Target detection by scanning a given search volume is a basic surveillance function of radar. The radar scans a search volume by steering beam to angular positions based on a pre-defined beam lattice (or grid) in pre-defined sequences. In mechanically scanning radar, target tracking is generally performed by associating detections in consecutive search frames. The advent of electronically scanning radar made it possible to allocate an independent beam to the target being tracked. Modern radars are composed of an active electronically scanning array (AESA) antenna that offers extreme beam agility [1] and a processor that allocates beam to perform multi-functions, such as search, acquisition, and track for multiple targets. In the case of airborne AESA radar, interleaving of air-to-air and air-to-surface operations is also possible [2]. Thus, to maximize the performance of each function, efficient resource management in AESA radars is critical [3].

Previous researches have focused on optimizing parameters for search task to maximize the radar performance with minimum radar resources [4–7]. In [4, 5], parameters such as search frame time, instrument range R_s (for a case of low pulse repetition frequency [PRF]), beam spacing, and duty factor were investigated to maximize track initiation range R_t with minimum power. With respect to the search frame time, ΔR⁄R_t against the required power to achieve a given R_t was analyzed (where ΔR is a radial distance moved by the target between successive looks) to find the condition that minimizes the required power. With respect to the instrument range, the combination of ΔR⁄R_t and R_s⁄R_t that also minimizes the required power was investigated. With respect to the beam spacing, the optimum beam spacing which minimizes the required power was analyzed by calculating the ratio between pulse integration gain per dwell and beamshape loss depending on beam spacing. In this analysis, a fixed search volume and search frame time were assumed. With respect to the duty factor, eclipse losses for medium and high PRF waveforms were considered (together with the increase in mean power as duty factor increases) to find the optimum value of the duty factor. In [6], similar parameter optimization to maintain the pre-determined R_t for ensuring minimum load for search task was presented. Specifically, a radar search load was defined as an average time required to conduct surveillance over a given search volume during a search frame time, and an optimization (minimization) process of the search load was proposed to satisfy a given R_t for the cumulative probability of 90% by tuning signal-to-noise ratio (SNR), beam overlap angle, and search frame time. In [7], optimizing beamwidth, search dwell time, and beam spacing for pre-defined single probability of detection with minimum search load was discussed. An analytical optimization by expressing the search load as a function of dwell time, beam width, beam spacing, search frame time, and SNR is proposed and compared with numerical optimization results.

While the previous works are interested in parameter optimization for search task, the work presented in this paper aims to provide a guideline for parameter selection when it is required to sacrifice radar resource for search task by reducing search frame time. In modern airborne AESA radars with multi-functionality, each function or task has its own priority, and it is desired to allocate sufficient radar resources to high-priority tasks (such as tracking high-threat targets, or air-to-surface operation in interleaved mode). Generally, search task has relatively low priority, and therefore, the time allocated to the search task is sacrificed first if other tasks with higher priority occur. In case when only search task exists, reducing the time allocated to the search task is equivalent to reducing search frame time. Also, modern airborne AESA radars provide operators with options for customizing radar functions. For instance, a radar operator can select or change (usually, reduce) search frame time either directly by selecting the desired search frame time or indirectly by changing the range-scale on multi-function display.

Reduction in the search frame time can be achieved either by increasing angular area for a single beam with broader beamwidth, by reducing the time allocated to a single beam with shorter dwell time, or by sparsely illuminating overall search volume with wider beam spacing. Regardless of which parameter is controlled, reducing search frame time from optimally designed search parameters results in degradation of detection range performance (which is defined as a cumulative probability of detection [8–12] in this work). In this paper, the detection range performance is analyzed for reduced search frame time by tuning either of the three parameters (beamwidth, dwell time, and beam spacing), and a guideline for selecting the parameter that minimizes detection performance degradation among the three parameters is presented.

