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radar:tracking [2018/06/05 10:41] – anubabyradar:tracking [2026/04/28 15:13] (current) – external edit 127.0.0.1
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  ===== Tracking Radar Principles =====  ===== 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. 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.
  
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 In early radars, the receive time was measured by observing the pulse return on a display, such as an A scope. Some later radars use automatic range measurement,using two contiguous range gates, called **Early-late gates or split-gate tracker**. When the energy in the two gates is equal, the crossover time between the gates is at the center of the received pulse. Many modern radars sample the received signal and determine the target range by fitting the sample data to a replica of the pulse. The range measurement of a target is obtained, in principle, by locating the center of the received pulse.The optimal estimation is achieved by passing the pulse received in the matched filter, calculating the derivative of the output signal and detecting the zero crossing instant.The derivative operation is usually replaced with an operation called Early-Late Gate. In early radars, the receive time was measured by observing the pulse return on a display, such as an A scope. Some later radars use automatic range measurement,using two contiguous range gates, called **Early-late gates or split-gate tracker**. When the energy in the two gates is equal, the crossover time between the gates is at the center of the received pulse. Many modern radars sample the received signal and determine the target range by fitting the sample data to a replica of the pulse. The range measurement of a target is obtained, in principle, by locating the center of the received pulse.The optimal estimation is achieved by passing the pulse received in the matched filter, calculating the derivative of the output signal and detecting the zero crossing instant.The derivative operation is usually replaced with an operation called Early-Late Gate.
 +
  
 <figure label>{{ :media:earlylate.png?300*50 }}<caption> Illustration of split-range gate[(cite:Image2>> title: http://dsp-book.narod.ru/RSAD/C1828_PDF_C11.pdf)]</caption></figure> <figure label>{{ :media:earlylate.png?300*50 }}<caption> Illustration of split-range gate[(cite:Image2>> title: http://dsp-book.narod.ru/RSAD/C1828_PDF_C11.pdf)]</caption></figure>
 +
  
 The concept of split gate tracking is illustrated in Figure 2, where a sketch of a typical pulsed radar echo is shown in the figure. The early gate opens at the anticipated starting time of the radar echo and lasts for half its duration. The late gate opens at the center and closes at the end of the echo signal. For this purpose, good estimates of the echo duration and the pulse centertime must be reported to the range tracker so that the early and late gates can be placed properly at the start and center times of the expected echo. This reporting process is widely known as the “designation process.” The early gate produces positive voltage output while the late gate produces negative voltage output. The outputs of the early and late gates are subtracted, and the difference signal is fed into an integrator to generate an error signal. If both gates are placed properly in time, the integrator output will be equal to zero. Alternatively, when the gates are not timed properly, the integrator output is not zero, which gives an indication that the gates must be moved in time, left or right depending on the sign of the integrator output. The concept of split gate tracking is illustrated in Figure 2, where a sketch of a typical pulsed radar echo is shown in the figure. The early gate opens at the anticipated starting time of the radar echo and lasts for half its duration. The late gate opens at the center and closes at the end of the echo signal. For this purpose, good estimates of the echo duration and the pulse centertime must be reported to the range tracker so that the early and late gates can be placed properly at the start and center times of the expected echo. This reporting process is widely known as the “designation process.” The early gate produces positive voltage output while the late gate produces negative voltage output. The outputs of the early and late gates are subtracted, and the difference signal is fed into an integrator to generate an error signal. If both gates are placed properly in time, the integrator output will be equal to zero. Alternatively, when the gates are not timed properly, the integrator output is not zero, which gives an indication that the gates must be moved in time, left or right depending on the sign of the integrator output.
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 **Conical scanning** is a system used in early radar units to improve their accuracy, as well as making it easier to steer the antenna properly to point at a target. Conical scanning is similar in concept to the earlier lobe switching concept used on some of the earliest radars, and many examples of lobe switching sets were modified in the field to conical scanning during World War II, notably the German Würzburg radar. Antenna guidance can be made entirely automatic, as in the American SCR-584. Potential failure modes and susceptibility to deception jamming led to the replacement of conical scan systems with monopulse radar sets. **Conical scanning** is a system used in early radar units to improve their accuracy, as well as making it easier to steer the antenna properly to point at a target. Conical scanning is similar in concept to the earlier lobe switching concept used on some of the earliest radars, and many examples of lobe switching sets were modified in the field to conical scanning during World War II, notably the German Würzburg radar. Antenna guidance can be made entirely automatic, as in the American SCR-584. Potential failure modes and susceptibility to deception jamming led to the replacement of conical scan systems with monopulse radar sets.
 +
 +
  
