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kpc2
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"solution to an eigenvalue problem" ?
I am trying to reproduce the results from a paper. The essence of the paper is hidden in just one equation (eq. 11) and some lines of text. For me this is going somewhat too fast.
Below are the essential parts of the paper, describing the problem (and solution?)
It may be something trivial, but I cannot get how to solve the equations. According to the text it is some kind of eigenvalue problem. I can follow the text up to the point of "Not only ...", from there I am lost.
I am aiming for a way to solve this problem numericly.
I hope someone can help.
Given are:
Yi: generator admittance (complex)
Ti: noise temperature
Wi: Output power spectrum
Pi: Output power
k: Boltzmann constant
Unknowns:
Wu: voltage noise power spectrum
Wi: current noise power spectrum
Wui: Wu/Wi correlation (complex)
Yin: input admittance (complex)
http://ieeexplore.ieee.org//xpls/abs_all.jsp?arnumber=4139081
Stolle, Reinhard; Schiek, Burkhard; , "The Complete and Accurate Determination of Two-Port Noise Parameters without Seperate Measurements of the Two-Port Input Impedance," Microwave Conference, 1998. 28th European , vol.1, no., pp.241-246, Oct. 1998[cut]
Each of these methods [cut] requires the knowledge of the input admittance Yin
of the device under test (DUT). This paper presents a method to overcome this problem. [cut] The input admittance Yin of the device under test is found as the solution to an eigenvalue problem. The determinant of a matrix which is composed of the noise measurement values and of the input admittance Yin can be set to zero and solved for Yin.
[cut]
According to linear circuit theory, the output power spectrum W1 [cut] is given by
Wi = mu / |Yin+Yi|^2 * (
4 k Ti Re{Yi}
+ Wu |Yi|^2
+ Wi
+ Re{Wui} 2 Re{Yi}
+ Im{Wui} 2 Im{Yi}
) . . . (3)
where mu denotes a real conversion factor and is assumed to be a constant and k is Boltzmann's
constant. The elimination of mu is achieved with the introduction of
wij = Pi/Pj |Yin+Yi|^2 / |Yin+Yj|^2 . . . (4)
and yields
4k(wij Tj Re{Yj} - Ti Re{Yi}) =
Wu (|Yi|^2 - wij |Yj|^2)
+ Wi (1 - wij)
+ Re{Wui} 2*(Re{Yi} - wij Re{Yj})
+ Im{Wui} 2*(Im{Yi} - wij Im{Yj}) . . . (5)
which constitutes a linear equation in the four unknowns Wu, Wi, Re{Wui} and Im{Wui}.
Hence, there are five admittances Yi needed to make up a system of four equations to solve for the four unknowns:
A * [Wu, Wi, Re{Wui}, Im{Wui}]^T = b . . . (6)
where the 4 x 4-matrix A and the 4 x 1-vector b consist of the measurement data Wi, Yi for
i = 1, ..., 5 and of Yin.
[cut]
At least one of the five source temperatures Ti corresponding to each Yi has to be different
from the others, otherwise the 4 equations will be dependent. This can be seen from the fact, that if Ti = To for every i, then the left side of Eq. (5) becomes proportional to the coefficient of Re{Wuj} on the right side. In this case the right-hand vector b of the corresponding matrix equation (6) becomes proportional to the third column of A, such that the matrix equation (6) can be transformed into a homogeneous matrix equation with the same solution as (6). As this equation is homogeneous, it is not solvable and its determinant becomes zero.
Not only does this indicate the necessity of at least one measurement at an excess temperature, but it implies a way to determine the input admittance Yin of the device under test with the same noise measurements that are needed to solve for the four noise parameters. For Yi, unknown, four equations of the type (5) yield a matrix A, the coefficients of which are parameterized by Yin. If Ti = To for every i, its determinant det(A(Yin)) is zero. This determinant is a surface in the Yin plane which cuts the zero plane. By means of one more equation of the type (5) one obtains another solution matrix B which cuts the zero plane as well. The intersection point of both cuts is Yin and is the solution to
det(A(Yin))^2 + det(B(Yin))^2 = 0. . . . (11)
For numerical benefits the matrices A and B should be normalized to their norms.
Finally, at least seven measurements of noise power have to be performed, six of them at
the temperature To and one of them at a different temperature, in order to obtain the noise
parameters without separate measurements of the input impedance.
