The SAWTOOTH WAVE (or SAW WAVE) is a kind of nonsinusoidal waveform . It is so named based on its resemblance to the teeth of a plaintoothed saw with a zero rake angle . The convention is that a sawtooth wave ramps upward and then sharply drops. However, 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 . The piecewise linear function x ( t ) = t t = t floor ( t ) {displaystyle x(t)=tlfloor trfloor =toperatorname {floor} (t)} or x ( t ) = t ( mod 1 ) {displaystyle x(t)=t{pmod {1}}} 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 ( t a 1 2 + t a ) {displaystyle x(t)=2left({t over a}leftlfloor {1 over 2}+{t over a}rightrfloor right)} = 2 ( t a floor ( 1 2 + t a ) ) {displaystyle =2left({t over a}operatorname {floor} left({1 over 2}+{t over a}right)right)} This sawtooth function has the same phase as the sine function. Another function in trigonometric terms with period p and amplitude a: y ( x ) = 2 a arctan ( cot ( x p ) ) {displaystyle y(x)={frac {2a}{pi }}arctan left(cot left({frac {xpi }{p}}right)right)} While a 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 harmonics 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 slipstick behavior of the bow drives the strings with a sawtoothlike motion. Additive Sawtooth Demo 220Hz
Sawtooth wave
Problems playing this file? See media help . A sawtooth can be constructed using additive synthesis . The infinite Fourier series x r e v e r s e s a w t o o t h ( t ) = 2 A k = 1 ( 1 ) k sin ( 2 k f t ) k {displaystyle x_{mathrm {reversesawtooth} }(t)={frac {2A}{pi }}sum _{k=1}^{infty }{(1)}^{k}{frac {sin(2pi kft)}{k}}} converges to a reverse (inverse) sawtooth wave. A conventional sawtooth can be constructed using x s a w t o o t h ( t ) = A 2 A k = 1 ( 1 ) k sin ( 2 k f t ) k {displaystyle x_{mathrm {sawtooth} }(t)={frac {A}{2}}{frac {A}{pi }}sum _{k=1}^{infty }{(1)}^{k}{frac {sin(2pi kft)}{k}}} where A is amplitude. Note: cot y = tan x 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 nonbandlimited form, such as y = x  floor (x), infinite harmonics are sampled and the resulting tone contains aliasing distortion. 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. Both bandlimited (nonaliased) and aliased tones are presented. Sawtooth aliasing demo Sawtooth waves played bandlimited and aliased at 440 Hz, 880 Hz, and 1760 Hz  Problems playing this file? See media help . CONTENTS * 1 Applications * 2 See also * 3 References * 4 External links APPLICATIONS * Sawtooth waves are known for their use in music. The sawtooth and square waves are among the most common waveforms 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 screens. 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 by various
means (transformers, capacitors, centertapped windings) so that the
halfway 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 centermounted
deflection yoke can use the whole screen area to depict a trace.
Frequency is 15.734 kHz on
NTSC
SEE ALSO Sine , square , triangle , and sawtooth waveforms *
List of periodic functions
*
Sine wave
*
Sound
REFERENCES * ^ "Fourier SeriesTriangle
Wave
