Sawtooth wave
The sawtooth wave is a kind of basic non-sinusoidal
waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a
saw. A sawtooth wave can be created by an instrument such as an
oboe.
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.
Encyclopedia
The
sawtooth wave is a kind of basic non-sinusoidal
waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a
saw. A sawtooth wave can be created by an instrument such as an
oboe.
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.
The
piecewise linear function
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
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
-
converges to a sawtooth wave. In digital synthesis, the series is only summed over
k such that the highest harmonic, N
max, is less than the Nyquist frequency . This summation can generally be more efficiently calculated using the 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 1760 Hz is available below. Both bandlimited and aliased tones are presented.
Applications
- 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 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 15.75kHz on NTSC, 16.75kHz for PAL and SECAM)
- The vertical deflection system operates the same way as the horizontal, though at a much lower frequency .
- 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 horizonal linearity though they have generally disappeared due to the greater temporal stability of modern electronic components.
See also
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