In airborne radar, the target angle is commonly measured through monopulse processing [11]. The following example illustrates how the angular measurement is distorted due to FSL detection. A linear array antenna composed of N isotropic elements with element spacing d is considered. The target angle of the antenna can be measured by comparing the outputs of the sum (∑) and difference (Δ) channels. Notably, the ∑ channel output s_∑ is the summation of the received signals from the total N elements, while the Δ channel output s_Δ is the difference between the summation of the received signals from one-half of the N elements and that from the other half of the N elements. Fig. 2(a) shows the s_∑ and s_Δ with respect to the measured angle θ when N = 30 and d is half wavelength. Note that both values are normalized to the peak voltage of s_∑. The monopulse ratio (the ratio between s_Δ and s_∑, which is ideally a real number) within the half-power beamwidth of 3.4° is shown in Fig. 2(b). In conventional monopulse processing, when detection occurs, the monopulse ratio is calculated from the received signal and then compared with the curve in Fig. 2(b). For instance, supposing that the calculated monopulse ratio is 0.15, the radar processor decides that the measured monopulse angle is 0.36°, which is the angle marked with the “x” symbol in Fig. 2(b). This process assumes that the echo signal has been received through the mainlobe of the antenna. However, supposing that the signal is actually received through the FSL and not the mainlobe—for instance, at a 70° offset from the mainlobe peak direction—the monopulse ratio at the angle would be 0.15 (the same as that of 0.36°), which is marked with an “o” symbol in Fig. 2(c), the monopulse ratio for the entire angular range from −90° to +90°. However, since this process assumes that the echo signal has been received through the mainlobe of the antenna, the measured monopulse angle will still be the angle corresponding to the “x” symbol marked in Fig. 2(b) and 2(c), which is 0.36°, resulting in an angular measurement error θ_err = 69.64°.

The target velocity is measured using Doppler frequency. In airborne radar, the Doppler frequency is measured from the movements of both the target and the radar platform. Therefore, the radar platform velocity must be compensated to find the velocity of the target itself. Supposing that the radar platform is heading north with a constant altitude with velocity v_r, and a target at angle θ_t is approaching the radar with aspect angle θ_asp and velocity v_t, as shown in Fig. 3, the measured Doppler frequency f_d can be expressed as follows:

From Eq. (1), the radial velocity of the target v_t,r can be calculated using the following equation:

The second term on the right-hand side of Eq. (2) compensates for the radar platform velocity, assuming that the target echo signal is received through the mainlobe of the antenna. However, if the signal is received through the FSL at θ_fsl, the true radial velocity of the target would be as follows:

From Eqs. (2) and (3), the velocity measurement error v_err can be obtained as follows:

In the previous sections, angular and velocity measurement errors resulting from interference caused by FSL were analyzed. In this section, the estimation process for the true location (based on angle) and velocity of the surface clutter in the FSL direction is described.

Although the angle and velocity of the interference caused by FSL are distorted, the range remains unaffected because it is measured by the time delay of the transmitted pulses. Suppose that the radar platform is flying with velocity v_r at altitude h_r, the radar beam is steered to θ with respect to the platform heading, as shown in Fig. 4. In the case of SLB failure, false alarms may occur due to the surface clutter return through FSL. If the detected range is r_m, the elevation angle of the clutter in space-stabilized coordinate θc,elss can be estimated as follows:

Furthermore, if the measured radial velocity and the azimuth and elevation angles are v_m, θ_m,az, and θ_m,el, respectively, the azimuth angle of the clutter in space-stabilized coordinate θc,azss can be estimated using the following equation:

where v_r,x and v_r,y are the velocities of the radar platform in the x-axis and y-axis, respectively. From (r_m, θc,azss, θc,elss), the latitude, longitude, and altitude (LLA) of the surface clutter can be estimated through coordinate transformation into NED, and then to LLA [12]. This process has not been described here for brevity.

Finally, the radial velocity of clutter v_c,r can be estimated from Eq. (4) as follows:

One of the most effective techniques for rejecting false alarms arising from surface clutter returns through FSLs is to refer to the look-up table (LUT) of FSL angles corresponding to the beam steering angles.

The LUT comprises a list of beam steering angles (both azimuth and elevation) within a beam steering limit and a list of FSL angles (both azimuth and elevation) corresponding to each beam steering angle that is not covered by the auxiliary antenna radiation pattern. When detection occurs, the FSL angles can be estimated using Eqs. (5) and (6). Since these angles are in space-stabilized coordinate, they can be transformed into antenna coordinate and then compared with the FSL angles in the LUT. If the estimated FSL angles are within the angular regions in which the gain of the FSLs of the LUT is higher than that of the auxiliary antenna, the detection is considered as a false alarm and therefore rejected.

In Fig. 6, an example of applying the LUT to reject interference from the FSL is presented. The black dots correspond to the list of beam-steering angles in a LUT. In this example, the resolution of the beam steering angles is assumed to be 5°, meaning that the LUT was built by measuring the FSLs while changing the beam steering angles with a 5° step (note that the black dots exist only within the beam steering angle limit, indicated using a solid line in Fig. 6. A smaller resolution value can be chosen to improve the precision of the LUT. However, as the resolution value decreases, the time and cost required to build the LUT increase).

