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radar:measurements [2018/06/05 14:58] – dipaoloradar:measurements [2026/04/28 16:58] (current) – mauro
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->it is ok for the moment. You can start  --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/04/19 11:25// 
->  Where do the figures come from? Please cite the document as decribed in [[:start|Welcome!]] --- //[[webmaster@localhost|DokuWiki Administrator]] 2018/05/03 16:16// 
  
 ====== Radar Measurement ====== ====== Radar Measurement ======
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 ===== Range ===== ===== Range =====
 +
 +//An overall discussion about radar techniques to detect the range of a given target//
  
 To start our dissertation, let's assume that we have a radar system that is able to emit pulses of energy in any direction of the space in order to detect some desired objects. The shape of the transmitted pulse can be modelled as in fig.1. It has a rectangular envelope which duration is $\tau$ and its carrier is a sinusoidal waveform which has a wavelength equal to $\lambda$ and so frequency $f_0 = c/\lambda$. To start our dissertation, let's assume that we have a radar system that is able to emit pulses of energy in any direction of the space in order to detect some desired objects. The shape of the transmitted pulse can be modelled as in fig.1. It has a rectangular envelope which duration is $\tau$ and its carrier is a sinusoidal waveform which has a wavelength equal to $\lambda$ and so frequency $f_0 = c/\lambda$.
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 ===== Angle ===== ===== Angle =====
 +
 +//Angular detection of the target, starting from knowing how a given antenna is able to detect the presence of a taget in the space//
  
 To univocally identify the position of a target we must measure not only the distance but also the angle from which the scattered energy comes from. For this reason, very directive antennas are used, i.e. antennas that perform a radiation diagram with a maximum along a given direction.  To univocally identify the position of a target we must measure not only the distance but also the angle from which the scattered energy comes from. For this reason, very directive antennas are used, i.e. antennas that perform a radiation diagram with a maximum along a given direction. 
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 According to the requirements, we have to build in a proper way the antenna to get the right beamshaping for azimuth and elevation parameters. If we want a high angular resolution on both planes ($\theta$ and $\phi$) we need an antenna with a very narrow beam usually known as //Pencil Beam//. This kind of structure requires that $\theta = \phi$. Otherwise, if we don't need so much accuracy in measuring on the vertical plane we can realize an antenna with a narrow beam only on the horizontal plane. In this case we are dealing with the so-called //Fan Beam// antenna. If we use this last kind of beamshaping we are able to measure with a great precision only in distance and azimuth.\\ Such devices are used in the control of the air traffic, but only when the target can directly communicate its altitude. According to the requirements, we have to build in a proper way the antenna to get the right beamshaping for azimuth and elevation parameters. If we want a high angular resolution on both planes ($\theta$ and $\phi$) we need an antenna with a very narrow beam usually known as //Pencil Beam//. This kind of structure requires that $\theta = \phi$. Otherwise, if we don't need so much accuracy in measuring on the vertical plane we can realize an antenna with a narrow beam only on the horizontal plane. In this case we are dealing with the so-called //Fan Beam// antenna. If we use this last kind of beamshaping we are able to measure with a great precision only in distance and azimuth.\\ Such devices are used in the control of the air traffic, but only when the target can directly communicate its altitude.
  
-It is quite easy to recognize which kind of beam an antenna is using just watching to its physical dimensions. For example, if an antenna is wider than longer its beam should be a Fan Beam. Otherwise is a squared antenna is used, it produces a Pencil Beam.+It is quite easy to recognize which kind of beam an antenna is using just watching to its physical dimensions. For example, if an antenna is wider than longer its beam should be a Fan Beam. Otherwise if a squared antenna is used, it produces a Pencil Beam.
  
