# Dictionary Definition

sawtooth n : a serration on a saw blade

# User Contributed Dictionary

## English

### Alternative spellings

### Adjective

- Shaped like a sawtooth.
- In the context of "mathematics|engineering": Describing a function or waveform that consists of repeated triangular transitions when graphed.

# Extensive Definition

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 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.

- x(t) = t - \operatorname(t)

based on the floor
function 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

- x(t) = 2 \left( - \operatorname \left ( + \right ) \right )

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

A sawtooth wave's sound is harsh and clear and
its spectrum contains both even and odd harmonics of the fundamental
frequency. 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.

A sawtooth can be constructed using additive
synthesis. The infinite Fourier
series

- x_\mathrm(t) = \frac \sum_^ \frac

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

- x_\mathrm(t) = -\frac \sum_^ \frac

In digital synthesis, these series
are only summed over k such that the highest harmonic, Nmax, 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
= x - floor(x),
infinite harmonics are sampled and the resulting tone contains
aliasing
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 wave along with the square wave are the most common starting points used to create sounds with subtractive analog and Virtual analog music synthesizers.
- The sawtooth wave is the form of the vertical and horizontal
deflection signals
used to generate a raster on
CRT-based
television or monitor. Oscilloscopes
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 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, 15.625 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).
- 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.

## See also

sawtooth in German: Kippschwingung

sawtooth in Japanese: のこぎり波

sawtooth in Portuguese: Onda dente de
serra

sawtooth in Swedish: Sågtandskurva

sawtooth in Italian: Onda a dente di
sega