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radar:radarequation [2018/06/08 15:51] – [Derivation of Radar Equation] sidorettiradar:radarequation [2026/04/28 15:13] (current) – external edit 127.0.0.1
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 ======Radar Equation====== ======Radar Equation======
 +
 +In this chapter we will show how the basic radar equation is derived, that describes the conditions under which a target can be detected by a radar. We say "basic" because in the next chapters more advanced techniques to increase the performance of a radar will be introduced. After discussing various forms of the radar equation, we will focus on the losses that appear in the equation, in particular concerning thermal noise. Lastly, we will see how also the earth's horizon contributes to the maximum range at which a radar can work.
  
 =====Derivation of Radar Equation===== =====Derivation of Radar Equation=====
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 <figure farnearfields> <figure farnearfields>
 {{ :media:farnearfields.png?500 }} {{ :media:farnearfields.png?500 }}
-<caption>The far field is the region where the reactive fields have a neglectable power, while the radiative fields are predominant[(cite:Goran M Djuknic >> author: Goran M Djuknic (US Patent 6657596) [Public domain], via Wikimedia Commons src: https://upload.wikimedia.org/wikipedia/commons/5/5d/FarNearFields-USP-4998112-1.svg)]</caption>+<caption>The far field is the region where the reactive fields have a neglectable power, while the radiative fields are predominant[(cite:Goran_M_Djuknic >> author: Goran M Djuknic (US Patent 6657596) [Public domain], via Wikimedia Commons src: https://upload.wikimedia.org/wikipedia/commons/5/5d/FarNearFields-USP-4998112-1.svg)]</caption>
 </figure> </figure>
  
-The antenna directivity is the ratio between the power density measured in a certain point in space and the power density of an isotropic antenna would generate in the same point. $G_T$ can be written as:+The antenna directivity is the ratio between the power density measured in a certain point in space and the power density that an isotropic antenna would generate in the same point. $G_T$ can be written as:
  
 \begin{equation} \begin{equation}
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 <figure fig-2.2> <figure fig-2.2>
-{{ :media:fig-2.2.png?1050 }}+{{ :media:fig-2.2.png?600 }}
 <caption>Effective illumination time of target[(cite:TeoriaTecnicaRadar>> title     : Teoria e Tecnica Radar, authors   : Gaspare Galati, publisher : Texmat, published : 2009, pages     : 382 <caption>Effective illumination time of target[(cite:TeoriaTecnicaRadar>> title     : Teoria e Tecnica Radar, authors   : Gaspare Galati, publisher : Texmat, published : 2009, pages     : 382
 )]</caption> )]</caption>
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 ^ Reality $\downarrow$    | :::                     | :::                        | ^ Reality $\downarrow$    | :::                     | :::                        |
 | $H_1$                   | OK                      | Missed Detection ($1-p_D$) | | $H_1$                   | OK                      | Missed Detection ($1-p_D$) |
-| $H_1$                   | False Alarm ($p_{fa}$)  | OK                         |+| $H_0$                   | False Alarm ($p_{fa}$)  | OK                         |
 <caption>Decision Theory</caption> <caption>Decision Theory</caption>
 </table> </table>
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 <figure fig-2.3> <figure fig-2.3>
-{{ :media:fig-2.3.png?1050 }}+{{ :media:fig-2.3.png?500 }}
 <caption>Behaviour of the amplitude of received signal (V) and false alarms[(cite:TeoriaTecnicaRadar)]</caption> <caption>Behaviour of the amplitude of received signal (V) and false alarms[(cite:TeoriaTecnicaRadar)]</caption>
 </figure> </figure>
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 \end{equation} \end{equation}
  
-$S_{min}$ is the minimum signal needed to detect the target with given $p_D$ and $p_{fa}$. $S_{min}$ can be expressed ad the product $SNR_{min}P_N$, where $P_N$ is the noise power generated in the receiving subsystem. Figure {{ref>fig-2.4}} shows the typical scheme of a radar receiving subsystem.+$S_{min}$ is the minimum signal needed to detect the target with given $p_D$ and $p_{fa}$. $S_{min}$ can be expressed as the product $SNR_{min}P_N$, where $P_N$ is the noise power generated in the receiving subsystem. Figure {{ref>fig-2.4}} shows the typical scheme of a radar receiving subsystem.
  