 Using a Pencil Beam phased array it is possible to realize the so-called //3D Stacked Beams// radar. The electronic scanning can be performed both on elevation and azimuth planes, but usually we prefer to perform a mechanical scanning on the azimuth plane and the electronic one on the elevation plane. In this way, we are able to measure also the altitude of the target. For this reason, this radar is called 3D, because it is able to locate the target in the 3-dimensional space giving the $R, \theta$ and $\phi$ coordinates at the same time. Using a Pencil Beam phased array it is possible to realize the so-called //3D Stacked Beams// radar. The electronic scanning can be performed both on elevation and azimuth planes, but usually we prefer to perform a mechanical scanning on the azimuth plane and the electronic one on the elevation plane. In this way, we are able to measure also the altitude of the target. For this reason, this radar is called 3D, because it is able to locate the target in the 3-dimensional space giving the $R, \theta$ and $\phi$ coordinates at the same time.
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 \end{equation} \end{equation}
  
-If we take into account also antenna losses we must multiply the effective area for the loss factor $L_a$ that is less the 1. So we have:+If we take into account also antenna losses we must multiply the effective area for the loss factor $L_a$ that is less than 1. So we have:
  
 \begin{equation} \begin{equation}
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 ===== Velocity ===== ===== Velocity =====
 +
 +//Analysis of the velocity parameter of the target, with a main focus on the Doppler frequency//
  
 Once we have defined the previous parameters that allow us to measure the distance from the radar, another information that can be derived from the echo signal is the radial velocity of the target. Of course, this quantity is revealed by the radar only if the target is moving. Once we have defined the previous parameters that allow us to measure the distance from the radar, another information that can be derived from the echo signal is the radial velocity of the target. Of course, this quantity is revealed by the radar only if the target is moving.
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 === Maximum unambiguous velocity === === Maximum unambiguous velocity ===
  
-If a pulse radar sends a series of rectangular pulses with a period $PRT$, the output of the mixer is no more a sinusoidal signal of frequency $f_D$ as in the case of CW radar, but it is a sampling of this waveform at the instants $t_k = k \cdot PRT$. So, using a pulse radar is equivalent to introducing a sampling operation above the received signal with a period equal to $PRT$ equally spaced of multiples of $PRT$.\\ It is clear that if sampling is not performed in the right way we can incur the risk of //"aliasing"// over the value of $f_D$.\\ Let's suppose for example that the representative phasor of Doppler frequency has the following behaviour:+If a pulse radar sends a series of rectangular pulses with a period $PRT$, the output of the mixer is no more a sinusoidal signal of frequency $f_D$ as in the case of CW radar, but it is a sampling of this waveform at the instants $t_k = k \cdot PRT$. So, using a pulse radar is equivalent to introduce a sampling operation above the received signal with a period equal to $PRT$ equally spaced of multiples of $PRT$.\\ It is clear that if sampling is not performed in the right way we can incur the risk of //"aliasing"// over the value of $f_D$.\\ Let's suppose for example that the representative phasor of Doppler frequency has the following behaviour:
   *$f_D = 3/4PRF$, so $2\pi f_D T = \frac{3}{2}\pi$.\\ In this case, the phasor that rotates counterclockwise with an angular speed of $\omega_D = 2\pi f_D$ is confused with another phasor that rotates with an angular speed $\frac{3\pi}{2T} - \frac{2\pi}{T} = -\frac{\pi}{2T}$ i.e, it seems to rotate clockwise and with a speed that is one-third of the real one. This is shown in fig.17.   *$f_D = 3/4PRF$, so $2\pi f_D T = \frac{3}{2}\pi$.\\ In this case, the phasor that rotates counterclockwise with an angular speed of $\omega_D = 2\pi f_D$ is confused with another phasor that rotates with an angular speed $\frac{3\pi}{2T} - \frac{2\pi}{T} = -\frac{\pi}{2T}$ i.e, it seems to rotate clockwise and with a speed that is one-third of the real one. This is shown in fig.17.
  
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 </figure> </figure>
  
-In general, between the "folded" Doppler frequency $f^*$ comprised between $-\frac{1}{2T}$ and $\frac{1}{2T}$ and the real Doppler frequency $f_D = -\frac{2v_R}{\lambda}$ stands the Nyquist relation:+In general, between the "folded" Doppler frequency $f^*$ (comprised between $-\frac{1}{2T}$ and $\frac{1}{2T}$) and the real Doppler frequency $f_D = -\frac{2v_R}{\lambda}$ stands the Nyquist relation:
  
 \begin{equation} \begin{equation}
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 \end{equation} \end{equation}
  
