We want to show it is always possible to solve

for and , given for . For each component sinusoid, we can write

(A.3) |

Applying this expansion to Eq. (A.2) yields

Equating coefficients gives

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 , and (2) forming a ratio of both sides of Eq. (A.4) to eliminate . The results are

which has a unique solution for any values of and .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University