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radar:radardetection [2018/06/06 18:21] – dicristofaroradar:radardetection [2026/04/28 15:13] (current) – external edit 127.0.0.1
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 Due to the continuous nature of the noise, we can suppose that the dimensionality of $\Omega$ is infinite even if the size of //**$M$**//  and //**$S$**// are not.  Due to the continuous nature of the noise, we can suppose that the dimensionality of $\Omega$ is infinite even if the size of //**$M$**//  and //**$S$**// are not. 
  
-Finally we define the space of the possibile decisions //**$D$**//. Note that //**$D$**// and //**$M$**// must have the same dimensions due to the fact that the goal of the decision process is to extract the information contained in //**$M$**//.+Finally, we define the space of the possibile decisions //**$D$**//. Note that //**$D$**// and //**$M$**// must have the same dimensions due to the fact that the goal of the decision process is to extract the information contained in //**$M$**//.
  
-Due to the fact that the noise is a random process, the law that decide the transaction from the event's space to the space of the observable have a probabilistic behaviour. In particular, if $\vec{X}$ is the vector of the recieving observable, we define: +Due to the fact that the noise is a random process, the law that decide the transaction from the event's space to the space of the observable has a probabilistic behaviour. In particular, if $\vec{X}$ is the vector of the recieving observable, we define: 
  
 \begin{equation}  \begin{equation} 
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 The introduction of the probability distribution to describe the transaction from the space of the events and the space of the observable allow to define some decision procedures that can be of two types: parametric and non-parametric. The introduction of the probability distribution to describe the transaction from the space of the events and the space of the observable allow to define some decision procedures that can be of two types: parametric and non-parametric.
 We talk about parametric decision procedures when the probabilities of transition $p(x \mid M_i)$ , $i=0,...,m-1$ differs only from some values of one or more parameters. In this case the decision process coincides with finding the unknown parameters. We talk about parametric decision procedures when the probabilities of transition $p(x \mid M_i)$ , $i=0,...,m-1$ differs only from some values of one or more parameters. In this case the decision process coincides with finding the unknown parameters.
-We talk about //non parametric decisions// when the different hypotesis imply different behaviour of //transaction probabilities// $p(\vec(X) \mid M_i),i=0,...,m-1$. +We talk about //non parametric decisions// when the different hypothesis imply different behaviour of //transaction probabilities// $p(\vec(X) \mid M_i),i=0,...,m-1$. 
 An example of paremetric decision between the events $M_0$ and $M_1$ could be:  An example of paremetric decision between the events $M_0$ and $M_1$ could be: 
  
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 Once we have the observables' vector $\vec{X}$ we have to keep a decision by interpreting it. Once we have the observables' vector $\vec{X}$ we have to keep a decision by interpreting it.
 The decision rules allow to go from the space of the observable $\Omega$ to the space of the decisions $D$.  The decision rules allow to go from the space of the observable $\Omega$ to the space of the decisions $D$. 
-If the space of the events is continous the problem of decision become a problem of estimation.+If the space of the events is continuous the problem of decision become a problem of estimation.
 For example if $M$ is the space of the possible values of a parameter, to estimate this value we can use decision theory metodologies. For example if $M$ is the space of the possible values of a parameter, to estimate this value we can use decision theory metodologies.
-The definition of decision rules determinates a partition of the observable's space. This partition is realized with rules that operates in the following way: Let's consider the case when the space of the events $M$ have a cardinality equal to two.+The definition of decision rules determinates a partition of the observable's space. This partition is realized with rules that operates in the following way: Let's consider the case when the space of the events $M$ has a cardinality equal to two.
  
  
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 </figure> </figure>
  
-The above scheme (figure {{ref>binary_decision_problem}}) consider the particular case of __binary decision on single observation__ that is the case when the space of the events $M$ is only made by two elements $M_0$ and $M_1$; in the figure we omitted for simplicity the space of the signal $S$ because is supposed to be in a biunivocal correspondence (1:1) with the space of the events.+The above scheme (figure {{ref>binary_decision_problem}}) consider the particular case of __binary decision on a single observation__ that is the case when the space of the events $M$ is only made by two elements $M_0$ and $M_1$; in the figure we omitted for simplicity the space of the signal $S$ because is supposed to be in a biunivocal correspondence (1:1) with the space of the events.
  
 Let's suppose to have received the vector $\vec{X}$, the decision rule is: Let's suppose to have received the vector $\vec{X}$, the decision rule is:
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-__The radar detection procedures are particular cases of the decision problems__ and we are going to analize it; The scheme in figure {{ref>radar_detection_decision_theory}} represents a radar detection scenario.+__The radar detection procedures are particular cases of the decision problems__ and we are going to analyze it; The scheme in figure {{ref>radar_detection_decision_theory}} represents a radar detection scenario.
 From what we said until here, we have to decide considering two events: From what we said until here, we have to decide considering two events:
  
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 Note that we can also associate a cost to the correct decisions. Note that we can also associate a cost to the correct decisions.
  
-Since we operate in an uncertainity condition we have to refer to the medium cost $E(L)$ (mean of the cost, called also as risk) that in the binary case is defined as:+Since we operate in an uncertainty condition we have to refer to the medium cost $E(L)$ (mean of the cost, called also as risk) that in the binary case is defined as:
  
  
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-We remember that to estabilish the rules of the decision we have to find $\Omega_1$ knowing the weight values $L_{ij}$ and the functions $p(\vec{X} \mid M_i)$.+We remember that to establish the rules of the decision we have to find $\Omega_1$ knowing the weight values $L_{ij}$ and the functions $p(\vec{X} \mid M_i)$.
  
 The space $\Omega_1$ is defined such that the result of (11) is minimum. We are going to analyze this problem in the unidimensional case for semplicity: The space $\Omega_1$ is defined such that the result of (11) is minimum. We are going to analyze this problem in the unidimensional case for semplicity:
-Let's consider the geneirc function $f(x)$ in figure {{ref>generic_function}}.+Let's consider the generic function $f(x)$ in figure {{ref>generic_function}}.
  
 <figure generic_function> <figure generic_function>
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 \begin{equation}  \begin{equation} 
-l(\vec{X}) \geq \eta \Rightarrow M_1 \text{ hypotes is true}+l(\vec{X}) \geq \eta \Rightarrow M_1 \text{ hypothesis is true}
 \end{equation}  \end{equation} 
  
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 \end{equation}  \end{equation} 
  
-Due to the fact that the main difficulty in the application of the Bayes method consists in defining the values of costs $L_{ij}$ and in the knowledege of the a priori probability $p_0$ and $p_1$ in general this method is not used for radar detection while other methods are preferred.+Due to the fact that the main difficulty in the application of the Bayes method consists in defining the values of costs $L_{ij}$ and in the knowledge of the a priori probability $p_0$ and $p_1$ in general this method is not used for radar detection while other methods are preferred.
  
  
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 \end{equation}  \end{equation} 
  
-and we can obtain (24). The probability error criterion consist in comparing the likelihood ratio with the ratio $p_1 / p_0$.+and we can obtain (24). The probability error criterion consists in comparing the likelihood ratio with the ratio $p_1 / p_0$.
  
