Fourier Synthesis

Square Wave

Here is the sequence of adding together sinusoidal waves to make a square wave.

We start with the fundamental with unit amplitude.

We add to this the third harmonic ( taking the definition of the first harmonic to be the same as the fundamental-- a practice endorsed by the text but at variance with most music terminology) with 1/3 amplitude and with a phase shift of 180 degrees.

to get this. Note that this addition has flattened out the tops and bottoms and increased the slope as the wave passes through zero.

We now add some of the fifth harmonic with amplitude 1/5

to get this. A closer approximation to the square wave. The top is flatter, and the sides are more nearly vertical.

Adding the seventh harmonic, again with an amplitude of 1/7 and a phase of 180degrees.

We get an even better approximation to a square wave. We can continue this process with higher and higher harmonics coming closer and closer to a square wave. Adding the first thirty harmonics ( the odd ones each with amplitude of one over the harmonic number and each second odd harmonic with a phase shift of 180 degrees) we get

a much closer approximation to the square wave. The "ringing" ( the wiggles near the vertical parts of the square wave) are a consequence of trying to approximate a vertical line ( a "discontinuity") with a sum of sinusoidal waves.

Sawtooth

WE can do the same thing with the sawtooth wave. In this case all of the harmonics are non-zero, the amplitudes are one over the harmonic number and the phases are all 90 degrees (ie all start at zero going down>

Adding the seventh harmonic, again with an amplitude of 1/7 and a phase of 180degrees.

We get an even better approximation to a square wave. We can continue this process with higher and higher harmonics coming closer and closer to a square wave. Adding the first thirty harmonics ( the odd ones each with amplitude of one over the harmonic number and each second odd harmonic with a phase shift of 180 degrees) we get

a much closer approximation to the square wave. The "ringing" ( the wiggles near the vertical parts of the square wave) are a consequence of trying to approximate a vertical line ( a "discontinuity") with a sum of sinusoidal waves.

Sawtooth

WE can do the same thing with the sawtooth wave. In this case all of the harmonics are non-zero, the amplitudes are one over the harmonic number and the phases are all 90 degrees (ie all start at zero going down>

Adding the first 5 components gives us

which is a reasonable approximation to a sawtooth wave. Adding even more harmonics, again up to the thirtieth gives us

Again, because of the vertical line, we get ringing in the wave near that vertical line.

"Oboe"

Here is the waveform of a sound which sounds something like an oboe. Below the waveform, on the sliders are the amplitudes of the various harmonics which make up this waveform. Note that there is only a small amount of the fundamental, a lot of the second and third harmonics, less of the fourth and fifth and then nothing up to the 11,12,and 13th.

Voice

Here is an example of the synthesis of a voiced vowel sound -- something like me saying "oooh".

Note the strong third harmonic, and the secondary peak around the seventh harmonic in the amplitude of the various components. These "peaks" in the amplitude are called formants in the voice and arise due to resonances of the air tract between the larynx and the mouth.

These images were produced using the program FourierSynth, written by W. Unruh.