[cut]
I am trying to reproduce the results from a paper. The essence of the paper is hidden in just one equation (eq. 11) and some lines of text. For me this is going somewhat too fast.
Below are the essential parts of the paper, describing the problem (and solution?)
It may be something trivial, but I cannot get how to solve the equations. According to the text it is some kind of eigenvalue problem. I can follow the text up to the point of "Not only ...", from there I am lost.
I am aiming for a way to solve this problem numericly.
I hope someone can help.
Given are:
Yi: generator admittance (complex)
Ti: noise temperature
Wi: Output power spectrum
Pi: Output power
k: Boltzmann constant
Unknowns:
Wu: voltage noise power spectrum
Wi: current noise power spectrum
Wui: Wu/Wi correlation (complex)
Yin: input admittance (complex)
http://ieeexplore.ieee.org//xpls/abs_all.jsp?arnumber=4139081
Stolle, Reinhard; Schiek, Burkhard; , "The Complete and Accurate Determination of Two-Port Noise Parameters without Seperate Measurements of the Two-Port Input Impedance," Microwave Conference, 1998. 28th European , vol.1, no., pp.241-246, Oct. 1998[cut]
Each of these methods [cut] requires the knowledge of the input admittance Yin
of the device under test (DUT). This paper presents a method to overcome this problem. [cut] The input admittance Yin of the device under test is found as the solution to an eigenvalue problem. The determinant of a matrix which is composed of the noise measurement values and of the input admittance Yin can be set to zero and solved for Yin.
[cut]
According to linear circuit theory, the output power spectrum W1 [cut] is given by
Wi = mu / |Yin+Yi|^2 * (
4 k Ti Re{Yi}
+ Wu |Yi|^2
+ Wi
+ Re{Wui} 2 Re{Yi}
+ Im{Wui} 2 Im{Yi}
) . . . (3)
where mu denotes a real conversion factor and is assumed to be a constant and k is Boltzmann's
constant. The elimination of mu is achieved with the introduction of
wij = Pi/Pj |Yin+Yi|^2 / |Yin+Yj|^2 . . . (4)
and yields
4k(wij Tj Re{Yj} - Ti Re{Yi}) =
Wu (|Yi|^2 - wij |Yj|^2)
+ Wi (1 - wij)
+ Re{Wui} 2*(Re{Yi} - wij Re{Yj})
+ Im{Wui} 2*(Im{Yi} - wij Im{Yj}) . . . (5)
which constitutes a linear equation in the four unknowns Wu, Wi, Re{Wui} and Im{Wui}.
Hence, there are five admittances Yi needed to make up a system of four equations to solve for the four unknowns:
A * [Wu, Wi, Re{Wui}, Im{Wui}]^T = b . . . (6)
where the 4 x 4-matrix A and the 4 x 1-vector b consist of the measurement data Wi, Yi for
i = 1, ..., 5 and of Yin.
[cut]
At least one of the five source temperatures Ti corresponding to each Yi has to be different
from the others, otherwise the 4 equations will be dependent. This can be seen from the fact, that if Ti = To for every i, then the left side of Eq. (5) becomes proportional to the coefficient of Re{Wuj} on the right side. In this case the right-hand vector b of the corresponding matrix equation (6) becomes proportional to the third column of A, such that the matrix equation (6) can be transformed into a homogeneous matrix equation with the same solution as (6). As this equation is homogeneous, it is not solvable and its determinant becomes zero.
Not only does this indicate the necessity of at least one measurement at an excess temperature, but it implies a way to determine the input admittance Yin of the device under test with the same noise measurements that are needed to solve for the four noise parameters. For Yi, unknown, four equations of the type (5) yield a matrix A, the coefficients of which are parameterized by Yin. If Ti = To for every i, its determinant det(A(Yin)) is zero. This determinant is a surface in the Yin plane which cuts the zero plane. By means of one more equation of the type (5) one obtains another solution matrix B which cuts the zero plane as well. The intersection point of both cuts is Yin and is the solution to
det(A(Yin))^2 + det(B(Yin))^2 = 0. . . . (11)
For numerical benefits the matrices A and B should be normalized to their norms.
Finally, at least seven measurements of noise power have to be performed, six of them at
the temperature To and one of them at a different temperature, in order to obtain the noise
parameters without separate measurements of the input impedance.
[cut]
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