 Alternatively, we can derive (24) from (14) imposing $L_{01} - L_{00} = L_{10} - L_{11}$, that means that the probability error criterion is a particular case of the Bayes criterion. Alternatively, we can derive (24) from (14) imposing $L_{01} - L_{00} = L_{10} - L_{11}$, that means that the probability error criterion is a particular case of the Bayes criterion.
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 \begin{equation}  \begin{equation} 
 \begin{array} \begin{array}
-\text{p}(\vec{X} \mid M_0) = p(\vec{X}) p(M_0 \mid \vec{X})     &\text{,}&     p(\vec{X} \mid M_1) = p(\vec{X}) p(M_1 \mid \vec{X})  +\text{p}(\vec{X} \mid M_0)\text{p_0} = p(\vec{X}) p(M_0 \mid \vec{X})     &\text{,}&     p(\vec{X} \mid M_1)\text{p_1} = p(\vec{X}) p(M_1 \mid \vec{X})  
 \end{array} \end{array}
 \end{equation} \end{equation}
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-We obtain that the cost is minimum if the decision region for each hypotesis ${M_k}$ , ${k=1,2,...,m}$ is:+We obtain that the cost is minimum if the decision region for each hypothesis ${M_k}$ , ${k=1,2,...,m}$ is:
  
  
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-Therefore, the MAP decisor is: the calculation of the likelihood function $p(\vec{X} \mid M_k)$ for each hypotesis, multiplied by a priori probability of that hypotesis and finally the choise of the maximum.+Therefore, the MAP decisor is: the calculation of the likelihood function $p(\vec{X} \mid M_k)$ for each hypothesis, multiplied by a priori probability of that hypothesis and finally the choice of the maximum.
  
 <figure map_reciever> <figure map_reciever>
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 \end{equation}  \end{equation} 
  
-where $\eta$ is a parameter that depends from the costs and from the a priori probabilities. Due to the fact that often we don't know such values, in the hypotesis of maximum uncertainity, we can assume $\eta = 1$ (Maximum Likelihood Criterion (ML)).+where $\eta$ is a parameter that depends from the costs and from the a priori probabilities. Due to the fact that often we don't know such values, in the hypothesis of maximum uncertainty, we can assume $\eta = 1$ (Maximum Likelihood Criterion (ML)).
  
-If we assume $\eta = 1$ (maximum uncertainity) we can rewrite the decision rule (29) as:+If we assume $\eta = 1$ (maximum uncertainty) we can rewrite the decision rule (29) as:
  
 \begin{equation}  \begin{equation} 
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-The Bayes criterion, MAP and Neyman-Pearson are equavalent from the algorithmic point of view and are exploited comparing the likelihood ratio with a threshold chosen conveniently, the difference between them is how this threshold value is defined.+The Bayes criterion, MAP and Neyman-Pearson are equivalent from the algorithmic point of view and are exploited comparing the likelihood ratio with a threshold chosen conveniently, the difference between them is how this threshold value is defined.
  
 In general the most used criterion is the Neyman-Pearson's. In general the most used criterion is the Neyman-Pearson's.
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-This problem is a tipical case of a deterministic signal as we suppose that $\mu$ is constant.+This problem is a typical case of a deterministic signal as we suppose that $\mu$ is constant.
 We need to identify the regions $\Omega_0$ and $\Omega_1$. The problem is to find the threshold value $X_T$ that delimits the two decision regions. We need to identify the regions $\Omega_0$ and $\Omega_1$. The problem is to find the threshold value $X_T$ that delimits the two decision regions.
  
-If we adopt the maximum likelihood criterion (for which $\lambda = 1$) the threshold value $X_T$ is the point $X_T=\mu/2$ that is at the intercption point of the two curves defined by $p(x \mid M_0)$ and $p(x \mid M_1)$.+If we adopt the maximum likelihood criterion (for which $\lambda = 1$) the threshold value $X_T$ is the point $X_T=\mu/2$ that is at the interception point of the two curves defined by $p(x \mid M_0)$ and $p(x \mid M_1)$.
  
  
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 \end{equation}  \end{equation} 
  
-where the fucntion $\Phi(x)$ is $\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2} \,dt$.+where the function $\Phi(x)$ is $\Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^2/2} \,dt$.
  
  
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 //Application of Neyman-Pearson's criterion for detection of a fixed target using a single pulse.// //Application of Neyman-Pearson's criterion for detection of a fixed target using a single pulse.//
  
-Let's suppose that we have a target which produce an echo with a constant amplitude $A$ (the received signal is a sine with amplitude $A$) and the RMS voltage of the gaussian noise is $\sigma$. We suppose to take a decision on a single pulse recived in output from an enevelope detector.+Let's suppose that we have a target which produce an echo with a constant amplitude $A$ (the received signal is a sine with amplitude $A$) and the RMS voltage of the gaussian noise is $\sigma$. We suppose to take a decision on a single pulse recived in output from an envelope detector.
 Using the theory illustrated in the previous section we can say that: Using the theory illustrated in the previous section we can say that:
  
-a) If only noise is present ($M_0$ hypotesis) the probability density function of the noise envelope is a Rayleigh:+a) If only noise is present ($M_0$ hypothesis) the probability density function of the noise envelope is a Rayleigh:
  
 \begin{equation} \begin{equation}
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-b) If the target is present ($M_1$ hypotesis) the probability density associated to the envelope of the received signal is Rice type+b) If the target is present ($M_1$ hypothesis) the probability density associated to the envelope of the received signal is of type Rice 
  
 \begin{equation} \begin{equation}
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-The decisor will decide that the hypotesis that the target is present (event $M_1$) if we have:+The decisor will decide for the hypothesis that the target is present (event $M_1$) if we have:
  
 \begin{equation} \begin{equation}
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 The value $T$ can be calculated once we choose the decision criterion. The value $T$ can be calculated once we choose the decision criterion.
-If the decision criterion is the Neyman-Pearson's $T$ must be calculated such that+If the decision criterion is the Neyman-Pearson's, $T$ must be calculated such that
  
 \begin{equation} \begin{equation}
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 Note that the calculation indicated in the previous equations is optimal due to the Neyman-Pearson criterion such that it give us the maximum $P_d$. Note that the calculation indicated in the previous equations is optimal due to the Neyman-Pearson criterion such that it give us the maximum $P_d$.
 Due to the fact that the value of $T$ is not dependent from $A$ the procedure is optimal also if $A$ is unknown. Due to the fact that the value of $T$ is not dependent from $A$ the procedure is optimal also if $A$ is unknown.
-So, in case of fixed target when the SNR ratio is unknown (remember that $SNR = \frac{A^2}{2\sigma^2}$), the Neyman-Pearson preocedure is optimal only if the RMS voltage of the noise $\sigma$ is known.+So, in case of fixed target when the SNR ratio is unknown (remember that $SNR = \frac{A^2}{2\sigma^2}$), the Neyman-Pearson procedure is optimal only if the RMS voltage of the noise $\sigma$ is known.
  
 ====== Neyman-Pearson (single pulse + SW2 target) ====== ====== Neyman-Pearson (single pulse + SW2 target) ======
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 //Application of Neyman-Pearson's criterion for detection of a fluctuating target modeled as Swerling 2 using a single pulse.// //Application of Neyman-Pearson's criterion for detection of a fluctuating target modeled as Swerling 2 using a single pulse.//
  
-The Neyman-Pearson criterion can be succesfully applicated also in the case of a fluctuating target (type SW2) operating on a single impulse and with a non coherent detection.+The Neyman-Pearson criterion can be successfully applied also in the case of a fluctuating target (type SW2) operating on a single pulse and with a non coherent detection.
 We assume $x(t)$ is the amplitude of the received voltage, sum of the interesting signal $s(t)$ and the noise $n(t)$, $\sigma$ is the variance of the noise process and $\sigma^2_s$ the power of the signal.  We assume $x(t)$ is the amplitude of the received voltage, sum of the interesting signal $s(t)$ and the noise $n(t)$, $\sigma$ is the variance of the noise process and $\sigma^2_s$ the power of the signal. 
  