 <figure label> {{ :media:conical_scan.png?350*50 }}<caption>variation of the echo signal in a conical scanning.[(cite:Image3>> title: https://www.google.it/search?q=variation+of+signal+conical+scanning)]</caption></figure> <figure label> {{ :media:conical_scan.png?350*50 }}<caption>variation of the echo signal in a conical scanning.[(cite:Image3>> title: https://www.google.it/search?q=variation+of+signal+conical+scanning)]</caption></figure>
 +
 +
  
 Conical scanning addresses this problem by "moving" the radar beam slightly off center from the antenna's midline, and then rotating it. Given an example antenna that generates a beam of 2 degrees width – fairly typical – a conical scanning radar might move the beam 1.5 degrees to one side of the centerline by offsetting the feed slightly. The resulting pattern, at any one instant in time, covers the midline of the antenna for about 0.5 degrees, and 1.5 degrees to the side. By spinning the feed horn with a motor, the pattern becomes a cone centered on the midline, extending 3 degrees to the sides in our example Figure 4. Conical scanning addresses this problem by "moving" the radar beam slightly off center from the antenna's midline, and then rotating it. Given an example antenna that generates a beam of 2 degrees width – fairly typical – a conical scanning radar might move the beam 1.5 degrees to one side of the centerline by offsetting the feed slightly. The resulting pattern, at any one instant in time, covers the midline of the antenna for about 0.5 degrees, and 1.5 degrees to the side. By spinning the feed horn with a motor, the pattern becomes a cone centered on the midline, extending 3 degrees to the sides in our example Figure 4.
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 Conical scan, where the antenna scans a small cone around the target position.Lobing technique was extended to continuous rotation of the beam around the target (conical scan) as in Figure 5. Angle-error-detection circuitry is provided to generate error voltage outputs proportional to the tracking error and with a phase or polarity to indicate the direction of errors. The error signal actuates a servosystem to drive the antenna in the proper direction to reduce the error to zero.Because of the rotation of the squinted beam and the target’s offset from the rotation axis, the amplitude of the echo signal will be modulated at a frequency equal to the beam rotation frequency. Conical scan, where the antenna scans a small cone around the target position.Lobing technique was extended to continuous rotation of the beam around the target (conical scan) as in Figure 5. Angle-error-detection circuitry is provided to generate error voltage outputs proportional to the tracking error and with a phase or polarity to indicate the direction of errors. The error signal actuates a servosystem to drive the antenna in the proper direction to reduce the error to zero.Because of the rotation of the squinted beam and the target’s offset from the rotation axis, the amplitude of the echo signal will be modulated at a frequency equal to the beam rotation frequency.
 +
 +
  
 <figure label> {{ :media:conicalscan.png?400*60 }}<caption>Conical-scan tracking. [(cite:Image4>> title: Radar handbook / editor in chief, Merrill I. Skolnik. — 2nd ed.,Copyright © 1990 by McGraw-Hill, Inc)]</caption></figure> <figure label> {{ :media:conicalscan.png?400*60 }}<caption>Conical-scan tracking. [(cite:Image4>> title: Radar handbook / editor in chief, Merrill I. Skolnik. — 2nd ed.,Copyright © 1990 by McGraw-Hill, Inc)]</caption></figure>
 +
  
  
 The amplitude of the modulation depends on angular distance between the target direction and the rotation axis. The amplitude of the modulation depends on angular distance between the target direction and the rotation axis.
 žThe location of the target in two angle coordinates determines the phase of the **conical scan modulation** relative to conical scan beam rotation. Continuous beam scanning is accomplished by mechanically moving the feed of an antenna since the antenna beam will move off axis as the feed is moved off the focal point. The feed is typically moved in a circular path around the focal point, causing a corresponding movement of the antenna beam in a circular path around the target. A typical block diagram is shown in Figure 6. A range tracking system is included which automatically follows the target in range, with range gates that turn on the radar receiver only during the time when the echo is expected from the target under track. Range gating excludes undesired targets and noise. The system also includes an **automatic gain control (AGC)** necessary to maintain constant angle sensitivity (volts of error-detector output per degree of error) independent of the amplitude of the echo signal. This provides the constant gain in the angle-tracking loops necessary for stable angle tracking. žThe location of the target in two angle coordinates determines the phase of the **conical scan modulation** relative to conical scan beam rotation. Continuous beam scanning is accomplished by mechanically moving the feed of an antenna since the antenna beam will move off axis as the feed is moved off the focal point. The feed is typically moved in a circular path around the focal point, causing a corresponding movement of the antenna beam in a circular path around the target. A typical block diagram is shown in Figure 6. A range tracking system is included which automatically follows the target in range, with range gates that turn on the radar receiver only during the time when the echo is expected from the target under track. Range gating excludes undesired targets and noise. The system also includes an **automatic gain control (AGC)** necessary to maintain constant angle sensitivity (volts of error-detector output per degree of error) independent of the amplitude of the echo signal. This provides the constant gain in the angle-tracking loops necessary for stable angle tracking.
 +
 +
  
