Differences

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

--- radar:tracking [2018/06/05 10:45] – anubaby
+++ radar:tracking [2026/04/28 15:13] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
  ===== Tracking Radar Principles =====
-**Tracking radar** systems are used to measure the target’s relative position in
+Tracking radar systems are used to measure the target’s relative position in
 range, azimuth angle, elevation angle, and velocity. A typical tracking radar has a pencil beam to receive echoes from a single target and track the target in angle, range, and/or doppler.Its resolution cell defined by its antenna beamwidth, transmitter pulse length, and/or doppler bandwidth is usually small compared with that of a search radar and is used to exclude undesired echoes or signals from other targets, clutter, and countermeasures.The primary output of a tracking radar is the target location determined from the pointing angles of the beam and position of its range-tracking gates. The angle location is the data obtained from synchros or encoders on the antenna tracking axes shafts (or data from a beam positioning computer of an electronic-scan phased array radar). In some cases, tracking lag is measured by converting tracking-lag-error voltages from the tracking loops to units of angle.
@@ Line 199: / Line 199: @@
-<figure label> {{:media:phase_comparison.png?450*50}}<caption>Single coordinate phase comparison monopulse antenna.</caption></figure>
+<figure label> {{ :media:phase_comparison.png?300*40 }}<caption>Single coordinate phase comparison monopulse antenna.[(cite:Image2>)]</caption></figure>
-Consider Figure 15 since the angle α is equal to $ϕ + \frac{π}{2}$, it follows that
+Consider Figure 13 since the angle α is equal to $ϕ + \frac{π}{2}$, it follows that
-${R_1}^2$ = $R^2$ + ($\frac{d}{2}^2$) - 2$\frac{d}{2}$Rcos(ϕ + $\frac{π}{2}$)
+\begin{equation}
+{R_1}^2 = R^2 + (\frac{d}{2}^2) - 2\frac{d}{2}Rcos(ϕ + \frac{π}{2})
+\end{equation}
+\begin{equation}
+ =R^2 + \frac{d^2}{4} - dRsinϕ
+\end{equation}
- =$R^2$ + $\frac{d^2}{4}$ - dRsinϕ
 and since d << R we can use the binomial series expansion to get
-${R_1}$ ≅ R(1 + ($\frac{d}{2R}$)sinϕ)
+\begin{equation}
+{R_1} ≅ R(1 + (\frac{d}{2R})sinϕ)
+\end{equation}
 Similarly,
-${R_2}$ ≅ R(1 - ($\frac{d}{2R}$)sinϕ)
+\begin{equation}
+{R_2} ≅ R(1 - (\frac{d}{2R})sinϕ)
+\end{equation}
 The phase difference between the two elements is then given by
-ϕ = $\frac{2π}{λ}$(R1 – R2) = $\frac{2π}{λ}$dsinϕ
+\begin{equation}
+ϕ = \frac{2π}{λ}(R1 – R2) = \frac{2π}{λ}dsinϕ
+\end{equation}
-where λ is the wavelength. The phase difference ϕ is used to determine the angular target location.Note that if ϕ=0, then the target would be on the antenna’s main axis. The problem with this phase comparison monopulse technique is that it is quite difficult to maintain a stable measurement of the off boresight angle ϕ, which causes serious performance degradation. This problem can be overcome by implementing a phase comparison monopulse system as illustrated in Figure 16.
-<figure label> {{:media:phase_comparison1.png?550*70}}<caption>Single coordinate phase monopulse antenna, with sum and difference channels.</caption></figure>
+where λ is the wavelength. The phase difference ϕ is used to determine the angular target location.Note that if ϕ=0, then the target would be on the antenna’s main axis. The problem with this phase comparison monopulse technique is that it is quite difficult to maintain a stable measurement of the off boresight angle ϕ, which causes serious performance degradation. This problem can be overcome by implementing a phase comparison monopulse system as illustrated in Figure 14.