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-Since $\frac{1}{\sigma^2} - \frac{1}{s^2} > 0$ and the log function is monotonically increasing, the relation (56) can be expressed taking the natural log; So, we can write the quadratic detector:+Since $\frac{1}{\sigma^2} - \frac{1}{s^2} > 0$ and the log function is monotonically increasing, the relation (56) can be expressed by taking the natural log; So, we can write the quadratic detector:
  
 \begin{equation} \begin{equation}
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-(Obviusly it is possible to take the square root of the last equation and use a linear detector, if we decide on the single pulse).+(Obviously it is possible to take the square root of the last equation and use a linear detector, if we decide on the single pulse).
  
 The decision threshold $T$ must be chosen such that: The decision threshold $T$ must be chosen such that:
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 \begin{equation} \begin{equation}
-\Phi_x(\omega) = E[e^{j \omega x}] = \int_{-\infty}^{\infty} f_x(x) e^{j \omega x} \,dx = exp(j \omega x)exp(-\sigma^2 \omega^2 / 2)+\Phi_x(\omega) = E[e^{j \omega x}] = \int_{-\infty}^{\infty} f_x(x) e^{j \omega x} \,dx = exp(j \omega \eta)exp(-\sigma^2 \omega^2 / 2)
 \end{equation} \end{equation}
  
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 \end{equation} \end{equation}
  
-If we multiply the vector $\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ for the matrix $P = \frac{\sigma}{\sqrt{2}} \begin{bmatrix} \alpha & \beta  \\ -\alpha & \beta \end{bmatrix}$ , where $\alpha = \sqrt{1-\rho} \ \text{,}\ \beta=\sqrt{1+\rho}$, it's easy to verify the the pair of random gaussian variables $(x_1,x_2)$ have a correlation cohefficient equal to $\rho$ and every variable has a variance equal to $\sigma^2$ . So we have+If we multiply the vector $\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ for the matrix $P = \frac{\sigma}{\sqrt{2}} \begin{bmatrix} \alpha & \beta  \\ -\alpha & \beta \end{bmatrix}$ , where $\alpha = \sqrt{1-\rho} \ \text{,}\ \beta=\sqrt{1+\rho}$, it's easy to verify the the pair of random gaussian variables $(x_1,x_2)$ have a correlation coefficient equal to $\rho$ and every variable has a variance equal to $\sigma^2$ . So we have
  
 \begin{equation} \begin{equation}
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-In that way we can demostrate that $f(x_i)$ $i=1,2$ is gaussian. Then, from the (72) if the cross-correlation coefficient $\rho$ between the two random variables $x_1$ and $x_2$ is null then +In that way we can demonstrate that $f(x_i)$ $i=1,2$ is gaussian. Then, from the (72) if the cross-correlation coefficient $\rho$ between the two random variables $x_1$ and $x_2$ is null then 
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-this means that the two random variables are independent. For the gaussian random variable independece and scorrelation coincide.+this means that the two random variables are independent. For the gaussian random variable, independence and scorrelation coincide.
  
-The cross-correlation coefficient $\rho$, gives an idea of the dipendence between twe two variables. In particular, if $x_1$ and $x_2$ are for example a representation of a measure process, if $\left| \rho \right| \cong 1$ the pair $(x_1,x_2)$ tend to thicken along a straight line as indicated in figure {{ref>scattering_diagrams}}a.+The cross-correlation coefficient $\rho$, gives an idea of the dependence between the two variables. In particular, if $x_1$ and $x_2$ are for example a representation of a measure's process, if $\left| \rho \right| \cong 1$ the pair $(x_1,x_2)$ tend to thicken along a straight line as indicated in figure {{ref>scattering_diagrams}}a.
 If instead $\rho=0$ we can have the situation shown in the figure {{ref>scattering_diagrams}}b. If instead $\rho=0$ we can have the situation shown in the figure {{ref>scattering_diagrams}}b.
  
-The characteristic function assiociated to the gaussian bivariate is:+The characteristic function associated to the gaussian bivariate is:
  
 \begin{equation} \begin{equation}
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 <figure scattering_diagrams> <figure scattering_diagrams>
 {{ :media:scattering_diagrams.png?nolink |}} {{ :media:scattering_diagrams.png?nolink |}}
-<caption>Scattering diagrams associated to the pairs of random variables: a) correlated ; b) scorrelated</caption>+<caption>Scattering diagrams associated to the pairs of random variables: a) correlated ; b) uncorrelated</caption>
 </figure> </figure>
  
-Let's consider that a random vector $\vec{X} = [x_1,x_2...,x_n]^T$ where $x_i$ are random variables.+Let's consider a random vector $\vec{X} = [x_1,x_2,...,x_n]^T$ where $x_i$ are random variables.
 The vector $\vec{X}$ is described by the joint density function  The vector $\vec{X}$ is described by the joint density function 
  
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 $\vec{Y}$ is the vector with independent and gaussian components with zero mean and variance equal to one, that means that the density is of type negative exponential (except for the normalization constant) and the quadratic form is $\vec{Y}^T \vec{Y} = \sum_{i=1}^{n} y_i^2$;  $\vec{Y}$ is the vector with independent and gaussian components with zero mean and variance equal to one, that means that the density is of type negative exponential (except for the normalization constant) and the quadratic form is $\vec{Y}^T \vec{Y} = \sum_{i=1}^{n} y_i^2$; 
  
- $\boldsymbol{\vec{P}}$ is a coefficients matrix and is obtained, knwoing that $\vec{Y} = \vec{P}^{-1} \vec{X}$ \,\\ + $\boldsymbol{\vec{P}}$ is a coefficients matrix and is obtained, knowing that $\vec{Y} = \vec{P}^{-1} \vec{X}$ \\ 
- $\vec{Y}^T \vec{X}^T \boldsymbol{\vec{P}^{-T}} \boldsymbol{\vec{P}^{-1}} \vec{X} = X^T \boldsymbol{Q} \vec{X} $  + $\vec{Y}^T \vec{Y} = \vec{X}^T \boldsymbol{\vec{P}^{-T}} \boldsymbol{\vec{P}^{-1}} \vec{X} = X^T \boldsymbol{Q} \vec{X} $  
  
  
- We can repeat the procedure used to obtain the equation (70) and (71); the resulting density function is expect for a normalization constant, the exponential of the squared form: $X^T \boldsymbol{\vec{Q}} = P^{-T} P^{-1}$.+ We can repeat the procedure used to obtain the equation (70) and (71); the resulting density function is, except of a normalization constant, the exponential of the squared form: $X^T \boldsymbol{\vec{Q}} \vec{X}$ where $\boldsymbol{\vec{Q}} = \boldsymbol{P^{-T}} \boldsymbol{P^{-1}}$.
  
