Characterizing conditions for non-existence, nonuniqueness for a system of equations
$begingroup$
Given a noisy, finite-length signal $x in mathbb{R}^n$, we can estimate the SNR of this signal via the $M_2 M_4$ estimator:
begin{align*}
M_2 = S + N quad M_4 = k_a S^2 + 6SN + k_w N^2, S ge 0, N ge 0
end{align*}
where $M_2 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^2$ is the second moment of the data and $M_4 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^4$ is the fourth moment of the data, $k_a > 0$ is the assumed kurtosis of the signal, and $k_w > 0$ is the assumed kurtosis of the noise; for instance, if the noise is Gaussian then $k_w = 3$. See here for more.
What is a characterization of when the solution to this system either does not exist, or is not unique, or does not satisfy the constraint that $S> 0$ and $N> 0$?
I have come up with a hodge-podge of cases, but no unifying idea. For example, if $k_a = k_w = 3$, then the system becomes
begin{align*}
M_2 = S + N, quad M_4 = 3(S+N)^2
end{align*}
which either has no solution when $M_4 ne 3M_2^2$, or has infinitely many solutions when $M_4 = 3M_2^2$.
systems-of-equations signal-processing
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
$endgroup$
add a comment |
$begingroup$
Given a noisy, finite-length signal $x in mathbb{R}^n$, we can estimate the SNR of this signal via the $M_2 M_4$ estimator:
begin{align*}
M_2 = S + N quad M_4 = k_a S^2 + 6SN + k_w N^2, S ge 0, N ge 0
end{align*}
where $M_2 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^2$ is the second moment of the data and $M_4 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^4$ is the fourth moment of the data, $k_a > 0$ is the assumed kurtosis of the signal, and $k_w > 0$ is the assumed kurtosis of the noise; for instance, if the noise is Gaussian then $k_w = 3$. See here for more.
What is a characterization of when the solution to this system either does not exist, or is not unique, or does not satisfy the constraint that $S> 0$ and $N> 0$?
I have come up with a hodge-podge of cases, but no unifying idea. For example, if $k_a = k_w = 3$, then the system becomes
begin{align*}
M_2 = S + N, quad M_4 = 3(S+N)^2
end{align*}
which either has no solution when $M_4 ne 3M_2^2$, or has infinitely many solutions when $M_4 = 3M_2^2$.
systems-of-equations signal-processing
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
$endgroup$
add a comment |
$begingroup$
Given a noisy, finite-length signal $x in mathbb{R}^n$, we can estimate the SNR of this signal via the $M_2 M_4$ estimator:
begin{align*}
M_2 = S + N quad M_4 = k_a S^2 + 6SN + k_w N^2, S ge 0, N ge 0
end{align*}
where $M_2 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^2$ is the second moment of the data and $M_4 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^4$ is the fourth moment of the data, $k_a > 0$ is the assumed kurtosis of the signal, and $k_w > 0$ is the assumed kurtosis of the noise; for instance, if the noise is Gaussian then $k_w = 3$. See here for more.
What is a characterization of when the solution to this system either does not exist, or is not unique, or does not satisfy the constraint that $S> 0$ and $N> 0$?
I have come up with a hodge-podge of cases, but no unifying idea. For example, if $k_a = k_w = 3$, then the system becomes
begin{align*}
M_2 = S + N, quad M_4 = 3(S+N)^2
end{align*}
which either has no solution when $M_4 ne 3M_2^2$, or has infinitely many solutions when $M_4 = 3M_2^2$.
systems-of-equations signal-processing
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
$endgroup$
Given a noisy, finite-length signal $x in mathbb{R}^n$, we can estimate the SNR of this signal via the $M_2 M_4$ estimator:
begin{align*}
M_2 = S + N quad M_4 = k_a S^2 + 6SN + k_w N^2, S ge 0, N ge 0
end{align*}
where $M_2 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^2$ is the second moment of the data and $M_4 = frac{1}{n}sum_{i=0}^{n-1} (x[i]-mu)^4$ is the fourth moment of the data, $k_a > 0$ is the assumed kurtosis of the signal, and $k_w > 0$ is the assumed kurtosis of the noise; for instance, if the noise is Gaussian then $k_w = 3$. See here for more.
What is a characterization of when the solution to this system either does not exist, or is not unique, or does not satisfy the constraint that $S> 0$ and $N> 0$?
I have come up with a hodge-podge of cases, but no unifying idea. For example, if $k_a = k_w = 3$, then the system becomes
begin{align*}
M_2 = S + N, quad M_4 = 3(S+N)^2
end{align*}
which either has no solution when $M_4 ne 3M_2^2$, or has infinitely many solutions when $M_4 = 3M_2^2$.
systems-of-equations signal-processing
systems-of-equations signal-processing
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
edited Dec 17 '18 at 12:23
Harry Peter
5,47911439
5,47911439
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
asked Dec 13 '18 at 18:40
user14717user14717
3,8881120
3,8881120
add a comment |
add a comment |
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