RadarLab

User Tools

Site Tools

Trace: • scores • measurements • radarequation
radar:measurements

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
radar:measurements [2018/05/26 10:49] – dipaoloradar:measurements [2026/04/28 16:58] (current) – mauro
Line 1: Line 1:
->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 ======
  
-A radar system is able to measure with a great precision the distance of an object. It uses electromagnetic pulses to detect the presence of a target in the following way: a transmitter emits a pulse of energy towards a given direction and if it bumps into an object part of the energy will be sent back. This “retransmitted energy” is called ECHO and according to its amplitude, the receiver will decide whether the target is present or not (by the comparison of the received signal with a given threshold), its distance and, if the target is moving, also its radial velocity. The measuring techniques regarding these aspects are all analyzed in this chapter.+A radar system is able to measure with high accuracy the distance of an object. It uses electromagnetic pulses to detect the presence of a target in the following way: a transmitter emits a pulse of energy towards a given direction and if it bumps into an object part of the energy will be sent back. This “retransmitted energy” is called ECHO and according to its amplitude, the receiver will decide whether the target is present or not (by the comparison of the received signal with a given threshold), its distance and, if the target is moving, also its radial velocity. The measuring techniques regarding these aspects are all analyzed in this chapter.
  
 ===== Range ===== ===== Range =====
  
-To begin 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 $f_0 = c/\lambda$.+//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 $f_0 = c/\lambda$.
  
 <figure label> <figure label>
Line 30: Line 30:
 \end{equation} \end{equation}
  
-Once we know this value we can recover the value of $R_{T} + R_{R}$ that however does not give us the precise information about the distance of the target.\\ In practice, this kind of device is mainly used for weather radar. The bistatic radar technology has been in use for several years at the Institute of Atmospheric Physics at the German Aerospace Center. This system is also of some importance in military applications.+Once we know this value we can recover the value of $R_{T} + R_{R}$ that however does not give us the precise information about the distance of the target.\\ In practice, this kind of device is mainly used for weather radar. The bistatic radar technology has been in use for several years at the Institute of Atmospheric Physics at the German Aerospace Center. This system is also important in military applications.
  
 On the other hand, a lot of radar systems work with the same antenna on the $Tx$ and $Rx$ side. This is made possible to the presence of a special apparatus called //Duplexer// that disjoins the transmitting and receiving side. In this case, the distance that the e.m. wave has to cover in both paths is the same. We can then compute: On the other hand, a lot of radar systems work with the same antenna on the $Tx$ and $Rx$ side. This is made possible to the presence of a special apparatus called //Duplexer// that disjoins the transmitting and receiving side. In this case, the distance that the e.m. wave has to cover in both paths is the same. We can then compute:
Line 47: Line 47:
 == CW Radar systems == == CW Radar systems ==
  