-where $v_{max}$ is the maximum value of the target's speed.\\ Coversely, once a certain value of a $PRF$ is assigned, the maximum value of the Doppler frequency that we are able to measure without ambiguity is:+where $v_{max}$ is the maximum value of the target's speed.\\ Conversely, once a certain value of a $PRF$ is assigned, the maximum value of the Doppler frequency that we are able to measure without ambiguity is:
  
 \begin{equation} \begin{equation}
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 ===== Ambiguity cases ===== ===== Ambiguity cases =====
 +
 +//According to the characteristics of our radar system, we can incur some troubles about the accuracy of the measurements. Let's see it in detail//
  
 We can now understand that when we have to define the working characteristics of our radar system, we have to take into account the right choice of the $PRF$ value. According to its value, we have pointed out the following: We can now understand that when we have to define the working characteristics of our radar system, we have to take into account the right choice of the $PRF$ value. According to its value, we have pointed out the following:
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 If we choose a value for the $PRT$ like in the last equation and the maximum radial velocity of the target $v_{rlim}$ is greater than the value $v_{max}$, we have realized a system that is unambiguous in distance but not in velocity. If we choose a value for the $PRT$ like in the last equation and the maximum radial velocity of the target $v_{rlim}$ is greater than the value $v_{max}$, we have realized a system that is unambiguous in distance but not in velocity.
  
-So if we put on the plane in the next figure the distance and velocity values that identify the position of a given target in a given instant of time (points A, B, C), the radar will detect the situation showed in fig.b.+So if we put on the plane in the next figure the distance and velocity values that identify the position of a given target in a given instant of time (points A, B, C), the radar will detect the situation showed in fig.19b.
  
 <figure label> <figure label>
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 </figure> </figure>
  
-Fig.19b is obtained by "horizontally folding" upon the origin the rectangular portions on the $R-v$ plane that have a base equal to $\lambda PRF/2$ such to overlap them on the central portion, having $-\lambda PRF/4 < v < \lambda PRF/4$ in fig.a.\\ Fig.19c shows the distance-velocity plane with the velocity folded only on the positive plane.+Fig.19b is obtained by "horizontally folding" upon the origin the rectangular portions on the $R-v$ plane that have a base equal to $\lambda PRF/2$ such to overlap them on the central portion, having $-\lambda PRF/4 < v < \lambda PRF/4$ in fig.19a.\\ Fig.19c shows the distance-velocity plane with the velocity folded only on the positive plane.
  
 === Velocity unambiguous Radar === === Velocity unambiguous Radar ===
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 </figure> </figure>
  
-Fig.20b is obtained by a "vertical folding" over the rectangular portion centred on the origin and a length $R_{max} = c/(2PRF)$ of the other rectangular portion on the plane.+Fig.20b is obtained by a "vertical folding" over the rectangular portion centred on the origin and of a length $R_{max} = c/(2PRF)$ of the other rectangular portion on the plane.
  
 === Distance and velocity ambiguous Radar === === Distance and velocity ambiguous Radar ===
  
-Now suppose to choose a PRF value such that called $R_{max} = c/(2PRF)$ and $v_{rmax} = \lambda PRF/4$ the maximum unambiguous distance and velocity rispectively, and $R_{lim}$ and $v_{lim}$ the maximum target distance and velocity. If it happens that $v_{rmax} < v_{rlim}$ and $R_{max} < R_{lim}$ we get a radar system that is both ambiguous in distance and velocity.+Now suppose to choose a PRF value such that called $R_{max} = c/(2PRF)$ and $v_{rmax} = \lambda PRF/4$ the maximum unambiguous distance and velocity respectively, and $R_{lim}$ and $v_{lim}$ the maximum target distance and velocity. If it happens that $v_{rmax} < v_{rlim}$ and $R_{max} < R_{lim}$ we get a radar system that is both ambiguous in distance and velocity.
  
-This situation does not fit very well with our purposes to use the radar, because it is clear that we can easily incur in some error during the measurements. Once again it is shown that the choice of a right value for the $PRF$ is a key aspect when a radar system is developed united with the right choice of the working band.+This situation does not fit very well with our purposes to use the radar, because it is clear that we can easily incur in some errors during the measurements. Once again it is shown that the choice of a right value for the $PRF$ is a key aspect when a radar system is developed in conjunction with the right choice of the working band.
  
