Harmonic Addition Theorem

It is always possible to write a sum of sinusoidal functions

(1)

as a single sinusoid the form

(2)

This can be done by expanding (2) using the trigonometric addition formulas to obtain

(3)

Now equate the coefficients of (1) and (3)

			(4)
			(5)

			(6)
			(7)

and

			(8)
			(9)

giving

			(10)
			(11)

Therefore,

(12)

(Nahin 1995, p. 346).

In fact, given two general sinusoidal functions with frequency ,

			(13)
			(14)

their sum can be expressed as a sinusoidal function with frequency

Now, define

			(18)
			(19)

Then (17) becomes

(20)

Square and add (◇) and (◇)

(21)

Also, divide (◇) by (◇)

(22)

(23)

where and are defined by (◇) and (◇).

This procedure can be generalized to a sum of harmonic waves, giving

			(24)
			(25)

where

			(26)
			(27)

and

(28)

See also