-Until now it has been supposed that our radar system works with pulses of energy. Another typical waveform used by radar systems is the sinusoidal one. Radar using this approach are commonly known as //Continous Wave radar// (CW radar). It can be seen as a special type of bistatic radar because normally it uses two antennas for transmission. This radar schema was used in the first radar applications in history and nowadays it is almost deprecated because it doesn't allow to measure the distance of a target if the transmitted signal is not properly modulated. For this reason, FMCW radar (Frequency-Modulated Continuous Wave radar) is used.[(cite:Font >> title : http://www.radartutorial.eu, section : "Frequency-Modulated Continuous-Wave Radar (FMCW Radar)")] It is a special type of radar sensor which radiates continuous transmission power like a simple continuous wave radar. In contrast to this CW radar, FMCW radar can change its operating frequency during the measurement: that is, the transmission signal is modulated in frequency (or in phase). In this method, the transmitted signal increases or decreases the frequency periodically. When an echo signal is received, that change of frequency gets a delay $\Delta t = 2R/c$ (by runtime shift) like the one obtained using pulse radar technique. In FMCW radar are measured the differences in phase or frequency (according to the modulation performed) between the transmitted and the received signal. This mechanism is shown in fig.4.+Until now it has been supposed that our radar system works with pulses of energy. Another typical waveform used by radar systems is the sinusoidal one. Radar using this approach are commonly known as //Continous Wave radar// (CW radar). It can be seen as a special type of bistatic radar because normally it uses two antennas for transmission. This radar schema was used in the first radar applications in history and nowadays it is almost deprecated because it doesn't allow to measure the distance of a target if the transmitted signal is not properly modulated. For this reason, FMCW radar (Frequency-Modulated Continuous Wave radar) is used.[(cite:Font >> title : http://www.radartutorial.eu, section : "Frequency-Modulated Continuous-Wave Radar (FMCW Radar)")] It is a special type of radar sensor which radiates continuous transmission power like a simple continuous wave radar. In contrast to this CW radar, FMCW radar can change its operating frequency during the measurement: that is, the transmission signal is modulated in frequency (or in phase). In this method, the transmitted signal increases or decreases the frequency periodically. When an echo signal is received, that change of frequency gets a delay $\Delta t = 2R/c$ (by runtime shift) like the one obtained using pulse radar technique. In FMCW radar the differences in phase or frequency (according to the modulation performed) between the transmitted and the received signal are measured. This mechanism is shown in fig.4.
  
 <figure label> <figure label>
Line 110: Line 110:
 <figure label> <figure label>
 {{ :media:unambiguous.png?500 }} {{ :media:unambiguous.png?500 }}
-<caption>Ambiguouity problem in distance[(cite:Teoria)]</caption>+<caption>Ambiguity problem in distance[(cite:Teoria)]</caption>
 </figure> </figure>
  