+<figure label> {{ :media:phase_comparison1.png?400*50 }}<caption>Single coordinate phase monopulse antenna, with sum and difference channels.[(cite:Image2>)]</caption></figure>
 The (single coordinate) sum and difference signals are, respectively, given by
-Σ(ϕ) = ${s_1} + {s_2}$
+\begin{equation}
+Δ(ϕ) = {s_1} - {s_2}
+\end{equation}
-Δ(ϕ) = ${s_1} - {s_2}$
 where the ${s_1}$ and ${s_2}$ are the signals in the two elements. Now, since and have similar amplitude ${s_1}$ and ${s_2}$ are different in phase by ϕ, we can write
-${s_1} = {s_2}e^{-jϕ}$
+\begin{equation}
+{s_1} = {s_2}e^{-jϕ}
+\end{equation}
 It follows that
-Δ(ϕ) = ${S_2}(1 - e^{-jφ})$
+\begin{equation}
+Δ(ϕ) = {S_2}(1 - e^{-jφ})
+\end{equation}
+\begin{equation}
+Σ(ϕ) = {S_2}(1 + e^{-jφ})
+\end{equation}
-Σ(ϕ) = ${S_2}(1 + e^{-jφ})$
 The phase error signal is computed from the ratio $\frac{Δ}{Σ}$. More precisely,
-$\frac{Δ}{Σ}$ = $\frac{1-e^{-jφ}}{1 + e^{-jφ}}$ = jtan($\frac{ϕ}{2})$
+\begin{equation}
+\frac{Δ}{Σ}$= \frac{1-e^{-jφ}}{1 + e^{-jφ}} = jtan(\frac{ϕ}{2})
+\end{equation}
 which is purely imaginary. The modulus of the error signal is then given by
-$\frac{|Δ|}{|Σ|}$ = tan($\frac{ϕ}{2})$
+\begin{equation}
+\frac{|Δ|}{|Σ|} = tan(\frac{ϕ}{2})
+\end{equation}
 This kind of phase comparison monopulse tracker is often called the half-angle tracker.
-<figure label> {{:media:phase_radar.png?550*70}}<caption>Block diagram of a phase comparison monopulse radar (one angle coordinate).</caption></figure>
+<figure label> {{ :media:phase_radar.png?450*60 }}<caption>Block diagram of a phase comparison monopulse radar (one angle coordinate).[(cite:Image4>)]</caption></figure>
-Figure 17 shows the antenna and receiver for one angular-coordinate tracking by phase-comparison monopulse. Any phase shifts occurring in the mixer and IF amplifier stages cause a shift in the boresight of the system. The disadvantages of phase-comparison monopulse compared with amplitude-comparison monopulse are the relative difficulty in maintaining a highly stable boresight and the difficulty in providing the desired antenna illumination taper for both sum and difference signals. The longer paths from the antenna outputs to the comparator
+Figure 15 shows the antenna and receiver for one angular-coordinate tracking by phase-comparison monopulse. Any phase shifts occurring in the mixer and IF amplifier stages cause a shift in the boresight of the system. The disadvantages of phase-comparison monopulse compared with amplitude-comparison monopulse are the relative difficulty in maintaining a highly stable boresight and the difficulty in providing the desired antenna illumination taper for both sum and difference signals. The longer paths from the antenna outputs to the comparator
-circuitry make the phase-comparison system more susceptible to boresightchange due to mechanical loading or sag, differential heating, etc.
+circuitry make the phase-comparison system more susceptible to boresight change due to mechanical loading or sag, differential heating, etc.
@@ Line 266: / Line 297: @@
 comparison monopulse system converted by this method, and vice versa. Therefore, the fundamental accuracy performance is addressed here from the conceptual viewpoint of amplitude comparison monopulse.
-There are a variety of ways to implement monopulse processing on a sumdifference beam pair, depicted functionally in Figure 18, some of which have a substantial impact on the fundamental monopulse accuracy performance. In each
+There are a variety of ways to implement monopulse processing on a sumdifference beam pair, depicted functionally in Figure 16, some of which have a substantial impact on the fundamental monopulse accuracy performance. In each
-of these implementations, returns from a single transmission are received in simultaneously formed sum and difference beams and processed coherently. In the full-vector monopulse of Figure 18.a, two complex (I, Q) samples are fully utilized to calculate a complex monopulse ratio statistic. This calculated statistic, the measured monopulse ratio, provides the basis for a computer table lookup of the target angle of arrival relative to the null in the delta beam. The computer lookup function is simply a tabulated version of the assumed monopulse ratio consisting of the assumed delta beam antenna pattern to that of the assumed sum beam versus angle off-beam boresight. The tabulated monopulse ratio is inverted in the lookup process by entering the table with the measured monopulse ratio and finding the corresponding off-boresight angle. The full-vector monopulse processing in Figure 18.b differs somewhat from that in Figure 18.a, in that after low-noise amplification to establish the system noise figure, an RF quadrature hybrid device is used to combine the delta and sum beam signals 90° out of phase, i.e., as Σ + jΔ. The purpose of this combining in the difference channel is to bring the signal strength in the difference channel to approximately the same amplitude at that in the sum channel. This causes unavoidable receiver nonlinearities to have nearly the same effect in the two channels, resulting in less degradation in accuracy performance attributable to receive-string nonlinearities. In the absence of nonlinearities, the two techniques in Fig. 17.a and b are mathematically identical because
