Sawtooth wave - Citizendia

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. Non-sinusoidal waveforms are Waveforms that are not pure Sine waves They are usually derived from simple math functions It is named a sawtooth based on its resemblance to the teeth on the blade of a saw.

The usual convention is that a sawtooth wave ramps upward as time goes by 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'. The 2 orientations of sawtooth wave sound identical when other variables are controlled.

A bandlimited sawtooth wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A2).

The piecewise linear function

$x(t) = t - \operatorname{floor}(t)$

based on the floor function of time t, is an example of a sawtooth wave with period 1. In Mathematics, a piecewise linear function f \Omega \to V where V is a Vector space and \Omega In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The Frequency is a measure of the number of occurrences of a repeating event per unit Time.

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

$x(t) = 2 \left( {t \over a} - \operatorname{floor} \left ( {t \over a} + {1 \over 2} \right ) \right )$

This sawtooth function has the same phase as the sine function. 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

A sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer The fundamental tone, often referred to simply as the fundamental and abbreviated fo, 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 constructing other sounds, particularly strings, using 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

A sawtooth can be constructed using additive synthesis. Additive synthesis is a technique of audio synthesis which creates Musical Timbre. The infinite Fourier series

$x_\mathrm{sawtooth}(t) = \frac {2}{\pi}\sum_{k=1}^{\infin} \frac {\sin (2\pi kft)}{k}$

converges to an inverse sawtooth wave. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions A conventional sawtooth can be constructed using

$x_\mathrm{sawtooth}(t) = -\frac {2}{\pi}\sum_{k=1}^{\infin} \frac {\sin (2\pi kft)}{k}$

In digital synthesis, these series are only summed over k such that the highest harmonic, N_max, is less than the Nyquist frequency (half the sampling frequency). A digital system uses discrete (discontinuous values usually but not always Symbolized Numerically (hence called "digital" to represent information for The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the Sampling frequency Sampling theorem The Nyquist–Shannon sampling theorem states that perfect reconstruction 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 = x - floor(x), infinite harmonics are sampled and the resulting tone contains aliasing distortion. A bandlimited signal is a Deterministic or Stochastic signal whose Fourier transform or Power spectral density is zero above a certain finite In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The This article applies to signal processing including computer graphics

Animation of the additive synthesis of a sawtooth wave with an increasing number of harmonics

An audio demonstration of a sawtooth played at 440 Hz (A4) and 880 Hz (A5) and 1760 Hz (A6) is available below. A440 is the 440 Hz tone that serves as the standard for musical pitch. Both bandlimited (non-aliased) and aliased tones are presented.

Sawtooth aliasing demo

Sawtooth waves played bandlimited and aliased at 440 Hz, 880 Hz, and 1760 Hz

Problems listening to the file? See media help.

Applications

The sawtooth wave along with the square wave are the most common starting points used to create sounds with subtractive analog and Virtual analog music synthesizers. Subtractive synthesis is a method of subtracting Harmonic content from a sound via Sound synthesis, characterised by the application of an Audio filter An Analog Modeling Synthesizer is a Synthesizer that emulates the sounds of traditional Analog synthesizers using Digital signal processing components
The sawtooth wave is the form of the vertical and horizontal deflection signals used to generate a raster on CRT-based television or monitor. In Computer graphics, a raster graphics image or bitmap, is a Data structure representing a generally rectangular grid of Pixels Oscilloscopes also use a sawtooth wave for their horizontal deflection, though they typically use electrostatic deflection. An oscilloscope (commonly abbreviated to scope or O-scope) is a type of Electronic test equipment that allows signal Voltages to be viewed Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical
- 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. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J A scan line is one line or row in a Raster scanning pattern such as a video line on a Cathode ray tube (CRT display of a television or computer
- 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 through 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 will produce deflection in the other direction, allowing the whole screen to be covered by a center-mounted deflection yoke. Frequency is 15. 75 kHz on NTSC, 15. NTSC ( National Television System Committee) is the Analog television system used in the United States, Canada, Japan, Mexico 625 kHz for PAL and SECAM)
- The vertical deflection system operates the same way as the horizontal, though at a much lower frequency (60 Hz on NTSC, 50 Hz for PAL and SECAM). PAL, short for Phase Alternating Line, is a colour -encoding system used in Broadcast television systems in large parts of the world SECAM, also written SÉCAM ( Séquentiel couleur à mémoire, French for "Sequential Color with Memory" is an analog color television system NTSC ( National Television System Committee) is the Analog television system used in the United States, Canada, Japan, Mexico
- The ramp portion of the wave must be perfectly linear - if it isn't, it's an indication that the voltage isn't increasing linearly, and therefore that the magnetic field produced by the deflection yoke won't be linear. As a result, the electron beam will accelerate during the non-linear portions. On a television picture, this would result in the image being "squished" to the direction of the non-linearity. Extreme cases will show obvious brightness increases, since the electron beam spends more time on that side of the picture.
- Most TV sets used to have manual adjustments for vertical and/or horizontal linearity though they have generally disappeared due to the greater temporal stability of modern electronic components.

Applications

See also