I am struggling to understand the theory of parametric methods for power spectral estimations of line spectra.
And I want to find the solution of nonlinear least square (NLS) method for estimating the frequencies of line spectra. For example,
$Let,$ $$f(\omega, \alpha, \phi) = \sum_{t=1}^{N} |y(t) - \sum_{k=1}^{n} \alpha_{k} e^{i(\omega_{k}t+\phi_{k})}|^2,$$
$then,$
$$f = (Y-B\beta)^*(Y-B\beta) \\= [\beta-(B^*B)^{-1}B^*Y]^*[B^*B][\beta (B^*B)^{-1}B^*Y]+Y^*Y-Y^*B(B^*B)^{-1}B^*Y $$
$where$ $$Y = [y(1) ~y(2) ~... ~y(N)]^T$$ $$ \beta = [\beta_{1}~...~\beta_{n}]^T$$ $$B =\begin{bmatrix} e^{i\omega_{1}} & ... & e^{i\omega_{n}} \\ \vdots & & \vdots \\ e^{iN\omega_{1}} & ... & e^{iN\omega_{n}} \\ \end{bmatrix}.$$
The best choice of the $\omega = [\omega_{1}, \cdots, \omega_{n}]^T$ to minimize $f$ is,
$$\hat \omega = \underset{\omega}{\operatorname{argmax}} [Y^*B(B^*B)^{-1}B^*Y]$$
I have two questions for the above equations.
1)How can we derive $$f = [\beta-(B^*B)^{-1}B^*Y]^*[B^*B][\beta (B^*B)^{-1}B^*Y]+Y^*Y-Y^*B(B^*B)^{-1}B^*Y ?$$ 2)How can we solve the $\operatorname{argmax}$ problem for $n$ different parameters of $\omega$?
