Abstract
Monopulse radars are widely used in tracking systems, due to their relative simplicity and theoretical precision, but the presence of multipath impairs the tracking capabilities of these radars, especially when multipath signals are strong, as in a naval environment. A special monopulse setup, the crossfeed, has been proposed in the past to provide an automatic cancellation from smooth sea multipath. In this contribution, the performances of such a system are analyzed in presence of rough sea scattering and compared with those of a standard monopulse setup. Particular attention is devoted to performance degradations due to possible phase errors in the passive network implementing the comparator and due to ship rolling and pitching. This latter requires a full 3D monopulse simulator for its correct evaluation.
1. Introduction
Accurate tracking of an incoming target is a fundamental issue in defense systems. The preferred technique is that of monopulse radars [1–3]. These radars are small enough to mechanically track, in real time, a fast moving target yet very accurate in their pointing. Such accuracy is obtained by exploiting the null, rather than the main beam, of an appropriate pattern, the null being usually much sharper than the maximum. Real-time tracking is achieved via a control system evaluating and minimizing the signal received through the pattern realizing the null.
It is easy to comprehend that such devices are very sensitive to multipath as the multipath rays, even if received from a direction very close to the direct ray, can produce relevant signals in the pattern realizing the null.
The multipath problem is indeed very critical over the sea surface and for low elevation angles. In [4], an overview of the theories for the interaction of electromagnetic and oceanic waves is reported, and in [5–11], the problem of multipath effects on a “standard” monopulse both over smooth and rough sea has been addressed. In the literature, solutions have been proposed exploiting multiple radars [12, 13] or frequency agility [14] to overcome the depointing due to multipath. Some authors have also proposed a crossfeed monopulse [14] which is able, theoretically, to perfectly cancel out the sea multipath, hence, allowing a perfect tracking.
The crossfeed monopulse has been analyzed in [15] only in presence of a smooth isoreflective sea. In this paper, the configuration is tested over a realistic rough sea. Furthermore, its real-world performances are evaluated by taking into account possible errors in the waveguide beamforming network backing the antenna.
It is also worth noticing that nearly all papers present simulations in a vertical plane containing both the antenna and the target. This is not enough if ship rolling and pitching are to be taken into account, hence, in this paper, a full 3D monopulse simulator will be outlined. Some very preliminary results have been presented [16, 17] for what concerns a crossfeed over a rough sea in the 2D case. This paper presents the 3D case comprehensive of ship roll and pitch movements.
The paper is organized as follows. In Section 2, the basics of rough sea multipath will be recalled, while Section 3 will present the standard monopulse setup and show the crossfeed setup, which is more difficult to find in the literature. Section 4 will compare the performances of the two configurations for an ideal antenna in presence of rough sea. Sections 5 and 6 will present the effects of manufacturing tolerances and ship movements on tracking capabilities and some consideration over real antennas will be also presented. Finally, Section 7 will draw the conclusions.
2. Rough Sea Multipath
In a system comprising the radar, the target, and the sea surface (Figures 1 and 2), four possible paths are possible:(i) antenna-target-antenna, (ii) antenna-target-sea-antenna, (iii) antenna-sea-target-antenna, (iv) antenna-sea-target-sea-antenna.
In a low target configuration, all four paths are theoretically to be taken into account but, given the fact that the target is unknown, the way it is illuminated is irrelevant, hence, the problem can be reduced to a unitary source replacing the target and to the two paths leading from the target to the radar. Figure 1 shows indeed this case, for a flat earth, with just two rays, the directed and the reflected, traveling from the target ( meters above the sea) to the antenna ( meters above the sea). The mutual distance among the two is . The reflected signal impinges on the sea surface with an angle from the grazing direction. The direct and reflected rays are and meters long, respectively, and exhibit a difference in their direction of arrival at the antenna level equal to :
More accurate models do exist, taking into account earth curvature (Figure 2), and these are used in some papers [5] yet it is easy to show that the differences in , and are very small for targets low on sea and closer than m, so the flat-earth model can be safely assumed. In any case, it is easy to show that the effect of the spherical earth is merely a shift in distance of the tracking error and does not affect the error peak values. In both cases, the signal received by the antenna is being the pattern of the antenna, the appropriate reflection coefficient (horizontal or vertical polarization), as discussed here below, and the spreading factor which is introduced by the spherical earth, if the spherical model is considered. The reference system chosen is a spherical one, with the -axis pointing in the boresight direction of the antenna and the -axis pointing downward. Figure 3 shows this reference. It is important to note that this is fully 3D.