To scan a given search volume, radar illuminates search beam toward beam positions on pre-defined beam lattice in predefined sequence. For radar antenna beam pattern with one-way 3 dB beamwidth θ_bw, the beam spacing Δθ between the adjacent beam positions on the beam lattice is conventionally defined with a beam overlap ratio ρ as follows:

In the case of radar with an electronically scanned array antenna, θ_bw increases as beam steering angle with respect to the antenna boresight increases due to the reduced effective antenna aperture [13]. Considering such beam broadening of the antenna, the beam lattice in this work is designed to have uniform beam spacing in sine space, resulting in non-uniform beam spacing but uniform beam overlap ratio in antenna or radar coordinates. Also, instead of rectangular beam lattice, triangular beam lattice [14] is considered in this work.

For the beam lattice with uniform beam spacing in sine space, the number of beams N_b for a single search frame is approximately calculated as follows. Assuming a search volume centered at (az₀, el₀) with angular span of (az_w, el_w) in azimuth and elevation, the number of elevation bar N_el is given as

where Δv = ρ_el sin(θ_bw,el ), ρ_el and θ_bw,el are beam overlap ratio and one-way 3dB beamwidth in elevation, respectively. The upper and lower elevation limits of the search volume in sine-space are denoted by vlim+ and vlim-, respectively, where are vlim±=sin(el0±0.5elw). Eq. (2) assumes that the elevation offset of the first beam position from the boundary of the search volume is Δ_elΔv⁄ρ_el. Using N_el, N_b is given as follows:

where Δu=ρ_az sin(θ_bw,az), ρ_az and θ_bw,az are beam overlap ratio and one-way 3dB beamwidth in azimuth, respectively. The left and right azimuth limits of the search volume in sine-space are denoted by ulim,i+ and ulim,i-, respectively, where are ulim,i±=sin(az0±0.5azw) cos(eli). The elevation angle of i-th bar is denoted by el_i which is given as follows:

Note that Eq. (3) assumes that the azimuth offset of the first beam position from the boundary of the search volume is Δ_azΔu cos(el_i )⁄ρ_az. In Fig. 1, an example of a beam lattice is shown when (az₀, el₀) = (10°, −25°), (az_w, el_w) = (120°, 12°), (ρ_az, ρ_el) = (1, 1), and (θ_bw,az, θ_bw,el) = (3°, 4°) for both uv coordinate (sine space) and radar coordinate, respectively. The search volume boundary is shown with the red box, the search volume center is marked with the red-cross symbol, the beam steering angle of each search beam is marked with the black dot symbol, and the beam coverage corresponding to one-way beamwidth is shown with the gray ellipse. N_b calculated from Eq. (3) is 146, while the true total number of search beam N_b,true is 147. To prove the validity of Eq. (3), the error ratio ɛ_b from Eq. (5) is shown in Fig. 2 for az_w and el_w up to 120° with a fixed search volume center (az₀, el₀) = (0°, 0°), respectively, when (θ_bw,az, θ_bw,el ) and (ρ_az, ρ_el) are the same as those of the beam lattice in Fig. 1.

It is observed that ɛ_b is only less than 6.8% for whole values of (az_w, el_w).

Finally, with a dwell time of a single search beam t_d, the search frame time T_fr without interrupt of any other task is given as

In this section, the detection range performance is analyzed for reduced search frame time by tuning either of the three parameters described in the previous section: beamwidth, dwell time, and beam spacing. The detection range performance is defined as a cumulative probability of detection P_c [8–12, 21]. At the n-th search frame scan, P_c (n) is given as

where P_ds(m) is a single probability of detection at the m-th scan. Assuming that there are k search beams per single search frame, P_ds(m) is given as

where P_d(i, m) is a single probability of detection at the i-th search beam of the m-th scan. Note that, in this work, the detections of the search beams in the same search frame are assumed to be uncorrelated for the calculation of P_ds. For a target of Swerling case 1, P_d is given at range R as

where P_fa is a probability of false alarm. In Eq. (11), SNR is calculated as follows:

where P_t is the transmit power of each element in the AESA, N_array is the number of elements, δ is duty factor, σ is the radar cross section (RCS) of the target, λ is wavelength corresponding to RF carrier frequency, k is Boltzmann’s constant, T₀ is the standard temperature of 290 K, F is the noise figure, L_scan is scan loss, and L_sys is the overall system loss without L_bs and L_scan. L_bs is instantaneous beamshape loss (or antenna pattern loss [22]) which is calculated at each search beam as G₂(θ_t ), where θ_t is the angular offset of the target from the beam center.