+of these implementations, returns from a single transmission are received in simultaneously formed sum and difference beams and processed coherently. In the full-vector monopulse of Figure 16.a, two complex (I, Q) samples are fully utilized to calculate a complex monopulse ratio statistic. This calculated statistic, the measured monopulse ratio, provides the basis for a computer table lookup of the target angle of arrival relative to the null in the delta beam. The computer lookup function is simply a tabulated version of the assumed monopulse ratio consisting of the assumed delta beam antenna pattern to that of the assumed sum beam versus angle off-beam boresight. The tabulated monopulse ratio is inverted in the lookup process by entering the table with the measured monopulse ratio and finding the corresponding off-boresight angle. The full-vector monopulse processing in Figure 16.b differs somewhat from that in Figure 16.a, in that after low-noise amplification to establish the system noise figure, an RF quadrature hybrid device is used to combine the delta and sum beam signals 90° out of phase, i.e., as Σ + jΔ. The purpose of this combining in the difference channel is to bring the signal strength in the difference channel to approximately the same amplitude at that in the sum channel. This causes unavoidable receiver nonlinearities to have nearly the same effect in the two channels, resulting in less degradation in accuracy performance attributable to receive-string nonlinearities. In the absence of nonlinearities, the two techniques in Fig. 16.a and b are mathematically identical because
+\begin{equation}
+Im(\frac {Σ + jΔ} {Σ}) = Im|\frac{Δ}{Σ}|cosϕ)
+\end{equation}
-Im($\frac {Σ + jΔ} {Σ}$) = Im|$\frac{Δ}{Σ}|cosϕ)$
 Hence, they both provide the fundamental accuracy performance of full-vector monopulse processing, given by
-rmse = $\frac{||{W_Δ} - f{W_Σ}||}{|f|(2x)^{\frac{1}{2}}}$
+\begin{equation}
+rmse = \frac{||{W_Δ} - f{W_Σ}||}{|f|(2x)^{\frac{1}{2}}}
+\end{equation}
-<figure label> {{:media:monopulseaccuracy1.png?550*60}}{{:media:monopulseaccuracy2.png?550*60}}{{:media:monopulseaccuracy3.png?550*60}}{{:media:monopulseaccuracy4.png?550*60}}<caption>Functional monopulse processing implementations, (a) Full-vector monopulse processing,(b) Full-vector monopulse with prehybrid combining,( c) Amplitude-only monopulse processing,(d) Phase-only monopulse processing.</caption></figure>
+<figure label> {{ :media:monopulseaccuracy1.png?400*60 }}{{ :media:monopulseaccuracy2.png?400*60 }}{{ :media:monopulseaccuracy3.png?400*60 }}{{ :media:monopulseaccuracy4.png?400*60 }}<caption>Functional monopulse processing implementations, (a) Full-vector monopulse processing,(b) Full-vector monopulse with prehybrid combining,( c) Amplitude-only monopulse processing,(d) Phase-only monopulse processing.</caption></figure>
@@ Line 281: / Line 317: @@
 signal-to-noise ratio diminishes, causing the error probability again to approach 0.5. A minimum-error probability-maximum-accuracy condition is reached for intermediate angles.
-The last monopulse implementation illustrated (Figure 18) is termed phase only monopulse. This processing is to be distinguished from the technique of phase interferometry, which has also been called by some authors phase comparison monopulse. In Figure 18.d, RF or IF hybrids are used to combine the sum and delta channels in quadrature, i.e., with a 90° phase shift. An accurate phase detector then detects the phase difference between the two channels. The underlying principle is that this phase difference will be in one-to-one correspondence with the delta-to-sum ratio, as illustrated in the vector diagram accompanying Figure. 18.d. In phase-only monopulse, off-boresight accuracy is sacrificed to gain the benefit of identical amplitude signals in the two receiver-processor channels.
+The last monopulse implementation illustrated (Figure 16) is termed phase only monopulse. This processing is to be distinguished from the technique of phase interferometry, which has also been called by some authors phase comparison monopulse. In Figure 16.d, RF or IF hybrids are used to combine the sum and delta channels in quadrature, i.e., with a 90° phase shift. An accurate phase detector then detects the phase difference between the two channels. The underlying principle is that this phase difference will be in one-to-one correspondence with the delta-to-sum ratio, as illustrated in the vector diagram accompanying Figure. 16.d. In phase-only monopulse, off-boresight accuracy is sacrificed to gain the benefit of identical amplitude signals in the two receiver-processor channels.