 <figure fig-2.4> <figure fig-2.4>
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 </figure> </figure>
  
-If the range is not known, there is an iterative procedure that can be used for the attenuation.+If the range is not known, there is an iterative procedure that can be used to compute the attenuation.
   -If we have a constant specific attenuation $\alpha(R)$ (dB/Km), the attenuation factor is $A(R) = 10^{-\alpha R}$;   -If we have a constant specific attenuation $\alpha(R)$ (dB/Km), the attenuation factor is $A(R) = 10^{-\alpha R}$;
-  -The range is first computed using the radar equation without attenuation, the obtained range $R_0$ will be greater than the real range $R_1$; +  -The range is first computed using the radar equation without attenuation, the obtained range $R_0$ will be greater than the real range; 
-  -The 2-way attenuation $A_0 = 10^{-2\alpha R_0}$ is computed considering the range R_0 (in this case the range is too great, so the attenuation will be higher than reality too), R_1 is then computed by substituting the new attenuation factor $A_0$ in the radar equation;+  -The 2-way attenuation $A_0 = 10^{-2\alpha R_0}$ is computed considering the range $R_0$ (in this case the range is too great, so the attenuation will be higher than reality too), $R_1$ is then computed by substituting the new attenuation factor $A_0$ in the radar equation;
   -$A_1$ is computed again using the new range $R_1$;   -$A_1$ is computed again using the new range $R_1$;
   -iterate until $R_i \approx R_{i+1}$.   -iterate until $R_i \approx R_{i+1}$.
-The convergence of the results to an acceptable value is usually fast, after 2 or 3 iterations the obtained values can be considered optimum.\\+The convergence of the results to an acceptable value is usually fast, after 2 or 3 iterations the obtained values can be considered valid.\\
 The noise captured by the antenna also depends on other phenomena in the atmosphere and on cosmic sources. In general, this noise is characterized by simply giving an antenna noise temperature $T_a$, from which the noise power can be computed as $kT_a$ (being $k$ the Boltzmann constant). This under the assumptions that the antenna is ideal, with no losses, and that the radiation pattern has no lobes pointing on the ground. Figure {{ref>fig-2.11}} shows how the antenna temperature changes with frequency and elevation angle. The noise captured by the antenna also depends on other phenomena in the atmosphere and on cosmic sources. In general, this noise is characterized by simply giving an antenna noise temperature $T_a$, from which the noise power can be computed as $kT_a$ (being $k$ the Boltzmann constant). This under the assumptions that the antenna is ideal, with no losses, and that the radiation pattern has no lobes pointing on the ground. Figure {{ref>fig-2.11}} shows how the antenna temperature changes with frequency and elevation angle.
  
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 \end{equation} \end{equation}
  
-$T_E$ is the noise produced by the device, comparing the equations (54) and (57) such noise can becomes: $\Delta P_{on} = GkT_EB$. This is the temperature referred to the input terminals of the device, if the noise power in input has a temperature $T_1$, the overall device noise temperature will be $T_1+T_E$. Hence the noise power:+$T_E$ characterizes the noise produced by the device, comparing the equations (54) and (57) such noise becomes: $\Delta P_{on} = GkT_EB$. This is the temperature referred to the input terminals of the device, if the noise power in input has a temperature $T_1$, the overall device noise temperature will be $T_1+T_E$. Hence the noise power:
  
 \begin{equation} \begin{equation}
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 ===Over The Horizon Backscattering (OTH-B)=== ===Over The Horizon Backscattering (OTH-B)===
  
-At low frequencies (HF band), signals directed to the sky with a certain angle are reflected by the ionosphere (where the refractive index changes) towards the ground The OTH-B radar can detect signals backscattered from the target back to the ionosphere and reflected again towards the receiver (see Figure {{ref>fig-2.31}}). The frequencies used for this kind of radar are strictly dependant on the meteorological conditions of the atmosphere. The OTH-B radar then needs a continuous monitoring of such conditions to adapt dynamically to changes.+At low frequencies (HF band), signals directed to the sky with a certain angle are reflected by the ionosphere (where the refractive index changes) towards the ground. The OTH-B radar can detect signals backscattered from the target back to the ionosphere and reflected again towards the receiver (see Figure {{ref>fig-2.31}}). The frequencies used for this kind of radar are strictly dependant on the meteorological conditions of the atmosphere. The OTH-B radar then needs a continuous monitoring of such conditions to adapt dynamically to changes.
  
