I have a set of data points I want to approximate with the function
$ Q(w) = \frac{\sum_{j=0}^n a_jw^j}{1+\sum_{j=1}^nb_jw^j} $
for $n = 2, 4$, and I'm trying to set up a set of normal equations for the system.
Lets say you have a series of $n$ measurements $(t_i, y_i)$ that you want to approximate with a quadratic polynomial $y_i = x_0 + x_1t_i + x_2t_i^2$. The usual way to solve this would be to set up the equations
$ A = \left[ \begin{array}{ccc} 1 & t_1 & t_1^2 \newline 1 & t_2 & t_2^2 \newline & ... & \newline 1 & t_n & t_n^2 \end{array}\right], \ \ \ \hat{\mathbf{x}} = \left[ \begin{array}{c} x_0 \newline x_1 \newline x_2 \end{array}\right] \ \ \ \text{and} \ \ \ \mathbf{b} = \left[ \begin{array}{c} y_i \newline y_2 \newline ... \newline y_n \end{array}\right], $
and then solve $A^TA\hat{\mathbf{x}} = A^T\mathbf{b}$ for the least squares solution $\hat{\mathbf{x}}$ to the system system $A\mathbf{x} = \mathbf{b}$.
How would one go about finding a least squares solution with the form $Q(w)$ given above; ie, how do you set up $A$ and $\hat{\mathbf{x}}$?
Cheers!