 If the radar detects an echo in the position $A_2$, it cannot decide whether the object that generated the echo is due to the transmission of the first or second pulse because radar timing system reset to zero each time a pulse is radiated. In other words, we cannot decide whether the target is $ct_2/2$ or $(ct_2 + cT)/2$ far from the radar. The maximum range at which a target can be located so as to guarantee that it will be rightly detected is strictly related to the duration of the PRT. In other words, I want to receive the echo before I begin the transmission of the subsequent pulse. This range is called //maximum unambiguous range// or //first range ambiguity//. For this reason, pulse repetition time (PRT) of the radar is important when determining the maximum range because target return-times that exceed the PRT of the radar system appear at incorrect locations (ranges) on the radar screen. Returns that appear at these incorrect ranges are referred as ambiguous returns, second-sweep echoes or second time around echoes. Of course, if a target is placed exactly at a distance  $R = cPRT/2$ or something multiple than this quantity, it won't be revealed because its echo will arrive in the period of time that radar uses to transmit and the receiver is closed: this phenomenon is known as //"eclipsing loss"//. If the radar detects an echo in the position $A_2$, it cannot decide whether the object that generated the echo is due to the transmission of the first or second pulse because radar timing system reset to zero each time a pulse is radiated. In other words, we cannot decide whether the target is $ct_2/2$ or $(ct_2 + cT)/2$ far from the radar. The maximum range at which a target can be located so as to guarantee that it will be rightly detected is strictly related to the duration of the PRT. In other words, I want to receive the echo before I begin the transmission of the subsequent pulse. This range is called //maximum unambiguous range// or //first range ambiguity//. For this reason, pulse repetition time (PRT) of the radar is important when determining the maximum range because target return-times that exceed the PRT of the radar system appear at incorrect locations (ranges) on the radar screen. Returns that appear at these incorrect ranges are referred as ambiguous returns, second-sweep echoes or second time around echoes. Of course, if a target is placed exactly at a distance  $R = cPRT/2$ or something multiple than this quantity, it won't be revealed because its echo will arrive in the period of time that radar uses to transmit and the receiver is closed: this phenomenon is known as //"eclipsing loss"//.
 +
 +=== Range accuracy ===
 +
 +Accuracy is the degree of conformance between the estimated or measured position at a given time and its true position.\\ Accuracy **must not be confused with resolution**, that otherwise, is the radar capacity to distinguish between echoes that belong to different targets. So we can try to define what are the waveform parameters that our radar should have to reach good performances in terms of accuracy. 
 +Suppose to send a rectangular pulse whose width is $T$ at the instant $t = 0$ and that after a period $T_R$ we receive the echo associated to that pulse. Assuming that our receiving chain works with a matched filter approach (see next chapters) we can notice the situation described by the next figure.
 +
 +<figure label>
 +{{ :media:noise.png?500 }}
 +<caption>Noise effect on the range accuracy[(cite:Teoria)]</caption>
 +</figure>
 +
 +Fig.8a and fig.8b are the representation of the filtered signal and of the noise as they were treated separately. But when we filter a signal also an amount of noise enters the filter and at the exit of this one we have the situation shown by fig.8c. Due to the noise, we detect a distortion of the received signal. As we already know radar system is able to detect a target if the received signal overcomes a given threshold $V_T$ and at that instant we measure the distance of the object. It is clear now, looking at the fig.8c, that when the signal is distorted by the noise, we have some problems with the right detection of the delay because we are measuring in an instant of time that is different from the one that we use without the effect of the noise. So if we detect the target at the point C instead of B, we make a mistake in measuring the range of the target due to the wrong time estimation of the echo. Range accuracy is a function of the mean square error $\delta T_R$:
 +
 +\begin{equation}
 +\delta T_R = \sqrt{\overline{\Delta T_R}^2}
 +\end{equation}
 +
 +The overline indicates the statistical mean, while $\Delta T_R$ is the error in the measurements.
 +
 +Factors affecting measurement accuracy include not only noise and resolution but also signal and target characteristics and radar hardware considerations.\\ For example, any uncertainty in the antenna boresight angle, due for example to mounting or pattern calibration errors, will affect the accuracy of a location measurement. Radiofrequency (RF) hardware or antenna gain calibration errors (gain uncertainty) will affect the accuracy of target signal power measurements.[(cite:Accuracy >> title: Radar Measurements, authors: W. Dale Blair, Mark A. Richards, David A. Long, chapter 18)]\\ Signal propagation effects are another important factor in overall measurement accuracy: atmospheric refraction of the radar signal and multipath can introduce range measurement error. All of these effects shift the mean of the parameter measurement, thereby degrading the accuracy.
 +
 +The accuracy estimation is computed as the square root of the sum of the squared errors separately determined for
 +each error source. Specifically, given errors ${\delta_1, \delta_2,..., \delta_N}$ the RSS (root sum of squares) estimation of the total error $\delta$ is
 +
 +\begin{equation}
 +\delta = \sqrt{\delta_1^2 + \delta_2^2 + ... + \delta_N^2}
 +\end{equation}
 +
 +The RSS error effectively treats each error source as independent and Gaussian.
 +
 +Another source of accuracy and estimation error in surveillance radar is given to the fact that the received signal is sampled for further elaboration, in an asynchronous way with respect to the position of the echo. The bandwidth of the received signal is strictly related to the sampling frequency. To overcome this problem most of the times oversampling is performed, just to be sure that at most one of the samples can give us the maximum value of the filtered received signal. It is not always possible to choose a high number of samples because this will surely increase the manufacturing complexity and cost of the apparatus. 
  