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The difference in the direction of arrival is of course vertical since the incident and reflected rays always lie in a vertical plane. It is thus possible to compute the 3D direction of arrival of the reflected ray once the direction of arrival of the direct ray and are known:
For what concerns the reflection coefficient, the case of a smooth flat sea is trivial: being the index of refraction:
For the sea, typical values are Sm and , Fm being the free space permittivity.
For a rough sea, the electromagnetic model is that presented in [5], which, briefly, distinguishes in the reflection coefficient a specular component and a diffused component , so that with where is the standard deviation of the stochastic process defining the sea surface roughness, is a random number with uniform distribution in the range, is the free space wavelength at the radar working frequency, and is a heuristic coefficient developed on the basis of a best fit on experimental data:
Further details can be found in [5]; here it is worth noticing that effects of the roughness are more relevant for comparable with the radar wavelength.
More sophisticated models for sea scattering do exist, as models taking into account Bragg scattering and the other models enumerated in [4] or the two scale model described in [18]. These models are mainly used in remote sensing and polarimetry, and their aim is that to accurately model the backscattering of the sea since that is the main objective there. Here, the sea scattering is rather a noise and is the randomness in phase of the diffuse term, bound to the fine roughness, to have the larger impact in monopulse errors. In this framework, the model in [5, 6] for the conventional monopulse is perfectly fit.
3. Standard and Crossfeed Monopulses
The monopulse antenna in its standard configuration synthesizes three separate patterns.(i) A sum () pattern, exhibiting a maximum in the boresight direction; (ii) An elevation difference pattern (), exhibiting an horizontal line of nulls passing through the boresight direction; (iii) An azimuth difference pattern (), exhibiting a vertical line of nulls passing through the boresight direction.
These patterns are usually synthesized by creating four different beams, or primary channels named here , , and , by using four separate feeds on a reflector antenna or by subdividing an array into four subarrays. Each beam can be slightly squinted with respect to the boresight direction or simply placed at an offset with respect to the phase center of the whole antenna [1–3].
The primary channels are then combined in a comparator network, which usually comprises four magic-T junctions in the standard case. For what concerns a standard monopulse, the comparator network (Figure 4) produces the outputs
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The corresponding radiation diagrams are reported in Figure 5 for an ideal array of isotropic sources with a spacing.
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Tracking is attained by evaluating the error signals in elevation and azimuth: and minimizing them. It will be apparent form Section 4 how this approach is severely influenced by multipath.
For what concerns the crossfeed, the primary channels are arranged differently and are combined in a slightly different way, by using just 3 magic-T junctions (Figure 4):
The synthesized patterns, again for a array of isotropic sources with a spacing, are reported in Figure 6. The elevation error is now corrected by taking into account the quadrature component of the channel as well as both the in-phase and quadrature components of the new cross-channel:
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This leads theoretically to the complete cancellation of multipath contribution. In subsequent sections, it will be shown how this is strongly dependent on the accuracy and tolerances of the comparator network building the secondary channels from the primary ones, and which is usually manufactured in rectangular waveguide. It will also be shown how it is dependent on the roll and pitch of the ship. An enhanced correction will also be presented.
4. Ideal Behavior
As a first assessment of the behavior of the two feed setups an analysis of a system comprising an antenna m above sea level and a target 20 m above sea level closing to the radar from m down to 100 m is presented. The standard deviation of the sea roughness is , being the wavelength of the radar frequency, being this a worst case for (9).
Figure 7 reports the tracking error. This error is defined as the difference, in elevation and azimuth between the antenna pointing direction and the actual target direction . For an easier comparison between different antennas these errors are usually normalized with respect to the 3 dB beamwidth () of the antenna:
Since in the case at hand the rough sea reflection coefficient is a stochastic process a statistical error is computed and plotted. In Figure 7 the red curve is the average error whereas the two blue curves show the standard deviation limit around the average value. It is apparent how the sea roughness does not degrades the performances significantly.
A similar simulation is performed for the crossfeed monopulse, with the same configuration. No figure is reported for this case since the crossfeed monopulse leads, in the ideal case, to the complete cancellation of the multipath, hence no tracking error is present. Since the cancellation occurs without any explicit knowledge of the reflection coefficient, the presence of the stochastic process does not affect the error, which remains zero also in the presence of sea roughness.
5. Performances in Presence of Manufacturing Errors
The case presented in the former section, even if it presents a rough sea, is still ideal inasmuch the radar antenna and its comparator network are ideal. It is interesting, as a first investigation, to verify the effect of a nonideal comparator over the tracking capabilities.
