Lysario – by Panagiotis Chatzichrisafis
"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"
Archive for the ‘Solved Problems’ Category
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 35 exercise 2.14
Author: Panagiotis28 Mai
In [1, p. 35 exercise 2.14] we are asked to consider the solution to the set of linear equations
where
is a 
hermitian matrix given by:
and
is a complex 
vector given by
The complex vectors are defined in [1, p.22, (2.27)]. Furthermore we are asked to assumed that
, 
for 
are distinct integers in the range ![[-n/2,n/2-1]](https://lysario.de/wp-content/cache/tex_24916734a0a5230271c84429761581eb.png)
for 
even and ![[-(n-1)/2,(n-1)/2]](https://lysario.de/wp-content/cache/tex_b9ee4eee41d27a8e5eea256f738e00e2.png)
for odd. 
is defined to be a 
vector.
It is requested to show that is a singular matrix (assuming 
) and that there are infinite number of solutions.
A further task is to find the general solution and also the minimum norm solution of the set of linear equations. The hint provided by the exercise is to note that 
are eigenvectors of with nonzero eigenvalues and then to assume a solution of the form
where
for 
and solve for 
.
read the conclusion >
 | (1) | ||
where
is a hermitian matrix given by:
 | (2) | ||
and
is a complex vector given by
 | (3) | ||
The complex vectors are defined in [1, p.22, (2.27)]. Furthermore we are asked to assumed that
, for are distinct integers in the range for even and for odd. is defined to be a vector.
It is requested to show that is a singular matrix (assuming ) and that there are infinite number of solutions.
A further task is to find the general solution and also the minimum norm solution of the set of linear equations. The hint provided by the exercise is to note that are eigenvectors of with nonzero eigenvalues and then to assume a solution of the form
 | (4) | ||
where
for and solve for .
read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.13
Author: Panagiotis27 Apr
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\}](https://lysario.de/wp-content/cache/tex_9ceb2dd63d7268e57f2b85d862426d56.png)
by the sum of a dc signal and a sinusoid as:
. The complex dc level 
and the sinusoidal amplitude 
are unknown and we are notified that we may view the determination of , as the solution of the overdetermined set of equations.
It is asked to find the least squares solution for and . If furthermore 
, where 
is a nonzero integer in the range ![[-N/2,N/2-1]](https://lysario.de/wp-content/cache/tex_3a5bf53d3e9902f7fffe9d68b2de1428.png)
for 
even and ![[-(N-1)/2,(N-1)/2]](https://lysario.de/wp-content/cache/tex_dde8f99f860237fa0b573d2ad067620a.png)
for odd we are asked to determine again the least squares solution.
read the conclusion >
by the sum of a dc signal and a sinusoid as:
. The complex dc level and the sinusoidal amplitude are unknown and we are notified that we may view the determination of , 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].](https://lysario.de/wp-content/cache/tex_bf7d1a9edfb68f8a07beb5ca2929cbc6.png) | (1) | ||
It is asked to find the least squares solution for
and . If furthermore , where is a nonzero integer in the range for even and for odd we are asked to determine again the least squares solution.
read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.12
Author: Panagiotis1 Apr
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].
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Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.11
Author: Panagiotis24 Jan
In [1, p. 34 exercise 2.11] we are asked to find the eigenvalues of the circulant matrix given in [1, (2.27),p.22].
read the conclusion >
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.10
Author: Panagiotis7 Jan
In [1, p. 34 exercise 2.10] it is asked to prove that if 
is a complex positive definite matrix and 
is a full rank complex 
matrix with 
, then 
is also positive definite.
read the conclusion >
is a complex positive definite matrix and is a full rank complex matrix with , then is also positive definite.
read the conclusion >