 === RVP === === RVP ===
  
-We can now define a value that can help us to rightly choose the value of the $PRF$ in order to better satisfy the user requirements. It is the product of the maximum unambiguous quantity, it is called //Range Velocity Product (RVP)// and it is equal to:+We can now define a value that can help us to rightly choose the value of the $PRF$ in order to better satisfy the user requirements. It is the product of the maximum unambiguous quantities, it is called //Range Velocity Product (RVP)// and it is equal to:
  
 \begin{equation} \begin{equation}
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 ===== Dwell Time ===== ===== Dwell Time =====
 +
 +//Analysis of the Dwell time and the corresponding number of pulses available for the eleboration//
  
 Assume to work with a pulse radar whose beam is able to move to scan all the space. If there is an object that can be approximated by a point at a certain distance it will be lighted up by the -3dB beamwidth of the antenna for a certain period of time. This target will be hit by a certain number $N$ of pulses transmitted by the radar. Consequently, on the receiving side, we will detect $N$ different echoes, attenuated and delayed, associated to the target. By definition, this number $N$ is equal to: Assume to work with a pulse radar whose beam is able to move to scan all the space. If there is an object that can be approximated by a point at a certain distance it will be lighted up by the -3dB beamwidth of the antenna for a certain period of time. This target will be hit by a certain number $N$ of pulses transmitted by the radar. Consequently, on the receiving side, we will detect $N$ different echoes, attenuated and delayed, associated to the target. By definition, this number $N$ is equal to:
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 \end{equation} \end{equation}
  
-This new variable $t_D$ is defined as //Dwell Time// and is the timeframe during which the object is lighted up by the -3dB beamwidth.\\ If we perform an azimuthal scanning with a constant angular speed $\dot{\theta}$ and the width of the beam is equal to $\theta_B$, the definition of the dwell time is the sequent:+This new variable $t_D$ is defined as //Dwell Time// and is the timeframe during which the object is lighted up by the -3dB beamwidth.\\ If we perform an azimuthal scanning with a constant angular speed $\dot{\theta}$ and the width of the beam is equal to $\theta_B$, the definition of the dwell time is the following:
  
 \begin{equation} \begin{equation}
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 ===== Temporal scales for radar signal ===== ===== Temporal scales for radar signal =====
  
-In this chapter all the measurements that a radar system is able to perform have been defined, taking into account the presence of a target approximable to a point, i.e. the object is very small with respect to the dimensions of the radar cell.\\ They are the sequent:+//A brief recall of all the temporal concepts seen up to now// 
 + 
 +In this chapter all the measurements that a radar system is able to perform have been defined, taking into account the presence of a target approximable to a point, i.e. the object is very small with respect to the dimensions of the radar cell.\\ They are the following:
   -Range;   -Range;
   -Elevation angle;   -Elevation angle;
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   -Doppler frequency and radial velocity.   -Doppler frequency and radial velocity.
  
-The results that we can obtain with respect to angular measurements depend substantially on the width of the antenna beam, so they can be optimized by a further improving the antenna requirements obtaining narrower lobes. On the other hand, when we talk about range and velocity we can incur in some ambiguity related to multiples of $v_{rmax}$ and $R_{max}$.+The results that we can obtain with respect to angular measurements depend substantially on the width of the antenna beam, so they can be optimized by further improving the antenna requirements obtaining narrower lobes. On the other hand, when we talk about range and velocity we can incur in some ambiguity related to multiples of $v_{rmax}$ and $R_{max}$.
  
 If we analyze the reception phenom about the pulses that were previously transmitted by the radar, we can divide the temporal axis into three different scales used to better understand the radar behaviour.\\ The first one is the one related to the pulse length, of the order of microseconds; then there is the one related to pulse repetition time larger than the previous, of the order of milliseconds; at last, there is the dwell time, of the order of milliseconds too. The sketch in the next figure reports the aforementioned situation.    If we analyze the reception phenom about the pulses that were previously transmitted by the radar, we can divide the temporal axis into three different scales used to better understand the radar behaviour.\\ The first one is the one related to the pulse length, of the order of microseconds; then there is the one related to pulse repetition time larger than the previous, of the order of milliseconds; at last, there is the dwell time, of the order of milliseconds too. The sketch in the next figure reports the aforementioned situation.   
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