I am trying to determine the amplitude of an amplitude modulated sinus as accurate as possible. My sampling frequency is sufficently high. The entire model looks as follows:
$$ A*sin(2*pi*f_1*t+p_1)*sin(2*pi*f_2*t+p_2) + O + e(t) = y(t) $$
Where $e(t)$ is the error and $y(t)$ is my measured data.
I am interested in determining $A$ as good as possible, whilest I only know $f_1$ and $y(t)$.
Using the idea from here, I can determine $A$ (and $p_1, O$) for the not modulated case. However this reduced model is of course less accurate:
$$ A*sin(2*pi*f_1*t+p_1) + O + e(t) = y(t) $$
This can be LSQ-fitted similiar to a LSQ-polynom-fit using this approach:
$$ \frac{\partial}{\partial Parameter} (A_1*sin(f_1*t)+A_2*cos(f_1*t)+O-y)^2 = 0 $$
I implemented this and it works fine (so the constant term (offset) is not a problem at all).
Now if want to use the accurate model what options do I have? Can I add another parameter to reduce the error for a local fit (for example multiply a low degree taylorpolynom to model the influence of the lower frequent sinus)? What are the requirements for a new parameter to be separable? More general under what requirements can I orthogonalize parameters (for instance by using chebyshev polynomials)?