-The coviariance matrix of $\vec{X}$ (remember that the components of $\vec{Y}$ are scorrelated ) is:+The covariance matrix of $\vec{X}$ (remember that the components of $\vec{Y}$ are uncorrelated ) is:
 \begin{equation} \begin{equation}
 \boldsymbol{\vec{M}_{\vec{x}}} = E \left[\vec{X} \vec{X}^T \right] = \boldsymbol{\vec{T}} \boldsymbol{\vec{M}_y} \boldsymbol{\vec{P}^T} = \boldsymbol{\vec{P}} \boldsymbol{\vec{P}^T} \boldsymbol{\vec{M}_{\vec{x}}} = E \left[\vec{X} \vec{X}^T \right] = \boldsymbol{\vec{T}} \boldsymbol{\vec{M}_y} \boldsymbol{\vec{P}^T} = \boldsymbol{\vec{P}} \boldsymbol{\vec{P}^T}
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 is **gaussian**. is **gaussian**.
  
-From that definition result that the characteristic function of a gaussian random vector is expressed (if the means vector $\left< \vec{X} \right>$ is null):+From that definition result that the characteristic function of a gaussian random vector is expressed (if the means vector $\left< \vec{X} \right>$ is null) as:
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-where  $\boldsymbol{\vec{M}}$ is coaviariance matrix:+where  $\boldsymbol{\vec{M}}$ is covariance matrix:
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-If we antitransofrm using Fourier the equation (84) we obtain the density function of the gaussian multivariate. +If we antitransform using Fourier the equation (84) we obtain the density function of the gaussian multivariate. 
  
 \begin{equation} \begin{equation}
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 \begin{equation} \begin{equation}
-\boldsymbol{\vec{\hat{M}}} \frac{1}{v} \sum_{i=1}^{v} \vec{X_i} \vec{X_i}^T+\boldsymbol{\vec{\hat{M}}} = \frac{1}{v} \sum_{i=1}^{v} \vec{X_i} \vec{X_i}^T
 \end{equation}  \end{equation} 
  
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 where $v$ is the number of observation of the vector $\vec{X}$. where $v$ is the number of observation of the vector $\vec{X}$.
  
-The gaussian multivariate can be also obtained generalizing the procedure that allow to define the gaussian bivariate to the //n//-dimensional case.+The gaussian multivariate can be also obtained generalizing the procedure that allows to define the gaussian bivariate to the //n//-dimensional case.
  
 We can start from $n$ variables $y_i$ $i=1,...,n$ independent (vector $\vec{Y}$) that has a density function as We can start from $n$ variables $y_i$ $i=1,...,n$ independent (vector $\vec{Y}$) that has a density function as
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 \begin{equation} \begin{equation}
-f_{\vec{Y}}(\vec{Y}) = \frac{1}{  \left( 2 \pi \right)^{\frac{n}{2}} \left( det \vec{\Lambda} \right)^{\frac{1}{2}} } exp \left[ \frac{ -\left( \vec{Y} - \left< \vec{Y} \right> \right)^T  \vec{\Lambda^{-1}}   \left( \vec{Y} - \left< \vec{Y} \right> \right) }{2}  \right]+f_{\vec{Y}}(\vec{Y}) = \frac{1}{  \left( 2 \pi \right)^{\frac{n}{2}} \left( det \vec{\Lambda} \right)^{\frac{1}{2}} } exp \left[ \frac{ -\left( \vec{Y} - \left< \vec{Y} \right> \right)^T  \vec{\Lambda}^{-1}   \left( \vec{Y} - \left< \vec{Y} \right> \right) }{2}  \right]
 \end{equation}  \end{equation} 
  
  
-where $\vec{\Lambda}$ is a diagonal matrix that has variance of the random variables $y_i$ as elements, that means $l_{ii}=\text{var}(y_i) i=1,...,n$.+where $\vec{\Lambda}$ is a diagonal matrix that has variance of the random variables $y_i$ as elements, that means $\lambda_{ii}=\text{var}(y_i) \ i=1,...,n$.
  
 If we apply the linear transformation If we apply the linear transformation
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-where $\boldsymbol{\vec{U}}$ is a unitary orthonormal matrix, using the fondamental theorem for the random variables transformation, we have, supposing zero means:+where $\boldsymbol{\vec{U}}$ is a unitary orthonormal matrix, using the fundamental theorem for the random variables transformation, we have, supposing zero means:
  
 \begin{equation} \begin{equation}
 f_X(\vec{X}) = f_Y(\vec{Y} = \boldsymbol{\vec{U}} \vec{X}) = \\ f_X(\vec{X}) = f_Y(\vec{Y} = \boldsymbol{\vec{U}} \vec{X}) = \\
 = A exp \left \{ -\frac{1}{2} \left( \boldsymbol{\vec{U}} \vec{X} \right)^T  \boldsymbol{\vec{\Lambda}^{-1}} \left( \boldsymbol{\vec{U}} \vec{X} \right) \right \} = \\ = A exp \left \{ -\frac{1}{2} \left( \boldsymbol{\vec{U}} \vec{X} \right)^T  \boldsymbol{\vec{\Lambda}^{-1}} \left( \boldsymbol{\vec{U}} \vec{X} \right) \right \} = \\
-= A exp \left\{ -\frac{1}{2} \boldsymbol{\vec{X}}^T \  \vec{U}^T \ \boldsymbol{\vec{\Lambda}^{-1}} \  \boldsymbol{\vec{U}} \ \vec{X}  \right\} = \\ += A exp \left\{ -\frac{1}{2} \vec{X}^T \  \vec{U}^T \ \boldsymbol{\vec{\Lambda}^{-1}} \  \boldsymbol{\vec{U}} \ \vec{X}  \right\} = \\ 
-= A exp \left\{ -\frac{1}{2} \vec{X}^T  \boldsymbol{M}^{-1}  \vec{X} \right\}+= A exp \left\{ -\frac{1}{2} \vec{X}^T  \boldsymbol{\vec{M}}^{-1}  \vec{X} \right\}
 \end{equation} \end{equation}
      
  
-where $\boldsymbol{M^{-1}}$ is the inverse of the covariance matrix $\boldsymbol{M}$ of $\vec{X}$ that can be spectral decomposed as $\boldsymbol{\vec{M}} = \boldsymbol{\vec{U}} \ \boldsymbol{\vec{\Lambda}} \ \boldsymbol{\vec{U}^T} $.+where $\boldsymbol{M^{-1}}$ is the inverse of the covariance matrix $\boldsymbol{\vec{M}}$ of $\vec{X}$ that can be spectral decomposed as $\boldsymbol{\vec{M}} = \boldsymbol{\vec{U}} \ \boldsymbol{\vec{\Lambda}} \ \boldsymbol{\vec{U}^T} $.
  
 The probability density of the vector $\vec{X}$ is: The probability density of the vector $\vec{X}$ is:
Line 929: Line 929:
  
  
-So, the gaussian multivariate distribution of $\vec{X}$ is expressed +So, the gaussian multivariate distribution of $\vec{X}$ is expressed as
  
 \begin{equation} \begin{equation}
Line 936: Line 936:
  
  
-All we have done is necessary to extend the Neyman-Pearson criterion to the case when we have multiple pluses (or samples) associated to the target. +All we have done is necessary to extend the Neyman-Pearson criterion to the case when we have multiple pulses (or samples) associated to the target. 
-Since normally the compenents in phase and quadrature $I$ and $Q$ are elaborated we have to work with complex numbers and consecutively is necessary to define the __complex multivariate gaussian vectors__.+Since normally we elaborate the components in phase and quadrature $I$ and $Q$, we have to work with complex numbers and consecutively is necessary to define the __complex multivariate gaussian vectors__.
  
