Fourier Synthesizer

Fourier's Theorem: Any periodic wave with frequency f can be synthesized with sine waves having frequencies f, 2f, 3f, and so on (the harmonics or partials) by choosing unique amplitudes and phases for the harmonics.

Amplitude	A₁	A₂	A₃	A₄	A₅	A₆	A₇	A₈	A₉	A₁₀	A₁₁	A₁₂	A₁₃	A₁₄	A₁₅	A₁₆
Sine	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Triangle	1	0	1/9	0	1/25	0	1/49	0	1/81	0	1/121	0	1/169	0	1/225	0
Square	1	0	1/3	0	1/5	0	1/7	0	1/9	0	1/11	0	1/13	0	1/15	0
Ramp	1	1/2	1/3	1/4	1/5	1/6	1/7	1/8	1/9	1/10	1/11	1/12	1/13	1/14	1/15	1/16
Pulse Train	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Ramp is often called Sawtooth

Phase	ϕ₁	ϕ₂	ϕ₃	ϕ₄	ϕ₅	ϕ₆	ϕ₇	ϕ₈	ϕ₉	ϕ₁₀	ϕ₁₁	ϕ₁₂	ϕ₁₃	ϕ₁₄	ϕ₁₅	ϕ₁₆
Sine	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Triangle	0	0	180	0	0	0	180	0	0	0	180	0	0	0	180	0
Square	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Ramp	180	180	180	180	180	180	180	180	180	180	180	180	180	180	180	180
Pulse Train	90	90	90	90	90	90	90	90	90	90	90	90	90	90	90	90

The general formula is given below.

Using a trig identity,

the above equation can be expressed in terms of cosines and sine.

With the following assignments,

the familiar equation with cosines and sines is found.