I. Introduction
In airborne radar for fighters, it is tactically important to detect targets of interest prior to being intercepted by them. Therefore, one of the key performance factors of an airborne radar is a detection range, which is closely related to waveform design. A medium pulse repetition frequency (PRF) waveform is commonly used for surveillance mode since it is suitable for detect targets with both nose and tail aspects [1, 2]. However, the detection range performance of a high PRF waveform is generally superior to that of a medium PRF waveform because it offers a wide clutter-free region in the Doppler frequency domain, thereby allowing target detection in noise-only environments [2–4]. The costs of the improved detection range are limit on the detectable aspect angles of targets and poor unambiguous range [1]. As a target aspect angle shifts from nose-aspect to tail-aspect, the Doppler frequency of the target echo falls into sidelobe clutter region where strong interferences from close range sidelobe clutters or altitude returns are folded into the same range cells for targets at long ranges. Furthermore, since the unambiguous range of a common high PRF waveform in X-band is close to or even less than 1 km, the usage of the high PRF mode is limited to velocity search, unless other techniques to resolve range ambiguity are applied [5].
A number of techniques to resolve range ambiguity in high PRF have been reported [6–10]. The range ambiguity can be resolved with PRF staggering using multiple PRFs [6] which is also commonly used to resolve the range ambiguity of medium PRF [7, 8], or by using inter-pulse binary coding [9]. Another widely used technique to resolve range ambiguity in high PRF is called frequency modulation (FM) ranging [10]. Since the technique is similar to frequency modulated continuous wave (FMCW) in that the RF carrier frequency is linearly increased or decreased with, it is also called frequency modulated interrupted continuous wave [11]. In FM ranging, range is measured without ambiguity by extracting the difference between the RF carrier frequency of the transmitted pulse and that of the received pulse.
However, there are two significant disadvantages to FM ranging: poor range accuracy and degraded detection capability due to range-Doppler (RD) coupling. The range accuracy of FM ranging, which is dependent on its chirp rate, is typically on the order of a few kilometers. However, such poor range accuracy can be improved by applying range gating [12]. The degradation of detection capability due to RD coupling has been described in [13]. In FM ranging with high PRF, it is desirable to use positive chirp rate to avoid clutter signals spreading into Doppler frequency band of interest for closing targets [10]. However, with positive chirp rate, the Doppler frequency of target echo may fall into the clutter region as the target range increases. In [13], Doppler frequency threshold for high PRF FM ranging with positive chirp rate that satisfies clutter-free condition was derived in a closed form. Applying this threshold improves the minimum detectable velocity (MDV) at a given range, as well as the maximum detectable range (MDR) at a given velocity.
While the work in [13] aimed to determine a Doppler frequency threshold for the sidelobe clutter-free region in Doppler frequency domain, it is extended to deriving a threshold for the mainlobe clutter-free region in this work. As target azimuth angle increases, the minimum sidelobe clutter-free Doppler frequency in high PRF also increases under radar platform velocity compensation. This limits the MDV of a target if the signal processing for target detection is performed only for the Doppler frequency band above the minimum sidelobe clutter-free frequency. As a result, target visibility is severely degraded at large azimuth angles. However, if the target echo signal-to-clutter- and-noise ratio (SCNR) is sufficiently high, the target can be detected even in the sidelobe clutter region with a proper detection threshold. A constant false alarm rate (CFAR) threshold for detection in sidelobe clutter for high PRF was investigated in [14], and suppressing short range sidelobe clutters was studied in [15]. Although it is simple to find the Doppler frequency threshold for mainlobe clutter-free region in non-FM ranging radars, but the threshold cannot be easily determined in FM ranging radars due to clutter spreading.
In this work, the clutter rejection threshold that satisfies the mainlobe clutter-free for high PRF FM ranging airborne radars is derived. Also, MDV and MDR of high PRF FM ranging airborne radar are analyzed for each of the following four clutter rejection thresholds to show how the detection capability is improved: (i) the conventional threshold for sidelobe clutter-free without FM ranging, (ii) the threshold for sidelobe clutter-free proposed in [13], (iii) the conventional threshold for mainlobe clutter-free without FM ranging, and (iv) the threshold for mainlobe clutter-free with FM ranging derived in this work.
II. Clutter Rejection Threshold
To reject mainlobe or sidelobe clutters in Doppler frequency (or equivalently, velocity) domain in FM ranging, the RD coupling corresponding to chirp rate k should be considered. To illustrate this point, the following example is presented.