Suppose that a detection occurs when the radar radiates beams at a steering angle of (θ_s,az, θ_s,el) = (5°, 13°), and the estimated FSL angle is (θ_fsl,az, θ_fsl,el) = (−78°, −50°), which are marked using red “+” and “o” symbols in Fig. 6, respectively. From the list of beam steering angles in the LUT, the value nearest to the actual beam steering angle (θ_s,az, θ_s,el) is (θs,azLUT,θs,elLUT)=(5°,15°), marked by a blue “+” symbol in Fig. 6. In this example, it is assumed that there are two FSL angles in the LUT corresponding to the beam steering angle (θs,azLUT,θs,elLUT)=(5°,15°) (note that certain beam steering angles might have no FSL angle, while others might have one or several FSL angles. This depends on the array antenna characteristics). The first and second FSL angles obtained from the LUT are (θfsl#1,azLUT,θfsl#1,elLUT)=(-76.9°,-50.3°) and (θfsl#2,azLUT,θfsl#2,elLUT)=(-63.2°,-40.7°), shown using the blue “o” symbol in Fig. 6. The dotted blue lines covering the blue “o” symbols show the angular region where the gain of the first and second FSLs from the LUT are higher than the gain of the auxiliary antenna. It is observed that the estimated FSL angle falls within the angular region of the first FSL from the LUT. As a result, the detection is assumed as a false alarm due to the interference from the FSL and, therefore, is rejected.

A drawback of rejecting false alarms using the LUT is the time and cost involved in building the LUT. To build a LUT for FSL angles, radiation patterns of the main antenna need to be measured with full angular coverage (up to 90° from the antenna broadside). Since radars usually adopt planar array antennas with high directivity as their main antenna, radiation pattern measurement would require a near-field measurement system, which usually takes a considerable amount of time to measure the radiation pattern at a single beam steering angle to ensure full angular coverage. This measurement needs to be repeated for all beam steering angles within the beam steering limit for each frequency within the operational frequency band of the radar. Moreover, although finer resolutions of beam steering angles and frequency lead to more precise construction of the LUT, the time and cost required for the measurement increase.

In this section, an alternative technique for false alarm rejection is discussed for cases in which LUTs with sufficient angular and frequency resolutions are not available. This alternative approach is based on the characteristics of false alarms caused by surface clutter returns through FSL. Specifically, in this technique, false alarms are rejected if they satisfy a number of predefined criteria derived from their common characteristics.

The first criterion is that the measured range r_m should be longer than or the same as the radar platform altitude h_r.

If this criterion is not satisfied, the estimated elevation angle in the space-stabilized coordinate θc,elss from Eq. (5) will be a complex number. This criterion prevents the possibility of rejecting real targets within a close range, which are critical threats to fighters.

The second criterion is that the measured velocity v_m should be lower than the velocity threshold v_thr.

If a detection is actually a false alarm caused by surface clutter returns through the FSL, the estimated clutter velocity v_c,r in Eq. (7) should ideally be 0. The second term on the right-hand side in Eq. (7) is usually larger than the third term, since the measured angles (θ_m,az, θ_m,el) correspond to the beam steering angles within the beam steering limit, but the estimated angles ( θc,azss,θc,elss) correspond to the FSL angles outside the beam steering limit, which results in a negative value of v_m. Therefore, the second criterion prevents the possibility of rejecting real targets with nose-aspect angles, which are more critical threats to fighters than targets with tail-aspect angles.

Despite the first and second criteria, the possibility of rejecting real targets with tail-aspect angles at mid- and long-ranges remains. Unfortunately, it is difficult to perfectly sort out false alarms caused by real tail-aspect angle targets. However, by applying the third criterion, the probability of rejecting real tail-aspect angle targets is reduced. The third criterion is that the difference between the signal power level of ∑ channel p_∑ and Δ channel p_Δ, which is p_∑Δ, should be less than the threshold p_thr.

The third criterion is based on the fact that the gain of the ∑ channel pattern is lower than that of the Δ channel pattern for most FSLs. This can be illustrated using the example of a planar array antenna with radiation patterns of the ∑, Δ_az, and Δ_el channels, as shown in Fig. 7(a), 7(b), and 7(c), respectively (note that the planar array antenna considered here is a hypothetical one—it is not the real array antenna used in the flight test in Section III. The radiation patterns of the real array antenna are not provided for security reasons). The pattern levels are normalized to the peak value of the ∑ channel pattern and presented in dB scale. Fig. 7(d) shows the angular regions where p_∑Δ is higher than p_thr = 5 dB. The red-, blue-, and violet-colored regions represent the areas in which the third criterion is met for the Δ_az channel, Δ_el channel, and both channels, respectively. p_∑Δ is calculated only for the angles outside the beam steering limit corresponding to a 60° cone angle with respect to the broadside of the antenna, since the sidelobes of the main antenna within the cone angle can be covered by the radiation pattern of the auxiliary antenna. Also, since the probability of detection at low levels of the ∑ channel radiation pattern is extremely low, p_∑Δ is calculated only for the angles where the normalized ∑ channel radiation pattern level is higher than −60 dB.

