Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Sawtooth wave

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw.

The convention is that a sawtooth wave ramps upward and then sharply drops. However, there are also sawtooth waves in which the wave ramps downward and then sharply rises. The latter type sawtooth wave is called a "reverse sawtooth wave" or "inverse sawtooth wave".

The piecewise linear function

x(t) = t - lfloor t 
floor = t - operatorname{floor}(t)

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

x(t) = 2 left( {t over a} - leftlfloor {t over a} - {1 over 2} 
ight
floor 
ight)
= 2 left( {t over a} - operatorname{floor} left( {t over a} - {1 over 2} 
ight) 
ight)

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 synthesizing musical sounds, particularly bowed string instruments like violins and cellos, using subtractive synthesis.

A sawtooth can be constructed using additive synthesis. The infinite Fourier series

x_mathrm{sawtooth}(t) = frac {2}{pi}sum_{k=1}^{infin} {(-1)}^{k} frac {sin (2pi kft)}{k}

converges to an inverse sawtooth wave. A conventional sawtooth can be constructed using

x_mathrm{sawtooth}(t) = frac {2}{pi}sum_{k=1}^{infin} {(-1)}^{k+1} frac {sin (2pi kft)}{k}

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.

Animation of the additive synthesis of a sawtooth wave with an increasing number of harmonics

An audio demonstration of a sawtooth played at 440 Hz (A4) and 880 Hz (A5) and 1760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.

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