Fig. 1(a) shows the radar-target-clutter geometry of the example, where the radar platform is heading toward x-direction with velocity Vr = 150 m/s at height Hr = 10,000 ft. The radar is steering the mainlobe to (θAZ,0, θEL,0) = (45°, −5°), where θAZ,0 and θEL,0 denotes mainlobe steering azimuth and elevation angles, respectively. Thus, ψ0 = 45.218° since ψ0 = cos−1(cos(θAZ,0)cos(θEL,0)). Meanwhile, the target is at slant range Rt with velocity Vt, and its radial velocity toward the radar Vtr(= Vtcos(ψtr)) is assumed as 50 m/s, where ψtr is a line-of-sight angle toward the radar with respect to the target velocity vector.
In Fig. 1(b), mainlobe/sidelobe clutter boundaries for k = 0 and 5 MHz/s in range-velocity (RV) domain are shown with blue and red solid/dashed lines, respectively. Note that the clutter boundaries are presented in RV domain instead of RD domain, in order to provide more physical information, and that flat earth is assumed for simplicity. The crosses inside the mainlobe clutter boundaries are the points corresponding to the mainlobe steering angle. The range Rm and velocity Vm of the mainlobe clutter boundaries are calculated as
In Eqs. (1) and (2), θAZ,m and θEL,m refer to azimuth and elevation angles toward mainlobe clutter boundary which are calculated by converting the null-to-null beamwidth boundary in sine space into azimuth-elevation coordinate. In this example, the null-to-null beamwidth θNN at the boresight of the radar antenna is assumed as 6°. The beam broadening corresponding to beam steering angles is taken into account in Fig. 1, assuming that the radar antenna is a type of an active electronically scanned array (AESA). In Eq. (2)fRF denotes the RF carrier frequency. In this example, fRF = 10 GHz when k = 0. The last term in Eq. (2) is related to RD coupling due to FM ranging.
The range Rs and velocity Vs of the sidelobe clutter boundaries are calculated as
The black dotted line in Fig. 1(b) shows that Vtr shifts (decreases) as Rt increases due to k > 0. The four points labeled with (a), (b), (c), and (d) in Fig. 1(b) correspond to the following four clutter rejection thresholds, respectively.
1. Threshold for Sidelobe Clutter-Free when k = 0
The first threshold Vthr,b corresponds to the point (a) in Fig. 1(b) which is the velocity of the maximum sidelobe clutter frequency when k = 0. It is simply given as
If the threshold is applied for signal processing, the MDR is limited due to RD coupling when k > 0. For instance, the MDR is limited to 11.3 km in this example when k = 0.
2. Threshold for Sidelobe Clutter-Free when k > 0
The second threshold Vthr,b corresponds to the point (b) which is the velocity of the maximum sidelobe clutter frequency when k > 0. The derivation of Vthr,b is provided in [13], and only the results are presented in this paper for brevity as follows:
In Eq. (6), α and ρ0 are given as follows:
With the threshold Vthr,b, the MDR is improved to 32.5 km when k = 5 MHz/s.
3. Threshold for Mainlobe Clutter-Free when k = 0
The third threshold Vthr,c corresponds to the point (c) which is the velocity of the maximum mainlobe clutter frequency when k = 0. The approximated expression of Vthr,c can be derived from the Doppler extent of mainlobe clutter [16] as
With the threshold Vthr,c, the MDR is improved to 85.3 km when k = 0. Note that the target echo can be positioned in sidelobe clutter region if the threshold is lower than Vthr,b, and therefore it is desirable to apply a CFAR threshold that is different from the clutter-free region to suppress false alarms.
4. Threshold for Mainlobe Clutter-Free when k > 0
The fourth threshold Vthr,d corresponds to the point (d) which is the velocity of the maximum mainlobe clutter frequency when k > 0. Since Vthr,d is the maximum value of Vm in Eq. (2), it can be found from the derivation of Vm. While Vthr,b in Eq. (6) is derived as a closed form from the derivative of Vs in Eq. (4) as a function of sin(θEL,0)[13], an expression for Vm,max (the maximum value of Vm) in Eq. (2) cannot be simply derived as a closed form due to the following reason. Vm in Eq. (2) can also be expressed as
In Eq. (10)um = sin(θAZ,m)cos(θEL,m), vm = sin(θEL,m). For null-to-null beamwidth in sine space
θ N N S
In Eq. (11)u0 = sin(θAZ,0)cos(θEL,0), v0 = sin(θEL,0). Because of the relation in Eq. (11), the derivation of Vm cannot be simply solved as a closed form. Therefore, instead of finding a closed solution for Vm,max, alternative approaches of deriving equations for V̄thr,d (approximate values of Vthr,d) are proposed as follows.
