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Introduction to Digital Filters
    Background Fundamentals
       A Sum of Sinusoids at the Same Frequency is Another Sinusoid at that Frequency
          Proof Using Trigonometry

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Proof Using Trigonometry

We want to show it is always possible to solve

$\displaystyle A\cos(\omega t + \phi) = A_1\cos(\omega t + \phi_1) + A_2\cos(\omega t + \phi_2) + \cdots + A_N\cos(\omega t + \phi_N) \protect$

(A.2)

for

and $\phi$ , given $A_i, \phi_i$ for $i=1,\ldots,N$ . For each component sinusoid, we can write

$\displaystyle A_i\cos(\omega t + \phi_i)$	$\displaystyle =$	$\displaystyle A_i\cos(\omega t)\cos(\phi_i) - A_i\sin(\omega t)\sin(\phi_i)$
	$\displaystyle =$	$\displaystyle \left[A_i\cos(\phi_i)\right]\cos(\omega t) - \left[A_i\sin(\phi_i)\right]\sin(\omega t)$	(A.3)

Applying this expansion to Eq. $\,$ (A.2) yields

$\begin{eqnarray*} \left[A\cos(\phi)\right]\cos(\omega t) &-&\left[A\sin(\phi)\ri... ...a t) - \left[\sum_{i=1}^N A_i\sin(\phi_i)\right]\sin(\omega t). \end{eqnarray*}$

Equating coefficients gives

$\displaystyle A\cos(\phi)$	$\displaystyle =$	$\displaystyle \sum_{i=1}^N A_i\cos(\phi_i) \isdefs x$
$\displaystyle A\sin(\phi)$	$\displaystyle =$	$\displaystyle \sum_{i=1}^N A_i\sin(\phi_i) \isdefs y. \protect$	(A.4)

where

and

are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate $\phi$ , and (2) forming a ratio of both sides of Eq. $\,$ (A.4) to eliminate

. The results are

$\begin{eqnarray*} A &=& \sqrt{x^2+y^2}\\ \phi &=& \tan^{-1}\left(\frac{y}{x}\right) \end{eqnarray*}$

which has a unique solution for any values of and $\phi_i$ .

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Previous: A Sum of Sinusoids at the Same Frequency is Another Sinusoid at that Frequency
Next: Proof Using Complex Variables

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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