If a relatively large phase error of is introduced in any of the primary or secondary channels of the standard monopulse, no relevant effect is present and the error stays as that of Figure 7. is a relatively large phase error since current manufacturing capabilities for waveguide comparators can achieve accuracies of or . If, on the other hand, a relatively small error is introduced in the crossfeed monopulse radar, the effect is very evident and peaks of error on a discrete set of distances arise (Figure 8). This behavior is basically the same whichever channel is affected by the error and presents small range intervals where the average error is very high and, which is more relevant, the standard deviation of the error is extremely high.
The disruptive effect of even very small errors is evident in Figures 9 and 10. In the first figure, the error function for varying elevation angle is presented at two given distances for both the standard and the crossfeed ideal monopulse. It is evident how at 2000 m the standard monopulse radar behaves worse than at 1950 m, producing in both cases an appreciable error. The crossfeed monopulse, on the other hand, exactly points the target.
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Figure 10, on the other hand, shows how a small error introduced gives practically no effect at 1950 m and does not significantly degrade the standard monopulse at 2000 m, while the crossfeed monopulse behavior is completely spoiled. This can be ascribed to the presence in (7) of the ratio between quadrature components. The ratio becomes very critical, due to the small value of the denominator, when the difference between the direct and reflected path is an integer number of half wavelengths. Figure 11 indeed shows on the same graph the tracking error of a crossfeed monopulse and the fractional part, with respect to lambda, of the path difference between the direct and reflected rays. It is apparent how error peaks occur where such a path distance is an integer number of wavelengths.
Since the location of the distances at which the crossfeed behavior is critical is a function of wavelength, and hence frequency, it is possible to overcome this limitation by a frequency agility approach. Figure 12 shows the error as a function of distance for the crossfeed monopulse at and . It is evident how a 1% variation in the frequency shifts the critical distances significantly.
6. Performances in Presence of Roll and Pitch
As a last set of numerical tests, the sensitivity of the monopulse radars to ship's movements, namely, roll and pitch, has been considered.
The terms roll and pitch refer to standard ship movements along its main horizontal axes. Figure 13 shows roll and pitch movements and their consequences for a target at an azimuth of with respect to the ship bow. Let us consider the incidence plane, that vertical plane containing both the direct and the sea-reflected ray arriving to the antenna. The incidence plane is vertical with respect to the sea and, in absence of roll and pitch, is also vertical with respect to the ship and antenna reference. In the present case, it is also perpendicular to the ship axis.
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When rolls occur, the roll axis is perpendicular with respect to the incidence plane (Figures 13(a) and 13(b)) then the incidence plane stays vertical also in the rotated antenna reference. The roll effect is merely an elevation difference which does not affect multipath. On the other hand, when pitches occur, and the rotation axis is parallel to the incidence plane (Figures 13(c) and 13(d)), then the incidence plane is not vertical any more in the antenna reference. This implies that the sea-reflected signal arrives “sideways” with respect to the antenna and is detected also by the channel, hence the multipath also affects the azimuth tracking capabilities. If the target is not at with respect to the bow, roll and pitch movements interact in a more complex way but it is always possible to reduce them to an elevation difference and in a tilt in the horizontal plane reference in the antenna boresight direction. Only the latter of this needs to be considered and is investigated in this section.
A test case with a phase error affecting the channel and a pitch have been considered. The pitch consequence is that multipath affects also the azimuth tracking. Figure 14 shows how, with just a roll, the elevation tracking error basically remains the same, and is entirely due to the phase error in the network. On the other hand, the azimuth error, which is ideally zero if the antenna is not rolled, starts to be affected by the multipath and presents a behavior which is very similar, but on a more limited range, due to the limited roll, to that shown in Figure 7 for the elevation error of a standard monopulse.
To try to overcome this issue a crossfeed correction on the azimuth channel is also suggested:
With these latter equations the crossfeed monopulse behavior is enhanced and the tracking error becomes that in Figure 15. It can be noticed how the error behavior is the same for both elevation and azimuth and that the elevation error is larger. This is due to the limited roll angle. By increasing the roll angle the elevation error decreases and the azimuth error increases up to roll, where they become the same.
7. Conclusions
The crossfeed monopulse has been studied and its performances assessed for a marine environment with rough sea scattering multipath. The crossfeed behavior has been studied in presence of small phase errors in the comparator network, showing its sensitivity to these errors and a possible solution by resorting to frequency agility. Finally, the performances have also been studied in presence of antenna roll and pitch, defining a new error with crossfeed correction also on the azimuth.