-We indicate with $\vec{Z}$ the vector such that his $n$ components are samples of the complex envelope  of the received signal.+We indicate with $\vec{Z}$ the vector such that its $n$ components are samples of the complex envelope  of the received signal.
 In radar applications this signal can be modelled as a stationary process and has narrow band. In radar applications this signal can be modelled as a stationary process and has narrow band.
  
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-where $\vec{x}$ and $\vec{Y}$ are two real gaussian vectors, and we can consider instead of $\vec{Z}$ the real vector with a double dimension:+where $\vec{X}$ and $\vec{Y}$ are two real gaussian vectors, and we can consider instead of $\vec{Z}$, the real vector with a double dimension:
  
 \begin{equation} \begin{equation}
Line 955: Line 955:
 \end{equation} \end{equation}
  
-The covariance matrix of $\vec{Z}$, if is valid the hypotesis of narrow band stationary process (see above) with zero mean, is equal to+The covariance matrix of $\vec{Z}$, if the hypothesis of narrow band stationary process (see above) with zero mean is valid, is equal to
  
 \begin{equation} \begin{equation}
-\boldsymbol{\vec{M}} = E \left[ \begin{bmatrix} \vec{X} \\ \vec{Y} \end{bmatrix}  \begin{bmatrix} \vec{X^T} & \vec{Y^T} \end{bmatrix}  \right] = \begin{bmatrix} \vec{V} & -\vec{W} \\ \vec{W} & \vec{V} \end{bmatrix}_{2n \ \times \ 2n}+\boldsymbol{\vec{M}} = E \left[ \begin{bmatrix} \vec{X} \\ \vec{Y} \end{bmatrix}  \begin{bmatrix} \vec{X^T} & \vec{Y^T} \end{bmatrix}  \right] = \begin{bmatrix} \boldsymbol{\vec{V}} & - \boldsymbol{\vec{W}} \\ \boldsymbol{\vec{W}} & \boldsymbol{\vec{V}} \end{bmatrix}_{2n \ \times \ 2n}
 \end{equation} \end{equation}
  
-The $\boldsymbol{\vec{M}}$ matrix contains blocks that are equal two by two, so the information on the $\vec{Z}$ process is only related to the knowledge of $2n^2$ parameters and not $4n^2$ because we need to have only the $\vec{V}$ and $\vec{W}$ matrixes. $\vec{V}$ is the covariance matrix of $\vec{X}$ and $\vec{W}$ is the mutual covariance matrix of $\vec{X}$ and $\vec{Y}$. +The $\boldsymbol{\vec{M}}$ matrix contains blocks that are equal two by two, so the information on the $\vec{Z}$ process is only related to the knowledge of $2n^2$ parameters and not $4n^2$ because we need to have only the $\boldsymbol{\vec{V}}$ and $\boldsymbol{\vec{W}}$ matrixes. $\boldsymbol{\vec{V}}$ is the covariance matrix of $\vec{X}$ and $\vec{W}$ is the mutual covariance matrix of $\vec{X}$ and $\vec{Y}$. 
  
 Note that the relations  Note that the relations 
  
 \begin{equation} \begin{equation}
-\vec{V^T} = \vec{V} \ \ \ \text{and} \ \ \ \vec{W^T} = -\vec{W}+\boldsymbol{\vec{V^T}} = \boldsymbol{\vec{V}} \ \ \ \text{and} \ \ \ \boldsymbol{\vec{W^T}} = -\boldsymbol{\vec{W}}
 \end{equation} \end{equation}
  
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 The (99) extends the concept of the gaussian multivariate to the case of complex random vectors. The (99) extends the concept of the gaussian multivariate to the case of complex random vectors.
  
-In case of a complex process the matrix $\vec{M}$ is Hermitian and so:+In case of a complex process the matrix $\boldsymbol{\vec{M}}$ is Hermitian and so:
  
 \begin{equation} \begin{equation}
Line 1027: Line 1027:
  
 \begin{equation} \begin{equation}
-f_{\vec{Z}}(\vec{Z}) = \frac{1}{ (2 \pi)^N \Delta } exp \left( -\frac{1}{2} \left( \vec{Z} - \left< \vec{Z} \right> \right)^{T} \ \boldsymbol{\vec{M}} \ \left( \vec{Z} - \left< \vec{Z} \right> \right)^{*}  \right)+f_{\vec{Z}}(\vec{Z}) = \frac{1}{ (2 \pi)^N \Delta } exp \left( -\frac{1}{2} \left( \vec{Z} - \left< \vec{Z} \right> \right)^{T} \ \boldsymbol{\vec{M}^{-1}} \ \left( \vec{Z} - \left< \vec{Z} \right> \right)^{*}  \right)
 \end{equation} \end{equation}
  
Line 1035: Line 1035:
  
 \begin{equation} \begin{equation}
-\vec{Z} = \boldsymbol{\vec{A}} \boldsymbol{\vec{Z_0}}  +\vec{Z} = \boldsymbol{\vec{A}} \vec{Z_0} 
 \end{equation} \end{equation}
  
Line 1059: Line 1059:
 where $\boldsymbol{\vec{M}}$ is the covariance matrix of the process $\vec{Z}$. where $\boldsymbol{\vec{M}}$ is the covariance matrix of the process $\vec{Z}$.
  
-This formula is useful when we have to work with energies (for example to evaluating the Signal/Noise ratio after filtering).+This formula is useful when we have to work with energies (for example to evaluate the Signal/Noise ratio after filtering).
  
 **//Proof of (82) //** //(in case of a random vector with zero mean)//: **//Proof of (82) //** //(in case of a random vector with zero mean)//:
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 and we obtain: and we obtain:
  
-$\Phi\left(\vec{\Omega}\right) = E \left[ e^{j \vec{\Omega^T} \vec{X}} \right] = exp \left( - \frac{1}{2} \vec{\Omega}^T \vec{M} \vec{\Omega} \right)$+$\Phi_{X}\left(\vec{\Omega}\right) = E \left[ e^{j \vec{\Omega^T} \vec{X}} \right] = exp \left( - \frac{1}{2} \vec{\Omega}^T \vec{M} \vec{\Omega} \right)$
  
 that is equal to (82). that is equal to (82).
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 ====== Coherent detector and Discrete-Time optimal processor ====== ====== Coherent detector and Discrete-Time optimal processor ======
  
-//Application of Neyman-Pearson criterieon on a coherent radar detector.//+//Application of Neyman-Pearson criterion on a coherent radar detector.//
  
  
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 so if we use a coherent radar we have I and Q samples taken at multiple instants of the PRT. so if we use a coherent radar we have I and Q samples taken at multiple instants of the PRT.
-These type of samples can be considerated as a complex vector+These type of samples can be considered as a complex vector
  
 \begin{equation} \begin{equation}
Line 1123: Line 1123:
  
 where $N$ is the number of the samples (or pulses in the dwell time). The samples contained in the vector $\vec{Z}$ have a component of the useful signal and a component of noise. where $N$ is the number of the samples (or pulses in the dwell time). The samples contained in the vector $\vec{Z}$ have a component of the useful signal and a component of noise.
-Let's suppose that the useful signal is represented by (111) and that the noise, produced for example from external factors, (**clutter**) is additive Gaussian with zero mean+Let's suppose that the useful signal is represented by (111) and that the noise, produced for example from external factors (**clutter**), is additive Gaussian with zero mean
 and with covariance matrix $\boldsymbol{\vec{M}}$.  and with covariance matrix $\boldsymbol{\vec{M}}$. 
  