 ===== 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. 
Line 125: Line 158:
   - **Radiation intensity**\\ It is the radiated power normalized with respect to the solid angle unit along a given direction\\ $I(\theta,\phi) = \frac{dP_r}{d\Omega}$\\ This value is independent from the distance to the antenna. In case of //Isotropic Source// it is also independent of the direction and it is simply $I_0 = P_r/4\pi$, i.e. the total radiated power divided by the solid angle. It is important to notice that for any given antenna this quantity represent the //average// of the radiation intensity.   - **Radiation intensity**\\ It is the radiated power normalized with respect to the solid angle unit along a given direction\\ $I(\theta,\phi) = \frac{dP_r}{d\Omega}$\\ This value is independent from the distance to the antenna. In case of //Isotropic Source// it is also independent of the direction and it is simply $I_0 = P_r/4\pi$, i.e. the total radiated power divided by the solid angle. It is important to notice that for any given antenna this quantity represent the //average// of the radiation intensity.
   - **Directivity**\\ It is the general radiation intensity normalized by the one of the isotropic radiator\\ $D(\theta,\phi) = \frac{I(\theta,\phi)}{P_r}4\pi$\\ It shows us how a given antenna is able to concentrate all the power towards a given angular position ($\theta,\phi$) with respect to the isotropic radiator assuming that the power radiated is the same.   - **Directivity**\\ It is the general radiation intensity normalized by the one of the isotropic radiator\\ $D(\theta,\phi) = \frac{I(\theta,\phi)}{P_r}4\pi$\\ It shows us how a given antenna is able to concentrate all the power towards a given angular position ($\theta,\phi$) with respect to the isotropic radiator assuming that the power radiated is the same.
-  - **Radiation diagram**\\ It is the graphic representation of the radiated power at a given angle. In fig.8 we can observe a section of the radiation diagram along the azimuth direction, $\theta$.\\ {{ :media:rad_diagram.png?500 }}\\ We are mainly interested in what happens when the gain is equal to $-3dB$ to univocally determine the characteristics of the antenna in terms of directivity. In fact, the lower the width of the main lobe the greater the value of the directivity.+  - **Radiation diagram**\\ It is the graphic representation of the radiated power at a given angle. In the following figure we can observe a section of the radiation diagram along the azimuth direction, $\theta$.\\ {{ :media:rad_diagram.png?500 }}\\ We are mainly interested in what happens when the gain is equal to $-3dB$ to univocally determine the characteristics of the antenna in terms of directivity. In fact, the lower the width of the main lobe the greater the value of the directivity.
  
 We need to be very careful because in radar theory we use to flip the role of $\theta$ and $\phi$ angles. All the considerations about radar theory follow the scheme in fig.9 in which $\theta$ is the //azimuth// angle, while $\phi$ is the //elevation// angle. We need to be very careful because in radar theory we use to flip the role of $\theta$ and $\phi$ angles. All the considerations about radar theory follow the scheme in fig.9 in which $\theta$ is the //azimuth// angle, while $\phi$ is the //elevation// angle.
Line 136: Line 169:
 To obtain a radar antenna we can use multiple solutions and each one has the aim to focus on the aperture an electric field with a certain amplitude and phase to reach a desired value for the far field. For example, in the microwaves field, we can realize two different systems as follows: To obtain a radar antenna we can use multiple solutions and each one has the aim to focus on the aperture an electric field with a certain amplitude and phase to reach a desired value for the far field. For example, in the microwaves field, we can realize two different systems as follows:
   ***Parabolic reflector**\\ Considering geometrical optic's models, the following property stands: if we put a feeder in a given point called //focus// all the electromagnetic rays that bump into the reflector are parallel one to another, towards the same direction. This allows obtaining the desired field distribution. {{ :media:parabolic.png?500 }}   ***Parabolic reflector**\\ Considering geometrical optic's models, the following property stands: if we put a feeder in a given point called //focus// all the electromagnetic rays that bump into the reflector are parallel one to another, towards the same direction. This allows obtaining the desired field distribution. {{ :media:parabolic.png?500 }}
-  ***Phased array**\\ The same can be done using an array that contains more than one radiating element. All the elements are fed by a //Beamforming Network// and with respect to the phase shift between one element and the other we are able to switch the direction of the beam. For example to obtain the same performance of a parabolic reflector all the elements must be in phase (fig.9), but if we want to point the beam towards another direction, different from the straightforward one, we must introduce a phase shift $\Delta\phi$ between the elements (fig.10). If the array performs this type of beam shaping it is called //Phased Array//. In this way, we can avoid using mechanical features to move the antenna because the beam moves according to the phase shift.+  ***Phased array**\\ The same can be done using an array that contains more than one radiating element. All the elements are fed by a //Beamforming Network// and with respect to the phase shift between one element and the other we are able to switch the direction of the beam. For example to obtain the same performance of a parabolic reflector all the elements must be in phase (fig.10), but if we want to point the beam towards another direction, different from the straightforward one, we must introduce a phase shift $\Delta\phi$ between the elements (fig.11). If the array performs this type of beam shaping it is called //Phased Array//. In this way, we can avoid using mechanical features to move the antenna because the beam moves according to the phase shift.
  