In Fig. 2, the mainlobe clutter boundaries corresponding to
θ N N S
Thus, the velocity of the solid line (Vm when k > 0) is the summation of the velocity of the dashed line (Vm when k = 0) and that of the dotted line (Vk when k > 0). As |θEL,m| increases, |vm| is increased and converges to unity. Thus, for large values of |θEL,m|, Vk is almost constant for vm within the mainlobe clutter boundary. As a result, vm for Vm,max when k > 0 can be approximated with vm for Vm,max with k = 0. From Eqs. (10) and (11), Vm can be expressed as follows when k = 0:
From Eq. (13), the derivative of Vm with respect to vm is given as follows:
By letting
d V m d v m = 0 , d V m d v m
Also,
d V m d v m
In Eq. (17), the sign of the second term is positive for v0 < 0 and negative for v0 > 0. From Eq. (11), corresponding um is given as follows:
In Eq. (18), the sign of the second term is positive for u0 < 0 and negative for u0 > 0. Note that, while Vthr,c in Eq. (9) is an approximated result, Vm,max with um and vm from Eqs. (17) and (18) for k = 0 becomes the optimum value for Vthr,c. The point corresponding to Vm with um and vm from Eqs. (17) and (18) when k > 0 is marked with the circle symbol in Fig. 2. As shown in Fig. 2(a) and 2(b), the velocities of the triangle and circle symbols are almost the same. The detailed values of Vm, Vm,max and corresponding vm are given in Table 1. The difference between Vm and Vm,max are only −0.005 m/s and −0.035 m/s for (θAZ,0, θEL,0) = (−30°, −30°) in Fig. 2(a) and (θAZ,0, θEL,0) = (40°, −20°) in Fig. 2(b).
However, for smaller values of |θEL,m|, the point for the circle symbol deviates from the point of the triangle symbol, as shown in Fig. 2(c) and 2(d). Vk is no more constant for vm within the mainlobe clutter boundary, but gradually decreases as vm increases (decreases) for v0< 0 (v0> 0). Since Vm for k > 0 is the result of summation of Vm for k = 0 and Vk, Vm for k > 0 corresponding to vm from Eq. (17) is reduced due to decreased value of Vk. Instead, the point of the asterisk symbol, which is a crossing point of Vm for k > 0 and Vk, is introduced as an alternative since it almost coincides with the point of the triangle symbol. The expression of vm for the crossing point can be derived by letting Vm in Eq. (10) be the same as Vk in Eq. (12). One of the solution for Vm = Vk is given as follows:
In Eq. (20),
η = ( 0.5 θ N N S ) 2 - 2 ( u 0 2 + v 0 2 )
In Eq. (21), the sign of the second term is positive for u0> 0 and negative for u0< 0. As shown in Fig. 2(c) and 2(d), the velocities of the triangle and asterisk symbols are almost the same. The detailed values of Vm, Vm,max and corresponding vm are given in Table 1. The difference between Vm and Vm,max are −0.002 m/s and −0.017 m/s for (θAZ,0, θEL,0) = (−8°, −10°) in Fig. 2(c) and (θAZ,0, θEL,0) = (15°, −5°) in Fig. 2(d).
Unfortunately, the above two approaches yield a large difference between Vm and Vm,max when both θAZ,0 and θEL,0 are close to or less than θNN, as shown in Fig. 2(e) and 2(f). In this case, Vm,max is given at um = u0. Corresponding vm and Vm are solved by Eqs. (11) and (10), respectively, and they are marked with the diamond symbol in Fig. 2. As shown in Fig. 2(e) and 2(f), the velocities of the triangle and diamond symbols are exactly the same. The detailed values of Vm, Vm,max and corresponding vm are given in Table 1.