-Note that we don't make the hypotesis that the noise is white, this is why in the radar field there is not only the contribution of the white noise but can exist also noises caused by clutter's echos (__clutter__: echo caused by fixed target) that have some correlation between them; This correlation can be expressed in the covariance matrix $\boldsymbol{\vec{M}}$ assuming that the matrix is not diagonal.+Note that we don't make the hypothesis that the noise is white, this is why in the radar field there is not only the contribution of the white noise, but can exist also noises caused by clutter's echoes (__clutter__: echo caused by ground, sea, rain, animals/insects, chaff and atmospheric turbulences, etc.) that have some correlation between them; This correlation can be expressed in the covariance matrix $\boldsymbol{\vec{M}}$ assuming that the matrix is not diagonal.
 Anyway, in the case of noise caused by clutter it's right to say that the distribution is Gaussian, at least for some kind of clutters. Anyway, in the case of noise caused by clutter it's right to say that the distribution is Gaussian, at least for some kind of clutters.
 The vector of the observable $\vec{Z}$ is The vector of the observable $\vec{Z}$ is
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 where the random vector $\vec{Z}$ is made by the sum of the noise clutter and the useful signal. where the random vector $\vec{Z}$ is made by the sum of the noise clutter and the useful signal.
-If the useful signal is statistically independent from the additive noise and if is a process with zero mean, the probability density function of the vector $\vec{Z}$ in the case of absence and presence of the target +If the useful signal is statistically independent from the additive noise and if it's a process with zero mean, the probability density function of the vector $\vec{Z}$, in the case of absence and presence of the target, are n-dimensional Gaussian.
-are n-dimensional Gaussian.+
  
 a) In the absence of target we have: a) In the absence of target we have:
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 where we suppose that the vector containing the means of the noise is null. where we suppose that the vector containing the means of the noise is null.
  
-b) In cases in which the target is deterministic (that are known in addition to the carrier f0, the values of $f_D$, and $A$) is possible to obtain the analytical expression of the useful signal $\vec{s}$. Since the process of the noise has zero mean $\vec{s}$ can also be seen as the vector of the mean values of the n-dimensional Gaussian that statistically describes the vector $\vec{Z}$. So we can write+b) In cases in which the target is deterministic (that are known in addition to the carrier $f_0$, the values of $f_D$, and $A$) is possible to obtain the analytical expression of the useful signal $\vec{s}$. Since the process of the noise has zero mean $\vec{s}$ can also be seen as the vector of the mean values of the n-dimensional Gaussian, that statistically describes the vector $\vec{Z}$. So we can write
  
 \begin{equation} \begin{equation}
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 \begin{equation} \begin{equation}
-\Lambda(\vec{Z}) = - \vec{s}^T \boldsymbol{\vec{M}^{-1}} \vec{s}^{*} + \sum_{ij}{m_{ij} \boldsymbol{s_i} \boldsymbol{z_j}^{*}} + \sum_{ij}{m_{ij} \boldsymbol{z_i} \boldsymbol{s_j}^{*}}+\Lambda(\vec{Z}) = - \vec{s}^T \boldsymbol{\vec{M}^{-1}} \vec{s}^{*} + \sum_{ij}{m_{ij} s_i {z_j}^{*}} + \sum_{ij}{m_{ij} z_i {s_j}^{*}}
 \end{equation} \end{equation}
  
-where $m_{ij}$ are the elements of the matrix $\boldsymbol{\vec{M}^{-1}}$. Since $\boldsymbol{\vec{M}^{-1}}$ is Hermitian (since $\boldsymbol{\vec{M}}$ is Hermitian) we have $m_{ij} = m_{ji}^{*}$ and $\vec{s}^T \boldsymbol{\vec{M}^{-1}} \vec{Z}^{*}$ is the complex conjugate of $\vec{Z}^T \ \boldsymbol{\vec{M}^{-1}} \ \vec{s}^{*}$, so $m_{ij} = m_{ji}^{*}$ and consecutively (119) becomes:+where $m_{ij}$ are the elements of the matrix $\boldsymbol{\vec{M}^{-1}}$. Since $\boldsymbol{\vec{M}^{-1}}$ is Hermitian (because $\boldsymbol{\vec{M}}$ is Hermitian) we have $m_{ij} = m_{ji}^{*}$ and $\vec{s}^T \boldsymbol{\vec{M}^{-1}} \vec{Z}^{*}$ is the complex conjugate of $\vec{Z}^T \ \boldsymbol{\vec{M}^{-1}} \ \vec{s}^{*}$, so $m_{ij} = m_{ji}^{*}$ and consecutively (119) becomes:
  
 \begin{equation} \begin{equation}
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 where $\vec{k} = \boldsymbol{\vec{M}^{-1}} \vec{s}^{*}$, and to compare the result with the threshold. where $\vec{k} = \boldsymbol{\vec{M}^{-1}} \vec{s}^{*}$, and to compare the result with the threshold.
  
-Note that $\vec{s}$ is the vector associated with the received signal in the absence of noise. In the case of white noise (less than a constant factor), $\vec{k} = \vec{s}^{*}$.+Note that $\vec{s}$ is the vector associated with the received signal in absence of noise. In the case of white noise (except for a constant factor), $\vec{k} = \vec{s}^{*}$.
  
-If the vector $\vec{s}$ is considered as the impulse response of a discrete type of FIR filter, there is a filtering operation that provides an output value at a precise instant.The filter described by the vector $\vec{s}^{*}$ assumes the form of discrete-time matched filter to the signal waveform emitted from the target in the case of white noise. In the continous case, if $s(t)$ is the waveform emitted by the target the adaprive filter to this waveform (remember that this filter allow us to obtain the maximum signal/noise ratio in output) is expressed as+If the vector $\vec{s}$ is considered as the impulsive response of a discrete type of FIR filter, there is a filtering operation that provides an output value at a precise instant.The filter described by the vector $\vec{s}^{*}$ assumes the form of discrete-time matched (to the signal waveform emitted from the target in the case of white noise) filter. In the continuous case, if $s(t)$ is the waveform emitted by the target, the adaptive filter to this waveform (remember that this filter allow us to obtain the maximum signal/noise ratio in output) is expressed as
  
 \begin{equation} \begin{equation}
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-It is noted that it is necessary to know the parameters of the target values $A$, $f_D$, $\Phi$. If the density functions are not Gaussian, the likelihood ratio may not be written easily.+Note that it is necessary to know the parameters of the target values $A$, $f_D$, $\Phi$. If the density functions are not Gaussian, the likelihood ratio may not be written easily.
  