 <figure label> <figure label>
Line 159: Line 192:
 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.
Line 170: Line 203:
 === Resolution cell === === Resolution cell ===
  
-All the considerations made so far lead us to define the //radar angular resolution// as the interval of the solid angle delimited by $-3dB$ width of the main lobe, as shown in fig.13.+All the considerations made so far lead us to define the //radar angular resolution// as the interval of the solid angle delimited by $-3dB$ width of the main lobe, as shown in fig.14.
  
 <figure label> <figure label>
Line 216: Line 249:
 \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}
Line 223: Line 256:
  
 ===== 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.
Line 333: Line 368:
 === 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.15.+  *$f_D = 3/4PRF$, 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.
  
 <figure label> <figure label>
Line 341: Line 376:
 </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}
Line 347: Line 382:
 \end{equation} \end{equation}
  
-According to this law if the Doppler frequency of the received signal is less or equal to $PRF/2$ we do not have aliasing phenom. On the other hand, if $\lvert f_D\rvert > PRF/2$ we incur in undersampling that generates the aliasing phenom that shows up as a "folding" of the line at the frequency $f_D > PRF/2$ in the main interval that goes from $-PRF/2$ to $PRF/2$ (see fig.17).+According to this law if the Doppler frequency of the received signal is less or equal to $PRF/2$ we do not have aliasing phenom. On the other hand, if $\lvert f_D\rvert > PRF/2$ we incur in undersampling that generates the aliasing phenom that shows up as a "folding" of the line at the frequency $f_D > PRF/2$ in the main interval that goes from $-PRF/2$ to $PRF/2$ (see fig.18).
  
 <figure label> <figure label>
Line 362: Line 397:
 \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}
Line 371: Line 406:
  
 ===== 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:
Line 398: Line 435:
 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>
Line 405: Line 442:
 </figure> </figure>
  
-Fig.18b 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.18c 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 ===
Line 424: Line 461:
 </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}
Line 454: Line 491:
  
 ===== 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:
Line 461: Line 500:
 \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}
Line 475: Line 514:
 remembering that $1 rpm = \frac{2\pi}{60}[\frac{rad}{S}]$ and that $1^° = \frac{\pi}{180}$. remembering that $1 rpm = \frac{2\pi}{60}[\frac{rad}{S}]$ and that $1^° = \frac{\pi}{180}$.
  
-The physical meaning of dwell time is illustrated in fig.20.+The physical meaning of dwell time is illustrated in fig.21.
  
 <figure label> <figure label>
Line 484: Line 523:
 All the pulses emitted during the dwell time are scattered back and received as echoes by the radar. The aim of the radar is the elaboration of these pulses so we have to identify a temporal window delimitating the sequence of pulses that bumps into the target. This window, of course, has a duration $t_D$. All the pulses emitted during the dwell time are scattered back and received as echoes by the radar. The aim of the radar is the elaboration of these pulses so we have to identify a temporal window delimitating the sequence of pulses that bumps into the target. This window, of course, has a duration $t_D$.
  