The three approaches described above are summarized as follows. The first approach calculates um and vm for Vm,max with k = 0 from Eqs. (18) and (17), respectively, then calculates Vm from Eq. (10). This approach provides V̄thr,d for large values of |θEL,m|. The second approach calculates um and vm for crossing point between Vm and Vk from Eqs. (21) and (20), respectively, then calculates Vm from Eq. (10). This approach provides V̄thr,d for small values of |θEL,m|. The third approach fixes um as u0 and calculates vm and Vm from Eqs. (11) and (10), respectively. This approach provides V̄thr,d when both θAZ,0 and θEL,0 are close to or less than θNN. However, the boundaries of θAZ,0 and θEL,0 among the three approach are difficult to be defined clearly. Thus, letting Vm from each approach as Vm1, Vm2, and Vm3, respectively, V̄thr,d can be calculated as maximum among Vm1, Vm2, and Vm3 for whole ranges of θAZ,0 and θEL,0.
In Fig. 3(a), Vm,max for ψ0 ≤ 60° are presented. The dashed line is the boundary for ψ0 = 60°. Since the results are symmetric with respect to both θAZ,0 = 0 and θEL,0 = 0, the results are shown only for 0 ≤ θAZ,0 ≤ 60° and −60° ≤ θEL,0 ≤ 0. Note that Vm,max rapidly decreases as |θEL,m| decreases when θEL,m is close to 0° due to the rapid decrease of Vk. The difference between V̄thr,d and Vm,max, ΔVm (=V̄thr,d–Vm,max) is shown in Fig. 3(b). For most values of θAZ,0 and θEL,0, ΔVm is not less than −0.3 m/s except for the shaded area. In Fig. 2(g) and 2(h), Vm1, Vm2, and Vm3 for the two sample points of the shaded area are shown. Especially, the sample point in Fig. 2(g) for (θAZ,0, θEL,0) = (59.5°, −3.5°) is the worst case with the largest error. In this point, the maximum among Vm1, Vm2, and Vm3 is Vm2 = −13.481 m/s, and Vm,max = −7.362 m/s, resulting in ΔVm = −6.119 m/s. To find V̄thr,d with smaller error from Vm,max, the fourth approach is introduced, where vm is chosen as the mid between vm for Vm1 and vm for Vm2, i.e., the mid-point between the two points marked with the circle and asterisk symbols, respectively. This mid-point is marked with the square symbol in Fig. 2. The detailed values of Vm, Vm,max and corresponding vm are given in Table 1. The differences between Vm and Vm,max are −0.258 m/s and −3.854 m/s for (θAZ,0, θEL,0) = (−59.5°, −3.5°) in Fig. 2(g) and (θAZ,0, θEL,0) = (59°, −1.5°) in Fig. 2(h). In Fig. 3(c), ΔVm for ψ0 ≤ 60° is shown where the largest error is reduced to ΔVm = −3.854 m/s.
Further error reduction can be achieved by iteration of calculating vm for the mid-point between the point for Vm2 and the previous mid-point. Defining N as the number of iteration where N = 1 is the case for Fig. 3(c), the result for N = 5 is shown in Fig. 3(d) where the largest error is reduced to ΔVm = −0.545 m/s.
Regardless of N, ΔVm is always less than zero, thus a margin should be added to V̄thr,d in order to avoid any leakage of mainlobe clutter. Also, large N increases the complexity of the derivation process of V̄thr,d. Thus, in this work, N is limited to 1 and velocity margin Vmar is chosen as follows.
III. Detection Capability
In this section, to show how the detection capability of high PRF FM ranging airborne radar can be improved by applying the clutter rejection threshold proposed in this work, MDV and MDR corresponding to the four clutter rejection thresholds (Vthr,a, Vthr,b, Vthr,c, and Vthr,d) introduced in the previous section are analyzed. Note that it does not guarantee that the target can be always detected when the target’s radial velocity (Vtr) is above MDV at a given range, or the target’s slant range (Rt) is below MDR at a given velocity. MDV (or MDR) is figure of merit that provides information of the minimum velocity (or maximum range) that can be processed by radar signal processor. The target may not be detected if its SCNR is not high enough even if Vtr > MDV or Rt < MDR. Selection of optimum CFAR threshold and analysis on the resultant probability of detection depending on the clutter rejection Doppler frequency (or velocity) threshold are out-of-scope of this paper.
In Fig. 4, MDVs with respect to θAZ,0 and θEL,0 for Rt = 150 km and the same values of Vr, Hr, θNN, and fRF in Fig. 1 corresponding to the four clutter rejection thresholds are presented, respectively. By changing the threshold from Vthr,a to Vthr,b, it is observed that MDV is improved (reduced) by approximately 10 m/s for the same θAZ,0 and θEL,0. Changing the threshold from Vthr,b to Vthr,c, does not always guarantee the improvement of MDV. As both θAZ,0 and θEL,0 approach close to zero, MDV with Vthr,c is lower than MDV with Vthr,b since Vthr,c becomes lower than Vthr,c due to clutter spreading. The best MDV is obtained with Vthr,d.