 <figure label> {{ :media:conical_scanradar.png?400*60 }}<caption>Block diagram of a conical-scan radar. <figure label> {{ :media:conical_scanradar.png?400*60 }}<caption>Block diagram of a conical-scan radar.
 [(cite:Image4>)]</caption></figure> [(cite:Image4>)]</caption></figure>
 +
  
  
 **AGC** has the purpose of maintaining constant angle error sensitivity in spite of amplitude fluctuations or changes of the echo signal due to change in range. It is also important for avoiding saturation by large signals which could cause the loss of the scanning modulation and the accompanying error signal. **AGC** has the purpose of maintaining constant angle error sensitivity in spite of amplitude fluctuations or changes of the echo signal due to change in range. It is also important for avoiding saturation by large signals which could cause the loss of the scanning modulation and the accompanying error signal.
 Monopulse Elaboration Monopulse Elaboration
 +
 +
  
 <figure label> {{ :media:angle_error_conicalscan.png?400*60 }}<caption>(a) Angle error information contained in the envelope of the received pulses in a conical-scan radar. (b) Reference signal derived from the drive of the conical-scan feed.[(cite:Image4>)]</caption></figure> <figure label> {{ :media:angle_error_conicalscan.png?400*60 }}<caption>(a) Angle error information contained in the envelope of the received pulses in a conical-scan radar. (b) Reference signal derived from the drive of the conical-scan feed.[(cite:Image4>)]</caption></figure>
 +
  
  
 The radar video output contains the angle-tracking-error information in the envelope of the pulses, as shown in Figure 7. The percentage modulation is proportional to the **angle-tracking error**, and the phase of the envelope function relative to the beam-scanning position contains direction information. Angle tracking-error detection (error demodulation) is accomplished by a pair of phase detectors using a reference input from the scan motor. The phase detectors perform essentially as dot-product devices with sine-wave reference signals at the frequency of scan and of proper phases to obtain elevation error from one and azimuth error from the other. The radar video output contains the angle-tracking-error information in the envelope of the pulses, as shown in Figure 7. The percentage modulation is proportional to the **angle-tracking error**, and the phase of the envelope function relative to the beam-scanning position contains direction information. Angle tracking-error detection (error demodulation) is accomplished by a pair of phase detectors using a reference input from the scan motor. The phase detectors perform essentially as dot-product devices with sine-wave reference signals at the frequency of scan and of proper phases to obtain elevation error from one and azimuth error from the other.
 +
  
  
 <figure label> {{ :media:top_viewconical.png?400*60 }}<caption>Conical-scan-radar antenna beam 3 dB contour (solid circle) and path of rotation (dashed circle) of the beam center.[(cite:Image4>)]</caption></figure> <figure label> {{ :media:top_viewconical.png?400*60 }}<caption>Conical-scan-radar antenna beam 3 dB contour (solid circle) and path of rotation (dashed circle) of the beam center.[(cite:Image4>)]</caption></figure>
 +
  
  
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 Figure 9 shows a typical monopulse antenna pattern. The four beams A, B, C, and D represent the four conical scan beam positions. Four feeds, mainly horns, are used to produce the monopulse antenna pattern. Amplitude monopulse processing requires that the four signals have the same phase and different amplitudes. Figure 9 shows a typical monopulse antenna pattern. The four beams A, B, C, and D represent the four conical scan beam positions. Four feeds, mainly horns, are used to produce the monopulse antenna pattern. Amplitude monopulse processing requires that the four signals have the same phase and different amplitudes.
 +
 +
  