 ===== Maximization of the Signal / Noise ratio =====  ===== Maximization of the Signal / Noise ratio ===== 
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 //Finding the maximum Signal / Noise ratio using optimal linear processor for a known deterministic signal. //Finding the maximum Signal / Noise ratio using optimal linear processor for a known deterministic signal.
 Application of the optimal filtering for an unfluctuating moving target. Application of the optimal filtering for an unfluctuating moving target.
-optimal linear processor for a deterministic signal with unknown phase.//+Optimal linear processor for a deterministic signal with unknown phase.//
  
-==== optimal processor for a known deterministic signal ====+==== Optimal processor for a known deterministic signal ====
  
 In the general case, we prefer to operate with a suboptimal decision criterion based on the research of the maximum signal/noise ratio. In case of Gaussian random variables we will see that we will arrive to the same results as the Neyman-Pearson's one. In the general case, we prefer to operate with a suboptimal decision criterion based on the research of the maximum signal/noise ratio. In case of Gaussian random variables we will see that we will arrive to the same results as the Neyman-Pearson's one.
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 These samples must be processed in a linear manner (by using a FIR filter) in such a way that at a certain instant (//instant of decision//), is maximized the signal/noise ratio at the output. These samples must be processed in a linear manner (by using a FIR filter) in such a way that at a certain instant (//instant of decision//), is maximized the signal/noise ratio at the output.
  
-Let $\vec{h}$ be the vector of coefficients of the FIR filter and $v$ is the filter output at the instant of decision, that is (when the sequence of n samples received is aligned with that of the+Let $\vec{h}$ be the vector of coefficients of the FIR filter and $v$ is the filter output at the instant of decision, that is (when the sequence of n samples received, is aligned with the sequence of the
 coefficients) equal to coefficients) equal to
  
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 where  $\boldsymbol{\vec{M}}$ is the covariance matrix of the noise which has zero mean. where  $\boldsymbol{\vec{M}}$ is the covariance matrix of the noise which has zero mean.
-The power of the noise in output of the filter defined by the vector $\vec{h}$ is expressable as a function of the filter coefficients. Let's suppose that the covariance matrix of the noise $\boldsymbol{\vec{M}}$ is known (or estimated). Let's consider for now the case of the deterministic signal $\vec{s}$; The power of the useful signal $S$ at the output of the processor is, by definition, equal to +The power of the noise in output of the filter defined by the vector $\vec{h}$, can be expressed as a function of the filter coefficients. Let's suppose that the covariance matrix of the noise $\boldsymbol{\vec{M}}$ is known (or estimated). Let's consider for now the case of the deterministic signal $\vec{s}$; The power of the useful signal $S$ at the output of the processor is, by definition, equal to 
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-where $\vec{s}$ is the vector of the  samples of the useful signal. The indicated quantities are obtained sampling in time (and so in Range) the radar signal in the point where we expect that the target echo (point) is maximum. Since the useful signal is deterministic, the signal/noise ratio (SNR) at the output of the linear processor is expressed as:+where $\vec{s}$ is the vector of the  samples of the useful signal. The indicated quantities are obtained sampling in time (and so in Range) the radar signal in the point where we expect that the target (point) echo is maximum. Since the useful signal is deterministic, the signal/noise ratio (SNR) at the output of the linear processor is expressed as:
  
 \begin{equation} \begin{equation}
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 __Note__: __Note__:
  
-In general, the useful signal can be a random process. In this case, it can be described by the vector of expected values and with the $\boldsymbol{\vec{M_s}}$ covariance matrix. +In general, the useful signal can be a random process. In this case, it can be described by the vector of the expected values and with the $\boldsymbol{\vec{M_s}}$ covariance matrix. 
-If the vector of expected values is null, the signal/noise ratio at the filter output is equal to+If the vector of the expected values is null, the signal/noise ratio at the filter output is equal to
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-In case of the useful signal is "white" (this corresponds to the standard definition of the improvement factor) $\boldsymbol{\vec{M_s}} = \boldsymbol{\vec{I}}$, solving the problem of the optimal filtering is equal to the problem of finding the eigenvalues of the covariance matrix of the noise (clutter + thermal noise) $\boldsymbol{\vec{M}}$, and in particular, to find the vector of optimal coefficients $\vec{h_0}$  is the eigenvector of $\boldsymbol{\vec{M}}$ associated to the minimum eigenvalue.+In case of the useful signal is "white" (this corresponds to the standard definition of the improvement factor) $\boldsymbol{\vec{M_s}} = \boldsymbol{\vec{I}}$, solving the problem of the optimal filtering is equal to the problem of finding the eigenvalues of the covariance matrix of the noise (clutter + thermal noise) $\boldsymbol{\vec{M}}$, and in particular, to find the vector of optimal coefficients $\vec{h_0}$, that  is the eigenvector of $\boldsymbol{\vec{M}}$ associated to the minimum eigenvalue.
  
-Knwoing that, we can express this theorem:+Knowing that, we can express this theorem:
  
  
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-where $\mu$ is non relevant constant.+where $\mu$ is a non relevant constant.
  
 Note that (135) is the same as (122), result already obtained by applying the Neyman-Pearson criterion in the case in which the statistics are Gaussian. Note that (135) is the same as (122), result already obtained by applying the Neyman-Pearson criterion in the case in which the statistics are Gaussian.
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 where $\rho$ is the correlation coefficient and $\sigma$ represents the power of the noise. where $\rho$ is the correlation coefficient and $\sigma$ represents the power of the noise.
  
-$\rho = \rho(\tau)$ by definition, that represents the correlation coefficient between two consecutive pulses, one observed at time $t=0$ and the other at time $t=T=PRT$ (see figure {{ref>prt_pulses}})+$\rho = \rho(\tau)$ by definition, represents the correlation coefficient between two consecutive pulses, one observed at time $t=0$ and the other at time $t=T=PRT$ (see figure {{ref>prt_pulses}})
  
 <figure prt_pulses> <figure prt_pulses>
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 \begin{equation} \begin{equation}
-S(f) = \frac{1}{\sqrt{2 \pi} \sigma_f} e^{-\frac{f^2}{2 {\sigma_f}^2}}+S(f) = \frac{1}{\sqrt{2 \pi} \sigma_f} e^{-\frac{f^2}{2 {\sigma_f^2}}}
 \end{equation} \end{equation}
  
  
-The equation (148) is utilized, in general, in the metereological radars to represent the spectrum of the rainy phenomena.+The equation (148) is utilized, in general, in the meteorological radars to represent the spectrum of the rainy phenomena.
 If we antitransform the (148) we have the expression of the correlation coefficient $\frac{R(\tau)}{R(0)}$ to which we will refer as $\rho(\tau)$ If we antitransform the (148) we have the expression of the correlation coefficient $\frac{R(\tau)}{R(0)}$ to which we will refer as $\rho(\tau)$
  
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-Since the white thermal noise contribution is scorrelated from the clutter contribution (produced for example by the atmospheric agents), it can be added in power. Consecutively the covariance matrix $\boldsymbol{\vec{M}}$ of the overall noise process ($\sigma_n^2$ for the noise, $\sigma_c^2$ for the clutter and $\rho_c$ as correlation coefficient) is: +Since the white thermal noise contribution is uncorrelated from the clutter contribution (produced for example by the atmospheric agents), it can be added in power. Consecutively the covariance matrix $\boldsymbol{\vec{M}}$ of the overall noise process ($\sigma_n^2$ for the noise, $\sigma_c^2$ for the clutter and $\rho_c$ as correlation coefficient) is: 
  
  
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 \end{equation} \end{equation}
  