MDRs with respect to θAZ,0 and θEL,0 for Rt = 150 km and the same values of Vr, Hr, θNN, and fRF in Fig. 1 corresponding to the four clutter rejection thresholds are shown in Fig. 5, respectively. Similar to MDV, MDR is improved (increased) by approximately 20 km for the same θAZ,0 and θEL,0 by changing the threshold from Vthr,a to Vthr,b. MDR is also improved by changing the threshold from Vthr,b to Vthr,c, when ψ0 is larger than approximately 30°, but as both θAZ,0 and θEL,0 approach close to zero, MDR with Vthr,c is longer than MDR with Vthr,b. The best MDR is obtained with Vthr,d.
To further investigate the tactical advantage of applying the clutter rejection threshold proposed in this work, the following scenario is considered. Assuming that a radar platform is heading toward x-direction with a constant velocity of Vr = 150 m/s at Hr = 10,000 ft (3,048 m). A target is approaching toward the radar platform with a constant velocity of Vt = 200 m/s at Ht = 8,000 ft (2,438 m). In the beginning, the target is at Rt = 100 km and θAZ = −60° with a target aspect angle θasp = 92.866° [17], resulting in a target radial velocity toward the radar Vtr = 10 m/s. The flight trajectories of the target and the radar platform are shown in Fig. 6(a). The target is continuously changing its heading toward the radar platform while maintaining its velocity and height. The range and azimuth of the target from the radar is shown in Fig. 6(b). In Fig. 6(c) and 6(d), MDR and MDV for Vthr,a/Vthr,b are shown with blue/red dotted lines, and those for Vthr,c/Vthr,d are shown with blue/red dash-dotted lines, respectively. The four points labeled with (a), (b), (c), and (d) in each sub-figure of Fig. 6 correspond to the moments when Rt = MDR and Vtr = MDV for each of the four clutter rejection thresholds, respectively. In other words, the target is detectable for the first time as the target reaches the point (a), (b), (c), or (d) when the clutter rejection threshold is Vthr,a, Vthr,b, Vthr,c, or Vthr,d. The range, radial velocity, and azimuth of the target at each of the four points are listed in Table 2. It is observed that the first detectable range/velocity of the target is increased/decreased as the clutter rejection threshold is changed from Vthr,a to Vthr,d. In Table II, the time of the four points are provided. By changing the threshold from Vthr,a to Vthr,b, the first detectable time of the target is 16.6 seconds earlier. Assuming a search frame time of the radar as 4 seconds, it means that the radar has four more chances of look at the target by changing the threshold. If the threshold is Vthr,d, the first detectable time of the target is 150.2 seconds earlier, which means that the radar has 37 more chances of look at the target. As a result, the cumulative probability of detection of the target can be drastically improved [18–20]. Thus, the applying the clutter rejection threshold Vthr,d proposed in this work improves not only MDR and MDV, but also the cumulative probability of detection.
IV. Conclusion
One of the distinctive advantage of high PRF is that the clutter-free condition can be satisfied in a wide Doppler frequency band with a proper clutter rejection threshold. The selection of the clutter rejection threshold for conventional high PRF radar is simple. However, in high PRF FM ranging radar, clutter spreading due to RD coupling must be considered to select the optimum clutter rejection frequency, as shown in Section II.
In this work, the four clutter rejection thresholds (Vthr,a, Vthr,b, Vthr,c, and Vthr,d) corresponding to the maximum Doppler frequencies of sidelobe/mainlobe clutter signals without and with FM, respectively, were presented. In Section III, the detection capability (MDR and MDV) was analyzed as the clutter rejection threshold is changed from Vthr,a to Vthr,d, proving that the detection capability of high PRF FM ranging radar can be improved by applying the proposed clutter rejection threshold (Vthr,d) in this work.
To detect targets with the clutter rejection threshold Vthr,d while avoiding undesired false alarms, the CFAR threshold of the Doppler frequency band between Vthr,b and Vthr,d (i.e., the band with sidelobe clutter only and mainlobe clutter-free) should be selected so that it is different from that of the Doppler frequency band above Vthr,b (i.e., both mainlobe and sidelobe clutter-free) and suppress false alarms sufficiently, which remains as a future work.
PDF Links
PubReader
ePub Link
Full text via DOI
Download Citation
Print