 <figure fig-2.31> <figure fig-2.31>
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 <figure duga> <figure duga>
 {{ :media:duga.jpg?500 }} {{ :media:duga.jpg?500 }}
-<caption>Duga Radar Array near Chernobyl, Ukraine[(cite:Ingmar_Runge>> author: Ingmar Runge ,CC BY 3.0 (https://creativecommons.org/licenses/by/3.0), from Wikimedia Commons src: https://commons.wikimedia.org/wiki/File:DUGA_Radar_Array_near_Chernobyl,_Ukraine_2014.jpg], from Wikimedia Commons)]</caption>+<caption>Duga Radar Array near Chernobyl, Ukraine[(cite:Ingmar_Runge>> author: Ingmar Runge ,CC BY 3.0 (https://creativecommons.org/licenses/by/3.0), from Wikimedia Commons src: https://commons.wikimedia.org/wiki/File:DUGA_Radar_Array_near_Chernobyl,_Ukraine_2014.jpg)]</caption>
 </figure> </figure>
  
 Note that there are two different arrays, one is used as a transmitter and the other as a receiver. The power used by OTH-B radars can reach the order of few MegaWatts (10 MW in the case of the Duga). Those radars could only be built when the technology permitted the development of Doppler radars (see [[http://160.80.86.204/dokuwiki/doku.php?id=radar:doppler|Doppler Radar]]), since the power backscattered from the target is much lower than the clutter (often given by the sea). These Doppler radars are more complex to design with respect to the usual ones because the movement of the ionosphere generates a shift in the Doppler frequency of the clutter too. Note that there are two different arrays, one is used as a transmitter and the other as a receiver. The power used by OTH-B radars can reach the order of few MegaWatts (10 MW in the case of the Duga). Those radars could only be built when the technology permitted the development of Doppler radars (see [[http://160.80.86.204/dokuwiki/doku.php?id=radar:doppler|Doppler Radar]]), since the power backscattered from the target is much lower than the clutter (often given by the sea). These Doppler radars are more complex to design with respect to the usual ones because the movement of the ionosphere generates a shift in the Doppler frequency of the clutter too.
  
-One important characteristic of OTH-B radars is the poor resolution. Nowadays these radars are only used when precision is not needed, being less expensive than satellites. OTH-B radars built during the cold war instead, were built because there was no better technology at the time, now for such military purposes, the use of satellites has made radars like the Duga obsolete.+One important characteristic of OTH-B radars is the poor resolution. Nowadays these radars are only used when precision is not needed, being less expensive than satellites. OTH-B radars built during the cold war, instead, were built because there was no better technology at the time, now for such military purposes, the use of satellites has made radars like the Duga obsolete.
  
-Another way of exploiting the change of refractive index id the conduct effect (Figure {{ref>fig-2.35}}). In particular conditions, in fact, there is an inversion of the behavior of the refractive index, usually decreasing at higher altitudes, that instead starts increasing with the altitude. if the radar antenna is located inside this particular substrate of the atmosphere, the signals transmitted are also contained inside the substrate and can reach distances much greater than usual. In fact, the attenuation is directly proportional to the distance $R$, and not to $R^2$. The conduct effect can only be exploited when particular conditions arise, that depend on the aleatory behavior of the atmosphere. For this, the effect is not reliable.+Another way of exploiting the change of refractive index is the conduct effect (Figure {{ref>fig-2.35}}). In particular conditions, in fact, there is an inversion of the behavior of the refractive index, usually decreasing at higher altitudes, that instead starts increasing with the altitude. if the radar antenna is located inside this particular substrate of the atmosphere, the signals transmitted are also contained inside the substrate and can reach distances much greater than usual. In fact, the attenuation is directly proportional to the distance $R$, and not to $R^2$. The conduct effect can only be exploited when particular conditions arise, that depend on the aleatory behavior of the atmosphere. For this, the effect is not reliable.
  
 <figure fig-2.35> <figure fig-2.35>
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 </figure> </figure>
  
-This radar is mainly used for marine applications, such as surveillance from the coast. It can be used to detect ships as well as marine currents (see the [[http://www.radartutorial.eu/07.waves/wa52.en.html|Bragg effect]]), winds and other meteorogical conditions of the sea.+This radar is mainly used for marine applications, such as surveillance from the coast. It can be used to detect ships as well as marine currents (see the [[http://www.radartutorial.eu/07.waves/wa52.en.html|Bragg effect]]), winds and other meteorological conditions of the sea.
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