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Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous questionmy previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

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Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$$n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

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Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S$$S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

Question

Suppose I observe a vector $\mathbf{x}=[X_1 \ldots X_n]$, where each $X_i=m_i+n_i$, with $n_i$ being an independent zero-mean Gaussian random variable with variance $\sigma^2$ (i.e. $n_i\sim\mathcal{N}(0,\sigma^2)$ i.i.d.) and $\mathbf{m}=[m_1 \ldots m_n]$ is an unknown vector with a known Euclidean length to the origin $l(\mathbf{m})=\|\mathbf{m}\|_2=\sqrt{\sum_{i=1}^n m_i^2}$.

Suppose that the Euclidean length of the unknown vector $\mathbf{m}$ can equally likely be either $l_0$ or $l_1$, and, without loss of generality, $l_0<l_1$. Now, given the vector of noisy observations $\mathbf{x}$ I need to decide whether the length of $\mathbf{m}$ is $l_0$ or $l_1$. Note that I do not know (nor do I need to know for my answer) the values in $\mathbf{m}$, just the two possibilities for length.

The intuitive procedure is to compute the Euclidean length of the observed vector $S=\sqrt{\sum_{i=1}^nX_i^2}$, and then select whichever of $l_0$ or $l_1$ is closer to $S$ (based on some threshold that is related to the probabilities of error that you are willing to tolerate). Since $S^2$ is a non-central chi-square random variable, one can (numerically) obtain (at least bounds) on the probabilities of error of this approach.

However, is $S$ the "best" test statistic (in the sense of Neyman-Pearson optimality) for this problem? If it is not optimal, is a better test known? If $S$ is optimal, is there a proof of its optimality in the literature?

My prior effort

This seems like a problem that should be well-studied. In fact, it is related to the problem of non-coherent detection in communications theory, where $S^2$ is used by the square-law detector. John Proakis has a proof on pages 304-306 of "Digital Communications" (4th edition) that $S$ as defined is the N-P test statistic for when $n=2$ (detection of a complex-valued symbol with arbitrary phase offset). He essentially projects the problem into polar coordinate system, and takes the expectation of the test statistic over the uniform distribution for the angle.

I tried a naive approach of extending Proakis' proof to $n$-spherical coordinate system (since $\mathbf{m}$ can be thought of a coordinate on an $n$-sphere with radius $l_k$) and computing the likelihood function (from N-P) by taking the expectation over the uniform distribution on all the angles. However, while I can get a closed form expression $n=3$ which yields the N-P test statistic defined by $S$, the integration gets really nasty for $n>3$. Perhaps there is something more clever that one can do, maybe using the circular symmetry of joint distribution of i.i.d. Gaussians.

This question is related to my previous question.

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