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Concept
Sawtooth wave
Summary
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. 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 (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 equivalent piecewise linear functions
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 p, is
This sawtooth function has the same phase as the sine function.
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. For period p and amplitude a, the following infinite Fourier series converge to a sawtooth and a reverse (inverse) sawtooth wave:
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 non-bandlimited form, such as y = x − floor(x), infinite harmonics are sampled and the resulting tone contains aliasing distortion.
An audio demonstration of a sawtooth played at 440 Hz (A4) and 880 Hz (A5) and 1,760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.
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Fourier series
A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.
Square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions between minimum and maximum are instantaneous. The square wave is a special case of a pulse wave which allows arbitrary durations at minimum and maximum amplitudes. The ratio of the high period to the total period of a pulse wave is called the duty cycle.
Waveform
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time. Periodic waveforms are those that vary periodically – they repeat regularly at consistent intervals. In electronics, the term is usually applied to periodically varying voltages, currents, or electromagnetic fields. In acoustics, it is usually applied to steady periodic sounds — variations of pressure in air or other media.