-If the azimuthal scanning is performed in a mechanical way the amplitude of the received pulses are weighted by a proper function (as shown in the previous figure) that takes into account the fact that during the rotation the gain of the beam that light up the target is changing (fig.21).+If the azimuthal scanning is performed in a mechanical way the amplitude of the received pulses are weighted by a proper function (as shown in the previous figure) that takes into account the fact that during the rotation the gain of the beam that light up the target is changing (fig.22).
  
 <figure label> <figure label>
Line 519: Line 558:
 ===== 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;
Line 525: Line 566:
   -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.   
Line 536: Line 577:
 These scales are characterized by the values $\tau$, $T$, and $t_D$: the performance obtainable, in terms of accuracy and discrimination about velocity and range, strictly depends on these three values. In particular, it must be noticed the importance of the dwell time and so the number of pulses that the receiver can process to discriminate the Doppler frequency between different targets. These scales are characterized by the values $\tau$, $T$, and $t_D$: the performance obtainable, in terms of accuracy and discrimination about velocity and range, strictly depends on these three values. In particular, it must be noticed the importance of the dwell time and so the number of pulses that the receiver can process to discriminate the Doppler frequency between different targets.
  
-The corresponding frequency representation of the situation described by the last figure is shown in fig.23, assuming that N goes to infinity, i.e. the received signal has a period equal to $PRT$. +The corresponding frequency representation of the situation described by the last figure is shown in fig.24, assuming that N goes to infinity, i.e. the received signal has a period equal to $PRT$. The distance between two lines is equal to $PRF = 1/PRT$. 
  
 <figure label> <figure label>
Line 543: Line 584:
 </figure> </figure>
  
-If we assume to cut the signal off with a temporal window equal to $t_D$ with $N$ pulses, the spectrum in fig.23 makes a convolution with the Fourier transform of the temporal window that we can suppose to have a gaussian shape (and so its Fourier transform has a gaussian shape that lasts for $1/t_D$): in the next and last figure it is shown the spectrum of $R(f)$ that is the transform of each $r(n)$ obtained from eq.29. +If we use a pulse radar we can observe the received signal only in time instants associated to the pulse coming from a given echo. It is like to assume to cut the signal off with a temporal window equal to $t_D$ with $N$ pulses, and so the spectrum in fig.24 makes a convolution with the Fourier transform of the temporal window that we can suppose to have a gaussian shape (and so its Fourier transform has a gaussian shape that lasts for $1/t_D$): in the next and last figure it is shown the spectrum of $R(f)$ that is the transform of each $r(n)$ obtained from eq.29. 
  
 <figure label> <figure label>
Line 549: Line 590:
 <caption>Echo radar spectrum for a real case of an observation window that lasts for $t_D$. In the reality the width of $1/t_D$ for each lobe is much more smaller than $1/T$. [(cite:Teoria)]</caption> <caption>Echo radar spectrum for a real case of an observation window that lasts for $t_D$. In the reality the width of $1/t_D$ for each lobe is much more smaller than $1/T$. [(cite:Teoria)]</caption>
 </figure> </figure>
 +
 +This is just to say that if we consider that the Dwell time is finite and so also the number of pulses bumping into the target is limited, we cannot consider at the receiving side the whole spectrum resulting by the Fourier transform of the transmitted pulse. The window used to cut the signal lasts for a time equal to the Dwell time $t_D.$ This is why the resulting received signal has a shape like the one in fig.25: the result of the convolution between the sampled signal and the window implies a widening of every line in the spectrum equal to $1/t_d$.
 +
  
 More info in lecture notes FDR1 in diddatticaweb More info in lecture notes FDR1 in diddatticaweb
radar/measurements.1527331792.txt.gz · Last modified: (external edit)