Re: Sine Wave Curve Fit Question

Helmut Jarausch wrote:
Your model is A*sin(omega*t+alpha) where A and alpha are sought.
Let T=(t_1,...,t_N)' and Y=(y_1,..,y_N)' your measurements (t_i,y_i)
( ' denotes transposition )

First, A*sin(omega*t+alpha) =
A*cos(alpha)*sin(omega*t) + A*sin(alpha)*cos(omega*t) =
B*sin(omega*t) + D*cos(omega*t)

by setting B=A*cos(alpha) and D=A*sin(alpha)

Once, you have B and D, tan(alpha)= D/B A=sqrt(B^2+D^2)

This is all very true, but the equation tan(alpha)=D/B may fool you. This may lead you to believe that alpha=arctan(D/B) is a solution, which is not always the case. The point (B,D) may be in any of the four quadrants of the plane. Assuming B!=0, the solutions to this equation fall into the two classes

alpha = arctan(D/B) + 2*k*pi

and

alpha = arctan(D/B) + (2*k+1)*pi,

where k is an integer. The sign of B tells you which class gives you the solution. If B is positive, the solutions are those in the first class. If B is negative, the solutions are instead those in the second class. Whithin the correct class, you may of course choose any alternative.

Then we have the case B=0. Then the sign of D determines alpha. If D is positive, we have alpha=pi/2, and if D is negative, we have alpha=-pi/2.

Last if both B and D are zero, any alpha will do.

/MiO
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