="block">2\left({\frac t p} - \left\lfloor {\frac 1 2} + {\frac t p} \right\rfloor\right)</math>

This sawtooth function has the same [[Phase (waves)|phase]] as the [[sine]] function.

While a [[Square wave (waveform)|square wave]] is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd [[harmonic]]s of the [[fundamental frequency]]. Because it contains all the integer harmonics, it is one of the best waveforms to use for [[subtractive synthesis]] of musical sounds, particularly bowed string instruments like violins and cellos, since the [[Stick-slip phenomenon|slip-stick behavior]] of the bow drives the strings with a sawtooth-like motion.<ref>{{cite web|url=https://homepages.abdn.ac.uk/d.j.benson/pages/html/music.pdf|format=PDF|title=Music: A Mathematical Offering|author=Dave Benson|page=42|website=Homepages.abdn.ac.uk|access-date=26 November 2021}}</ref>

{{Listen|filename=Additive_220Hz_Sawtooth_Wave.wav|title=Additive Sawtooth Demo|description=220&nbsp;Hz sawtooth wave created by harmonics added every second over sine wave.}}

A sawtooth can be constructed using [[additive synthesis]].  For period ''p'' and amplitude ''a'', the following infinite [[Fourier series]] converge to a sawtooth and a reverse (inverse) sawtooth wave:

<math display="block"> f = \frac{1}{p}</math>
<math display="block">x_\text{sawtooth}(t) = - \frac{2 a}{\pi}\sum_{k=1}^{\infty} {(-1)}^{k} \frac{\sin(2\pi kft)}{k}</math>
<math display="block">x_\text{reverse sawtooth}(t) = \frac{2a}{\pi}\sum_{k=1}^{\infty} {(-1)}^{k} \frac{\sin(2\pi kft)}{k}</math>

In [[Digital data|digital]] synthesis, these series are only summed over ''k'' such that the highest harmonic, ''N''<sub>max</sub>, is less than the [[Nyquist frequency]] (half the [[sampling frequency]]).  This summation can generally be more efficiently calculated with a [[fast Fourier transform]].  If the waveform is digitally created directly in the time domain using a non-[[bandlimited]] form, such as ''y''&nbsp;=&nbsp;''x''&nbsp;−&nbsp;[[Floor function|floor]](''x''), infinite harmonics are sampled and the resulting tone contains [[aliasing]] distortion.

[[File:Synthesis sawtooth.gif|thumb|upright=1.6|right|Animation of the additive synthesis of a sawtooth wave with an increasing number of harmonics]]

An audio demonstration of a sawtooth played at [[440&nbsp;Hz]] (A<sub>4</sub>) and 880&nbsp;Hz (A<sub>5</sub>) and 1,760&nbsp;Hz (A<sub>6</sub>) is available below.  Both bandlimited (non-aliased) and aliased tones are presented.

{{Listen|filename=Sawtooth-aliasingdemo.ogg|title=Sawtooth aliasing demo|description=Sawtooth waves played bandlimited and aliased at 440&nbsp;Hz, 880&nbsp;Hz, and 1,760&nbsp;Hz}}

== Applications ==
* Sawtooth waves are known for their use in [[electronic music]]. The sawtooth and square waves are among the most common waveforms used to create sounds with subtractive [[subtractive synthesis|analog]] and [[virtual analog]] music synthesizers.
* Sawtooth waves are used in [[Switched-mode power supply|switched-mode power supplies]]. In the regulator chip the feedback signal from the output is continuously compared to a high-frequency sawtooth to generate a new duty cycle PWM signal on the output of the [[comparator]].
* In the field of computer science, particularly in automation and robotics, <math>\mathrm{sawtooth}(\theta)=2\arctan(\tan(\frac{\theta}{2}))</math> allows to calculate sums and differences of angles while avoiding discontinuities at 360° and 0°.{{Fact|date=October 2024}}
* The sawtooth wave is the form of the vertical and horizontal [[Deflection (physics)|deflection]] signals used to generate a [[raster graphics|raster]] on [[cathode-ray tube|CRT]]-based television or monitor screens. [[Oscilloscope]]s also use a sawtooth wave for their horizontal deflection, though they typically use [[electrostatic]] deflection.
** On the wave's "ramp", the magnetic field produced by the deflection yoke drags the [[electron]] beam across the face of the CRT, creating a [[scan line]].
** On the wave's "cliff", the magnetic field suddenly collapses, causing the electron beam to return to its resting position as quickly as possible.
** The current applied to the deflection yoke is adjusted by various means (transformers, capacitors, center-tapped windings) so that the half-way voltage on the sawtooth's cliff is at the zero mark, meaning that a negative current will cause deflection in one direction, and a positive current deflection in the other; thus, a center-mounted deflection yoke can use the whole screen area to depict a trace. The horizontal frequency is 15.734&nbsp;kHz on [[NTSC]], 15.625&nbsp;kHz for [[PAL]] and [[SECAM]].
** The vertical deflection system operates the same way as the horizontal, though at a much lower frequency (59.94&nbsp;Hz on [[NTSC]], 50&nbsp;Hz for PAL and SECAM).
** The ramp portion of the wave must appear as a straight line. If otherwise, it indicates that the current isn't increasing linearly, and therefore that the magnetic field produced by the deflection yoke is not linear. As a result, the electron beam will accelerate during the non-linear portions. This would result in a television image "squished" in the direction of the non-linearity. Extreme cases will show marked brightness increases, since the electron beam spends more time on that side of the picture. 
** The first television receivers had controls allowing users to adjust the picture's vertical or horizontal linearity. Such controls were not present on later sets as the stability of electronic components had improved.

== See also ==
[[File:Waveforms.svg|thumb|upright=1.8|[[sine wave|Sine]], [[Square wave (waveform)|square]], [[triangle wave|triangle]], and sawtooth waveforms]]

* [[List of periodic functions]]
* [[Sine wave]]
* [[Square wave (waveform)|Square wave]]
* [[Triangle wave]]
* [[Pulse wave]]
* [[Sound]]
* [[Wave]]
* [[Zigzag]]

== References ==
{{reflist}}

== External links ==
*{{cite book | author=Hugh L. Montgomery | author-link=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |author-link2=Robert Charles Vaughan (mathematician)  | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=536–537}}

{{Waveforms}}

[[Category:Waveforms]]
[[Category:Fourier series]]