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 . This gives

for any values of
and
. Since
, we have
. To impose
and
, a
four-quadrant arctangent
must be used, normally
written `atan2(y,x)` in computer languages.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

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