Sawtooth wave

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Sawtooth wave

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw.

The convention is that a sawtooth wave ramps upward and then sharply drops. However, there are also sawtooth waves in which the wave ramps downward and then sharply rises. The latter type of sawtooth wave is called a 'reverse sawtooth wave' or 'inverse sawtooth wave'.

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Encyclopedia

The piecewise linear function

based on the floor function

Floor function

In mathematics and computer science, the floor and ceiling function s map a real number to the next smallest or next largest integer. More precisely, floor is the largest integer not greater than x and ceiling is the smallest integer not less than x....
of time t, is an example of a sawtooth wave with period 1.

A more general form, in the range -1 to 1, and with period a, is

This sawtooth function has the same phase

Phase (waves)

The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0....
as the sine function.

A sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonic

Harmonic

In acoustics and telecommunication, a harmonic of a wave is a component frequency of the Signalling that is an integer multiple of the fundamental frequency....
s of the fundamental frequency

Fundamental frequency

The fundamental tone, often referred to simply as the fundamental and abbreviated f0 or F0, is the lowest frequency in a harmonic series ....
. Because it contains all the integer harmonics, it is one of the best waveforms to use for synthesizing musical sounds, particularly bowed string instruments like violins and cellos, using subtractive synthesis

Subtractive synthesis

Subtractive synthesis is a method of subtracting harmonic content from a sound via sound synthesis, characterised by the application of an audio filter to an audio signal....
.

A sawtooth can be constructed using additive synthesis

Additive synthesis

Additive synthesis is a technique of audio synthesis which creates musical timbre.The timbre of an instrument is composed of multiple harmonics or partials, in different quantities, that change over time....
. The infinite Fourier series

Fourier series

In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sine wave . The study of Fourier series is a branch of Fourier analysis....

converges to an inverse sawtooth wave. A conventional sawtooth can be constructed using

In digital

Digital

A digital system uses discrete values, usually but not always symbolized numerically to represent information for input, processing, transmission, storage, etc....
synthesis, these series are only summed over k such that the highest harmonic, N_max, is less than the Nyquist frequency

Nyquist frequency

The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist?Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system....
(half the sampling frequency). This summation can generally be more efficiently calculated with a Fast Fourier transform

Fast Fourier transform

A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex number to group theory and number theory; this article gives an overview of the available techniques and some of their general propert...
. If the waveform is digitally created directly in the time domain using a non-bandlimited

Bandlimited

A bandlimited signal is a deterministic or stochastic signal whose Fourier transform or power spectral density is zero above a certain finite frequency....
form, such as y = x - floor

Floor function

In mathematics and computer science, the floor and ceiling function s map a real number to the next smallest or next largest integer. More precisely, floor is the largest integer not greater than x and ceiling is the smallest integer not less than x....
(x), infinite harmonics are sampled and the resulting tone contains aliasing

Aliasing

In statistics, signal processing, computer graphics and related disciplines, aliasing refers to an effect that causes different continuous signals to become indistinguishable when sampling ....
distortion.

An audio demonstration of a sawtooth played at 440 Hz (A4) and 880 Hz (A5) and 1760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.

Applications

The sawtooth and square waves are the most common starting points used to create sounds with subtractive analog
Subtractive synthesis

Subtractive synthesis is a method of subtracting harmonic content from a sound via sound synthesis, characterised by the application of an audio filter to an audio signal....
and virtual analog music synthesizers.
The sawtooth wave is the form of the vertical and horizontal deflection
Deflection

Deflection or deflexion may refer to:*Deflection *Deflection *Deflection *Electrostatic deflection*Deflection ...
signals used to generate a raster
Raster graphics

In computer graphics, a raster graphics image or bitmap, is a data structure representing a generally Rectangle grid of pixels, or points of color, viewable via a Computer display, paper, or other display medium....
on CRT
CRT

CRT may refer to:In computing:* Transport_Layer_Security, in computing* The C runtime library , in programming* The C++ Curiously recurring template pattern, in programming....
-based television or monitor screens. Oscilloscope
Oscilloscope

An oscilloscope is a type of electronic test instrument that allows signal voltages to be viewed, usually as a two-dimensional graph of one or more electrical potential differences plotted as a function of time or of some other voltage ....
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
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
beam across the face of the CRT, creating a scan line
Scan line

A scan line is one line, or row, in a raster scanning pattern, such as a line on a cathode ray tube display of a television or computer.On older CRT screens the horizontal scan lines were visually discernible, even when viewed from a distance, as alternating colored lines and black lines....
.
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 voltage 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 voltage will cause deflection in one direction, and a positive voltage deflection in the other; thus, a center-mounted deflection yoke can use the whole screen area to depict a trace. Frequency is 15.734 kHz on NTSC
NTSC

NTSC is the analog television system used in most of the Americas, Japan, South Korea, Taiwan, the Philippines, Burma, and some Pacific island nations and territories ....
, 15.625 kHz for PAL
PAL

PAL, short for Phase Alternating Line, is a color-encoding system used in broadcast television systems in large parts of the world. Other common analog television systems are SECAM and NTSC....
and SECAM
SECAM

SECAM, also written S?CAM , is an analog television system first used in France.A team led by Henri de France working at Compagnie Fran?aise de T?l?vision invented SECAM....
)
The vertical deflection system operates the same way as the horizontal, though at a much lower frequency (59.94 Hz on NTSC
NTSC

NTSC is the analog television system used in most of the Americas, Japan, South Korea, Taiwan, the Philippines, Burma, and some Pacific island nations and territories ....
, 50 Hz for PAL and SECAM).
The ramp portion of the wave must appear as a straight line. If otherwise, it indicates that the voltage 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.

Sawtooth wave

Applications

See also