 <figure label> {{ :media:monopulse.png?400*60 }}<caption>Monopulse antenna pattern.[(cite:Image2>)]</caption></figure> <figure label> {{ :media:monopulse.png?400*60 }}<caption>Monopulse antenna pattern.[(cite:Image2>)]</caption></figure>
 +
 +
  
 A good way to explain the concept of amplitude monopulse technique is to represent the target echo signal by a circle centered at the antenna’s tracking axis, as illustrated by Figure 10(a), where the four quadrants represent the four beams. In this case, the four horns receive an equal amount of energy, which indicates that the target is located on the antenna’s tracking axis. However, when the target is off the tracking axis (Figure. 10(b-d)), an unbalance of energy occurs in the different beams.  A good way to explain the concept of amplitude monopulse technique is to represent the target echo signal by a circle centered at the antenna’s tracking axis, as illustrated by Figure 10(a), where the four quadrants represent the four beams. In this case, the four horns receive an equal amount of energy, which indicates that the target is located on the antenna’s tracking axis. However, when the target is off the tracking axis (Figure. 10(b-d)), an unbalance of energy occurs in the different beams. 
 +
 +
  
 <figure label> {{ :media:monopulseq.png?400*60 }}<caption>Illustration of monopulse concept. (a) Target is on the <figure label> {{ :media:monopulseq.png?400*60 }}<caption>Illustration of monopulse concept. (a) Target is on the
 tracking axis. (b) - (d) Target is off the tracking axis.[(cite:Image2>)]</caption></figure> tracking axis. (b) - (d) Target is off the tracking axis.[(cite:Image2>)]</caption></figure>
 +
  
  
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 <figure label> {{ :media:monopulseradar.png?500*60 }}<caption>Simplified amplitude comparison monopulse radar block diagram.[(cite:Image2>)]</caption></figure> <figure label> {{ :media:monopulseradar.png?500*60 }}<caption>Simplified amplitude comparison monopulse radar block diagram.[(cite:Image2>)]</caption></figure>
 +
 +
  
 The sum signal, elevation difference signal, and azimuth difference signal are each converted to intermediate frequency (IF), using a common local oscillator to maintain relative phase at IF. The IF sum-signal output is detected and provides the video input to the range tracker. The range tracker determines the time of arrival of the desired target echo and provides gate pulses which turn on portions of the radar receiver only during the brief period when the desired target echo is expected. The gated video is used to generate the dc voltage proportional to the magnitude of the ∑ signal or |∑| for the AGC of all three IF amplifier channels. The AGC maintains constant angle-tracking sensitivity (volts per degree error) even though the target echo signal varies over a large dynamic range by controlling gain or dividing by |∑|. AGC is necessary to keep the gain of the angle-tracking loops constant for stable automatic angle tracking. The sum signal, elevation difference signal, and azimuth difference signal are each converted to intermediate frequency (IF), using a common local oscillator to maintain relative phase at IF. The IF sum-signal output is detected and provides the video input to the range tracker. The range tracker determines the time of arrival of the desired target echo and provides gate pulses which turn on portions of the radar receiver only during the brief period when the desired target echo is expected. The gated video is used to generate the dc voltage proportional to the magnitude of the ∑ signal or |∑| for the AGC of all three IF amplifier channels. The AGC maintains constant angle-tracking sensitivity (volts per degree error) even though the target echo signal varies over a large dynamic range by controlling gain or dividing by |∑|. AGC is necessary to keep the gain of the angle-tracking loops constant for stable automatic angle tracking.
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 direction of error. direction of error.
  
-<figure label> {{:media:phase_comparison.png?450*50}}<caption>Single coordinate phase comparison monopulse antenna.</caption></figure> 
  
-Consider Figure 15 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 + $\frac{π}{2}$) 
  
- =$R^2$ + $\frac{d^2}{4}$ - dRsinϕ+<figure label> {{ :media:phase_comparison.png?300*40 }}<caption>Single coordinate phase comparison monopulse antenna.[(cite:Image2>)]</caption></figure> 
 + 
 + 
 + 
 +Consider Figure 13 since the angle α is equal to $ϕ + \frac{π}{2}$, it follows that 
 + 
 +\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} 
  
 and since d << R we can use the binomial series expansion to get 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, 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 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 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 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 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, 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 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. 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.
  
  
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 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. 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 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>
  
  
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 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. 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.
  
  
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