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 >

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