-using (135) the optimal filter coffieceints are+using (135) the optimal filter cofficients are
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-knwoing the vector of the observed samples $\vec{z}^T = \begin{bmatrix} z_1 \ & \ z_2 \end{bmatrix}$, the output of the filter is+knowing the vector of the observed samples $\vec{z}^T = \begin{bmatrix} z_1 \ & \ z_2 \end{bmatrix}$, the output of the filter is
  
 \begin{equation} \begin{equation}
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-To see the gain introduced by the optimal linear filter we observe that before the processing the signal/noise ratio on a single pulse $(S/N)_1$ is equal to +To see the gain introduced by the optimal linear filter we observe that before the processing, the signal/noise ratio on a single pulse $(S/N)_1$ is equal to 
  
  
 \begin{equation} \begin{equation}
-\left ( \frac{S}{N}_{1} \right) = \frac{1}{\sigma^2}  +\left ( \frac{S}{N} \right)_{1} = \frac{1}{\sigma^2}  
 \end{equation} \end{equation}
  
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 \begin{equation} \begin{equation}
-\left ( \frac{S}{N}_{OUT} \right) = \vec{s}^T \ \boldsymbol{\vec{M}^{-1}} \ \vec{s}^{*} = \frac{2}{{\sigma}^2 (1 + \rho)}+\left ( \frac{S}{N} \right)_{OUT} = \vec{s}^T \ \boldsymbol{\vec{M}^{-1}} \ \vec{s}^{*} = \frac{2}{{\sigma}^2 (1 + \rho)}
 \end{equation} \end{equation}
  
  
-We observe that the improvement obtained is lower compared to the case when we have only white noise, this is caused by the contribution of a colored noise.+We observe that the improvement obtained is lower compared to the case when we have only white noise, this is caused by the contribution of a coloured noise.
 As much as the coefficient $\rho$ is lower, the improvement will be bigger, if $\rho = 0$ we have an improvement equal to 2 (in general equal to $n$). As much as the coefficient $\rho$ is lower, the improvement will be bigger, if $\rho = 0$ we have an improvement equal to 2 (in general equal to $n$).
-If $\rho \rightarrow 1$, the spectrum of the noise and the spectrum of the useful signal tend to be overlapped and the improvement tends to zero. Note that the spectral situation in figure {{ref>spectrum_noise_signal_doppler}}, where we are supposing that the useful signal has a doppler frequency $f_D$ such that the representative raw of the signal is not overlapped with the spectrum of the noise. In this case, the output gain improvement of the optimal linear processor can be very high and this is due to the fact that the spectrums are not overlapped.+If $\rho \rightarrow 1$, the spectrum of the noise and the spectrum of the useful signal tend to be overlapped and the improvement tends to zero. Note that the spectral situation in figure {{ref>spectrum_noise_signal_doppler}}, where we are supposing that the useful signal has a doppler frequency $f_D$ such that the representative line of the signal is not overlapped with the spectrum of the noise. In this case, the output gain improvement of the optimal linear processor can be very high and this is due to the fact that the spectrums are not overlapped.
 For example, consider the case where the Doppler frequency of the useful signal is equal to half of the PRF value (see figure {{ref>spectrum_noise_signal_doppler}}) For example, consider the case where the Doppler frequency of the useful signal is equal to half of the PRF value (see figure {{ref>spectrum_noise_signal_doppler}})
  
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-To perform the test it is necessary to calculate $w$ and consequently it is necessary to know the initial phase $\Phi_0$ associated with the target signal.+To perform the test is necessary to calculate $w$ and consequently it is necessary to know the initial phase $\Phi_0$ associated with the target signal.
 If the base band representation of the useful signal is If the base band representation of the useful signal is
  
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 \end{equation} \end{equation}
  
-where $\Phi_i$ is the difference of phase between the sample $i$ and the first sample (with $i=0$) and it's related to the doppler effect.+where $\Phi_i$ is the difference of phase between the sample $i$ and the first sample (with $i=0$) and __it's related to the doppler effect__.
 In particular we have: In particular we have:
  
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-==== optimal processor for deterministic signal with $\Phi_0$ unknown ====+==== Optimal processor for deterministic signal with $\Phi_0$ unknown ====
  
-the phase $\Phi_0$, that is independent from the doppler frequency (see (170)) not known a priori.+the phase $\Phi_0$, that is independent from the doppler frequency (see (170)), is not known a priori.
  
 We are going to analize what is the optimal test to do when the initial phase $\Phi_0$ is unknown. We are going to analize what is the optimal test to do when the initial phase $\Phi_0$ is unknown.
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 that depends from $\Phi_0$ because $\vec{k}$ depends from $\Phi_0$. that depends from $\Phi_0$ because $\vec{k}$ depends from $\Phi_0$.
  
-Note that the joint probability density of $\rho$ and $\Phi$, given the event $M_0$ (that represents the absence of useful signal) can be written as+Note that the joint probability density of $\rho$ and $\Phi$, given the event $M_0$, (that represents the absence of useful signal) can be written as
  
 \begin{equation} \begin{equation}
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 When the phase information is not known, the optimum test consists of __linear__ signal processing that maximizes the S / N ratio (consistent integration), followed by envelope detection and threshold comparison. This type of detection is less efficient than the completely coherent detection (where we suppose to know the initial phase) (see figure {{ref>optimum_linear_processor_known_signal}}) When the phase information is not known, the optimum test consists of __linear__ signal processing that maximizes the S / N ratio (consistent integration), followed by envelope detection and threshold comparison. This type of detection is less efficient than the completely coherent detection (where we suppose to know the initial phase) (see figure {{ref>optimum_linear_processor_known_signal}})
  
-Unknowing the phase factor $\Phi_0$ results in a loss of sensitivity of the receiver thatcan be quantified by calculating the probability of correct revelation in both cases (knowledge of $\Phi_0$ and not).+Unknowing the phase factor $\Phi_0$ results in a loss of sensitivity of the receiver that can be quantified by calculating the probability of correct revelation in both cases (knowledge of $\Phi_0$ and not).
  
-The figure {{ref>reciever_sensitivity_phi_known_unknow}} represents the probability of detection in the two cases: $\Phi_0$ known (continous curve) and $\Phi_0$ unknown (dotted curve) and we can see that: for the same SNR, if $\Phi_0$ is unknown, the probability of detection is lower. The variation of the signal/noise ratio $\Delta SNR$ in the two cases, for the same probability of detection (and the same $p_fa$), indicates the loss of sensitivity of the reciever.+The figure {{ref>reciever_sensitivity_phi_known_unknow}} represents the probability of detection in the two cases: $\Phi_0$ known (continuous curve) and $\Phi_0$ unknown (dotted curve) and we can see that: for the same SNR, if $\Phi_0$ is unknown, the probability of detection is lower. The variation of the signal/noise ratio $\Delta SNR$ in the two cases, for the same probability of detection (and the same $p_{fa}$), indicates the loss of sensitivity of the reciever.
  
  
 <figure reciever_sensitivity_phi_known_unknow> <figure reciever_sensitivity_phi_known_unknow>
 {{ :media:reciever_sensitivity_phi_known_unknow.png?nolink |}} {{ :media:reciever_sensitivity_phi_known_unknow.png?nolink |}}
-<caption>Reciever sensitivity in the two cases: $\Phi_0$ known (continous curve) and $\Phi_0$ unknown (dotted curve)[(cite:RadarBook)]</caption>+<caption>Reciever sensitivity in the two cases: $\Phi_0$ known (continuous curve) and $\Phi_0$ unknown (dotted curve)[(cite:RadarBook)]</caption>
 </figure> </figure>
  
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