The value of p_thr should be decided considering the trade-off between two probabilities: the probability of failing to reject the false alarms caused by FSL (i.e., the probability that the third criterion is not satisfied within the FSL region) and the probability of failing to detect the real target in the mainlobe (i.e., the probability that the third criterion is satisfied within the mainlobe region). The first probability P_fail,FSL can be estimated from the ratio of the angles that do not satisfy the third criterion to the total angles outside the beam steering limit. Similarly, the second probability P_fail,ML can be estimated through the ratio of the angles that satisfy the third criterion to the total angles within the mainlobe beamwidth. Note that the value of P_fail,FSL (or P_fail,ML) for the Δ_az channel will not be the same as that of P_fail,FSL (or P_fail,ML) for the Δ_el channel. Fig. 8 shows the values of P_fail,FSL and P_fail,ML for the Δ_az and Δ_el channels in terms of p_thr. It is observed that with an increase in p_thr, P_fail,FSL decreases but P_fail,ML increases. When p_thr = 5 dB, the values of P_fail,FSL are 5.30% for the Δ_az channel and 4.58% for the Δ_el channel. This means that the probability of failing to reject false alarms caused by FSL when the third criterion is applied is less than 6%, which is sufficiently low. Meanwhile, the values of P_fail,ML are 14.03% for the Δ_az channel and 12.39% for the Δ_el channel, implying that the probability of rejecting real targets within the beamwidth is more than 12%.

To mitigate the risk of rejecting real targets within the beamwidth and to enhance the probability of rejecting false alarms caused by surface clutter returns through FSL, the M-out-of-N criterion is applied—the detection is rejected only if all the criteria are satisfied for at least M detections in N continuous detections. The characteristics that allow improvements in the detection of real targets and false alarms by FSL by applying the M-out-of-N criterion are noted below.

If detections originate from a real target with a sufficient signal-to-noise ratio, their measured angle will converge to the true angle of the target as detections are continued through tracking [13]. In other words, the measured monopulse angle θ_mon will converge to zero as detections continue. θ_mon for the real target during the flight test (not shown in Fig. 5 for brevity and security reasons) is presented in Fig. 9(a), where it is observed that θ_mon converges to zero as detections continue. Thus, p_∑Δ becomes higher than p_thr and converges to a certain value (corresponding to the peak gain of ∑ channel and the null depth of Δ channel). In Fig. 9(b), p_∑Δ for a real target during the flight test is presented, showing that it becomes higher than p_thr = 5 dB and converges to approximately 30 dB. However, occasionally, p_∑Δ is lower than p_thr. For various reasons, such as fluctuations of the target radar cross-section [14], interference signals, or radar hardware degradation, θ_mon can be increased, and consequently, p_∑Δ is reduced, as indicated by the detections marked in red dashed circles in Fig. 9(a) and 9(b). Without the M-out-of-N criterion, these detections would have been rejected on applying the third criterion, even though they are not false alarms caused by FSL.

In contrast, if the detection originates from interference through FSL, both θ_mon and p_∑Δ will continuously fluctuate and not converge. These phenomena were observed for the false alarms discussed in Section III, as shown in Fig. 9(c) and 9(d). Note that p_∑Δ of the Δ_az channel is mostly lower than that of the Δ_el channel since the azimuth angles of FSL angles are larger than those of the elevation angle, as shown in Fig. 5(b). It is observed that p_∑Δ fluctuates continuously, and occasionally exceeds p_thr. Thus, without the M-out-of-N criterion, these false alarms would not have been rejected, since the third criterion would be applicable.

The decision regarding the values of (M, N) should consider the following issues. As M increases, the risk of rejecting real targets within the beamwidth declines, and the probability of rejecting false alarms by FSLs increases. However, since at least M detections are required, the time required to decide whether the detection should be rejected also increases. As N for a given M increases, the probability of rejecting false alarms through FSL is enhanced, but the risk of rejecting real targets within the beamwidth increases as well. In Table 1, the rejection ratios (ratios of the number of rejected detections to that of total detections) for various combinations of (M, N) are provided for the cases in Fig. 9(b) (detections of the real target through mainlobe) and Fig. 9(d) (false alarms due to surface clutter through FSL). As previously mentioned, the rejection ratio for the detection of the real target declines as M increases. Therefore, for a given M, the rejection ratio of false alarms improves as N increases, but that for detections of the real target increases as well. Note that the optimum choice of (M, N) depends on the radar system requirements. For instance, if waiting for the decision of rejection until the 7th detection is allowable, (M, N) = (4, 7) is optimum. However, if the radar system requires faster decisions, (M, N) = (2, 4) or (3, 5) would be a better choice.