Thesawtooth wave (orsaw wave) is a kind ofnon-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothedsaw with a zerorake angle. A single sawtooth, or an intermittently triggered sawtooth, is called aramp waveform.
The convention is that a sawtooth wave ramps upward and then sharply drops. 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 asymmetrictriangle wave.[2]
A more general form, in the range −1 to 1, and with periodp, is
This sawtooth function has the samephase as thesine function.
While asquare wave is constructed from only odd harmonics, a sawtooth wave's sound is harsh and clear and its spectrum contains both even and oddharmonics of thefundamental frequency. Because it contains all the integer harmonics, it is one of the best waveforms to use forsubtractive synthesis of musical sounds, particularly bowed string instruments like violins and cellos, since theslip-stick behavior of the bow drives the strings with a sawtooth-like motion.[3]
A sawtooth can be constructed usingadditive synthesis. For periodp and amplitudea, the following infiniteFourier series converge to a sawtooth and a reverse (inverse) sawtooth wave:
Indigital synthesis, these series are only summed overk such that the highest harmonic,Nmax, is less than theNyquist frequency (half thesampling frequency). This summation can generally be more efficiently calculated with afast Fourier transform. If the waveform is digitally created directly in the time domain using a non-bandlimited form, such asy = x − floor(x), infinite harmonics are sampled and the resulting tone containsaliasing distortion.
Animation of the additive synthesis of a sawtooth wave with an increasing number of harmonics
An audio demonstration of a sawtooth played at440 Hz (A4) and 880 Hz (A5) and 1,760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.
Sawtooth waves are known for their use inelectronic music. The sawtooth and square waves are among the most common waveforms used to create sounds with subtractiveanalog andvirtual analog music synthesizers.
Sawtooth waves are used inswitched-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 thecomparator.
In the field of computer science, particularly in automation and robotics, allows to calculate sums and differences of angles while avoiding discontinuities at 360° and 0°.[citation needed]
The sawtooth wave is the form of the vertical and horizontaldeflection signals used to generate araster onCRT-based television or monitor screens.Oscilloscopes also use a sawtooth wave for their horizontal deflection, though they typically useelectrostatic deflection.
On the wave's "ramp", the magnetic field produced by the deflection yoke drags theelectron beam across the face of the CRT, creating ascan 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 current 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 current will cause deflection in one direction, and a positive current deflection in the other; thus, a center-mounted deflection yoke can use the whole screen area to depict a trace. The horizontal frequency is 15.734 kHz onNTSC, 15.625 kHz forPAL andSECAM.
The vertical deflection system operates the same way as the horizontal, though at a much lower frequency (59.94 Hz onNTSC, 50 Hz for PAL and SECAM).
The ramp portion of the wave must appear as a straight line. If otherwise, it indicates that the current 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.
