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Problem statement: The Matlab file mdt 24 . mat contains $ N=1000 $ realizations of a random vector $ \mathbf{Y} $. Each realization (each vector $ \mathbf{y} $ ) has $ M=16 $ elements corresponding to multiple observations. Your task is to design the detector, and make $ N=1000 $ decisions, one for each realization $ \mathbf{y} $. A decision is either 1 , corresponding to "signal present," or 0 , corresponding to "signal absent." The detection problem is specified as follows: \[ \begin{array}{l} H_{0}: \mathbf{Y}=\mathbf{Z} \\ H_{1}: \mathbf{Y}=A \mathbf{s}+\mathbf{Z} \end{array} \] The noise is circularly symmetric complex Gaussian, $ \mathbf{Z} \sim \mathcal{C N}\left(\mathbf{0}, \sigma_{Z_{1}}^{2} \mathbf{I}\right) $. In other words, each element of $ \mathbf{Z} $ consists of independent zero-mean Gaussian real/imaginary parts of variance $ \sigma_{Z_{1}}^{2} / 2 $. The elements (different noise observations) are uncorrelated, as evident from the diagonal structure of the covariance matrix $ \mathbf{C}_{\mathbf{Z}}=E\left\{\mathbf{Z Z}^{\prime}\right\}=\sigma_{Z_{1}}^{2} \mathbf{I} $ (prime denotes conjugate transpose). The amplitude $ A $ is real-valued, positive and constant, and the vector $ \mathbf{s} $ is defined in terms of a phase $ \phi $ as \[ \mathbf{s}=\left[\begin{array}{c} 1 \\ e^{-j \phi} \\ e^{-j 2 \phi} \\ \vdots \\ e^{-j(M-1) \phi} \end{array}\right] \] This type of problem is found in radar/sonar, or any type of sounding or probing, when an array of $ M $ receivers is used to detect presence/absence of a signal. If the array receivers are separated by a distance $ d $, and the signal of wavelength $ \lambda $ arrives to the array from direction $ \theta $, the phase $ \phi $ is given by \[ \phi=2 \pi \frac{d}{\lambda} \sin \theta \] In the present problem, the direction of arrival is $ \theta=20^{\circ} $, and $ d=\frac{\lambda}{2} $. The variance of the noise is $ \sigma_{Z_{1}}^{2}=1 $. 1 Specific tasks: 1) Specify the relevant probability density functions $ f_{0}(\mathbf{y}) $ and $ f_{1}(\mathbf{y}) $. 2) State the likelihood ratio test (LRT). 3) Identify the sufficient statistic and set the decision threshold to ensure probability of false alarm of $ 10 \% $. 4) Apply the resulting Neuman-Pearson detection rule to the data stored in mdt 24 . mat. How many "signal present" decisions have you made?

          Problem statement: The Matlab file mdt 24 . mat contains \( N=1000 \) realizations of a random vector \( \mathbf{Y} \). Each realization (each vector \( \mathbf{y} \) ) has \( M=16 \) elements corresponding to multiple observations. Your task is to design the detector, and make \( N=1000 \) decisions, one for each realization \( \mathbf{y} \). A decision is either 1 , corresponding to "signal present," or 0 , corresponding to "signal absent."

The detection problem is specified as follows:
\[
\begin{array}{l}
H_{0}: \mathbf{Y}=\mathbf{Z} \\
H_{1}: \mathbf{Y}=A \mathbf{s}+\mathbf{Z}
\end{array}
\]

The noise is circularly symmetric complex Gaussian, \( \mathbf{Z} \sim \mathcal{C N}\left(\mathbf{0}, \sigma_{Z_{1}}^{2} \mathbf{I}\right) \). In other words, each element of \( \mathbf{Z} \) consists of independent zero-mean Gaussian real/imaginary parts of variance \( \sigma_{Z_{1}}^{2} / 2 \). The elements (different noise observations) are uncorrelated, as evident from the diagonal structure of the covariance matrix \( \mathbf{C}_{\mathbf{Z}}=E\left\{\mathbf{Z Z}^{\prime}\right\}=\sigma_{Z_{1}}^{2} \mathbf{I} \) (prime denotes conjugate transpose). The amplitude \( A \) is real-valued, positive and constant, and the vector \( \mathbf{s} \) is defined in terms of a phase \( \phi \) as
\[
\mathbf{s}=\left[\begin{array}{c}
1 \\
e^{-j \phi} \\
e^{-j 2 \phi} \\
\vdots \\
e^{-j(M-1) \phi}
\end{array}\right]
\]

This type of problem is found in radar/sonar, or any type of sounding or probing, when an array of \( M \) receivers is used to detect presence/absence of a signal. If the array receivers are separated by a distance \( d \), and the signal of wavelength \( \lambda \) arrives to the array from direction \( \theta \), the phase \( \phi \) is given by
\[
\phi=2 \pi \frac{d}{\lambda} \sin \theta
\]

In the present problem, the direction of arrival is \( \theta=20^{\circ} \), and \( d=\frac{\lambda}{2} \). The variance of the noise is \( \sigma_{Z_{1}}^{2}=1 \).
1
Specific tasks:
1) Specify the relevant probability density functions \( f_{0}(\mathbf{y}) \) and \( f_{1}(\mathbf{y}) \).
2) State the likelihood ratio test (LRT).
3) Identify the sufficient statistic and set the decision threshold to ensure probability of false alarm of \( 10 \% \).
4) Apply the resulting Neuman-Pearson detection rule to the data stored in mdt 24 . mat. How many "signal present" decisions have you made?

$Problem statement: The Matlab file mdt 24 . mat contains N=1000 realizations of a random vector 𝐘. Each realization (each vector 𝐲 ) has M=16 elements corresponding to multiple observations. Your task is to design the detector, and make N=1000 decisions, one for each realization 𝐲. A decision is either 1 , corresponding to "signal present," or 0 , corresponding to "signal absent." The detection problem is specified as follows: H0: 𝐘=𝐙 H1: 𝐘=A 𝐬+𝐙 The noise is circularly symmetric complex Gaussian, 𝐙∼𝒞 𝒩(0, σZ1^2𝐈). In other words, each element of 𝐙 consists of independent zero-mean Gaussian real/imaginary parts of variance σZ1^2 / 2. The elements (different noise observations) are uncorrelated, as evident from the diagonal structure of the covariance matrix 𝐂𝐙=E{𝐙 𝐙^'}=σZ1^2𝐈 (prime denotes conjugate transpose). The amplitude A is real-valued, positive and constant, and the vector 𝐬 is defined in terms of a phase ϕ as 𝐬=[ 1 e^-j ϕ e^-j 2 ϕ ⋮ e^-j(M-1) ϕ ] This type of problem is found in radar/sonar, or any type of sounding or probing, when an array of M receivers is used to detect presence/absence of a signal. If the array receivers are separated by a distance d, and the signal of wavelength λ arrives to the array from direction θ, the phase ϕ is given by ϕ=2 π(d)/(λ)sinθ In the present problem, the direction of arrival is θ=20^∘, and d=(λ)/(2). The variance of the noise is σZ1^2=1. 1 Specific tasks: 1) Specify the relevant probability density functions f0(𝐲) and f1(𝐲). 2) State the likelihood ratio test (LRT). 3) Identify the sufficient statistic and set the decision threshold to ensure probability of false alarm of 10 %. 4) Apply the resulting Neuman-Pearson detection rule to the data stored in mdt 24 . mat. How many "signal present" decisions have you made?$

Added by Philip C.

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