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. The convention is that a sawtooth wave ramps upward and then sharply drops[citation needed]. 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.[1] The piecewise linear function x ( t ) = t − ⌊ t ⌋ ⏟ floor ( t ) displaystyle x(t)=t-underbrace lfloor trfloor _ operatorname 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 2 ( t a − ⌊ 1 2 + t a ⌋ ) displaystyle 2left( frac t a -leftlfloor frac 1 2 + frac t a rightrfloor 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 π arctan ( cot ( x π 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 slip-stick behavior of the bow drives the strings with a sawtooth-like motion.[2] Additive Sawtooth Demo 220Hz
Problems playing this file? See media help. A sawtooth can be constructed using additive synthesis. The infinite Fourier series x reverse sawtooth ( t ) = 2 A π ∑ k = 1 ∞ ( − 1 ) k sin ( 2 π k f t ) k displaystyle x_ text reverse sawtooth (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 π ∑ k = 1 ∞ ( − 1 ) k sin ( 2 π 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.
In digital synthesis, these series are only summed over k such that
the highest harmonic, Nmax, is less than the
Animation of the additive synthesis of a sawtooth wave with an increasing number of harmonics An audio demonstration of a sawtooth played at
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[edit] 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. Sawtooth waves are used in 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. 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, 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, 15.625 kHz for
See also[edit] Sine, square, triangle, and sawtooth waveforms List of periodic functions Sine wave Sound Square wave Triangle wave Wave Zigzag References[edit] ^ "Fourier Series-Triangle
External links[edit] Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. pp. 536–537. ISBN 0-521-84903-9. v t e Waveforms Sine wave Non-sinusoidal Rectangular wave Sawtooth wave Square |