# Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

## Archive for April, 2010

In [1, p. 34 exercise 2.13] we are asked to consider the problem of fitting the data $\left\{x[0],...,x[N-1]\right\}$ by the sum of a dc signal and a sinusoid as: $\hat{x}=\mu+A_c e^{j2\pi f_0n} \; n=0,1,...,N-1$. The complex dc level $\mu$ and the sinusoidal amplitude $A_c$ are unknown and we are notified that we may view the determination of $\mu$, $A_c$ as the solution of the overdetermined set of equations.
 $\left[\begin{array}{cc}1 & 1 \\ 1 & e^{j2\pi f_{0}} \\ \vdots & \vdots \\ 1 & e^{j2\pi f_{0}(N-1)}\end{array} \right] \cdot \left[\begin{array}{c}\mu \\ A_{c} \end{array} \right] = \left[\begin{array}{c}x[0] \\ \vdots \\ x[N-1] \end{array} \right].$ (1)

It is asked to find the least squares solution for $\mu$ and $A_c$. If furthermore $f_0=k/N$, where $k$ is a nonzero integer in the range $[-N/2,N/2-1]$ for $N$ even and $[-(N-1)/2,(N-1)/2]$ for $N$ odd we are asked to determine again the least squares solution. read the conclusion >
In [1, p. 34 exercise 2.12] we are asked to verify the equations given for the Cholesky decomposition, [1, (2.53)-(2.55)]. Furthermore it is requested to use these equations to find the inverse of the matrix given in problem [1, 2.7]. read the conclusion >