The sawtooth 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. A single sawtooth, or an intermittently triggered sawtooth, is called a ramp waveform. The convention is that a sawtooth wave ramps upward and then sharply drops. In a reverse 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 , or equivalently based on the floorfunction of timet 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 This sawtooth function has the same phase as the sine function. Another function in trigonometric terms with period p and amplitude a: 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. A sawtooth can be constructed using additive synthesis. The infinite Fourier series converges to a reverse sawtooth wave. A conventional sawtooth can be constructed using where A is amplitude. In digital synthesis, these series are only summed over k such that the highest harmonic, Nmax, is less than the Nyquist 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, infinite harmonics are sampled and the resulting tone contains aliasing distortion. An audio demonstration of a sawtooth played at 440 Hz and 880 Hz and 1,760 Hz is available below. Both bandlimited and aliased tones are presented.
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 horizontaldeflection 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 "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.
* The vertical deflection system operates the same way as the horizontal, though at a much lower frequency.
* The ramp portion of the wave must appear as a straight line. If otherwise, it indicates that the voltage 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.