For detection range performance analysis, the following situation is assumed. The radar is scanning a search volume of (az_w, el_w) = (120°, 20°) and (az₀, el₀) = (0°, 0°) with a search beam of (θ_bw,az, θ_bw,el ) = (4°, 4°) and t_d = 50 ms on a beam lattice with (ρ_az, ρ_el) = (1, 1). The number of search beams are 248, and therefore, T_fr = 12.4 seocnds. The rest of the parameters for the calculation of P_c are listed in Table 2.

From this case (Case#1), suppose that it is desired to reduce the search frame time to 50% of the original value, and 50% of the radar resource is allocated to other tasks. This can be accomplished either by broadening the beamwidth by 2 (Case#2), shortening the dwell time by 0.5 (Case#3), or increasing the beam overlap ratio by 2 (Case#4). Specifically, the search frame time of 12.4 seconds in Case#1 is reduced to 50% (thus, T_fr = 6.2 seconds) by broadening the beamwidth to (θ_bw,az, θ_bw,el) = (5.64°, 5.64°) in Case#2, by shortening the dwell time to t_d = 25 ms in Case#3, and by increasing beam overlap ratio to (ρaz,ρel)=(2,2)in Case#4. In Case#4, it is possible to apply 2D-MIS with (N_s,az, N_s,el) = (4, 4) as an option. In this work, Case#4 without MIS is denoted by Case#4(a), and Case#4 with MIS is denoted by Case#4(b).

To analyze the detection range performance, it is assumed that the radar platform (for instance, a fighter) is heading north at a constant velocity V_r = 200 m/s, while the target is moving at velocity V_t = 200 m/s. For each of the four cases (Case#1 to #4), P_c is analyzed using Eq. (9) for the target azimuth angle az_t spans from 0° to 60°. It is assumed that the target aspect angle asp_t with respect to the target’s heading is set as

so that both az_t and asp_t are kept constant as the target approaches to the radar platform [17]. The target altitude is the same as that of the radar platform.

In Fig. 6, the range R₉₀ corresponding to P_c = 90% is shown against az_t. In all of the four cases, ripples are observed as az_t changes. Such ripples are originated from variation of L_bs. With a fixed beam lattice, R₉₀ is maximized when a target is at the center of each beam since L_bs = 0, and minimized when it is at the mid-point between the centers of adjacent beams since L_bs is maximized. Besides the ripples, the gradual decrease in R₉₀ as az_t increases is due to the increase in L_scan, which is a common characteristic of electronically scanned arrays.

As expected, R₉₀ of Case#1 is superior to those of the other cases, since whole radar resources are allocated to search task in Case#1. Among the four cases, Case#2 shows the worst detection range performance due to the largest SNR loss, as observed in the previous section. R₉₀ of Case#3 and Case#4(a) are longer than R₉₀ of Case#2 for the entire range of az_t. However, it is unclear which is superior between Case#3 and Case#4(a). When the target is near the mid-point between two adjacent search beams, R₉₀ of Case#3 is longer than that of Case#4(a). While L_bs of Case#3 remains the same as that of Case#1, since there is no change of either beamwidth or beam spacing, L_bs is drastically increased in Case#4(a) when the target is near the mid-point between two adjacent search beams due to wider beam spacing. On the other hand, R₉₀ of Case#4(a) is superior to that of Case#3 when the target is close to the center of search beams. When the target is at the exact center of search beams, the SNR of Case#3 is half of the SNR of Case#1, while the SNR of Case#4(a) is the same as that of Case#1. Consequently, detection range performance of Case#4(a) highly fluctuates depending on az_t, as observed in Fig. 6. Such fluctuation is not desirable since the detection range performance varies drastically depending on the angular position of a target [18], and can be smoothed by applying MIS.

When MIS is applied, beam positions of beam lattice gradually shift as search frame is repeated, and therefore, L_bs at a fixed az_t within N_s,azN_s,el (= 16) search frames changes continuously. In Fig. 7(a) and 7(b), the values of L_bs at az_t = 0.20° (where L_bs is minimized in Case#1) and 2.15° (where L_bs is maximized in Case#1) within 2N_s,azN_s,el search frames are shown for Case#1, Case#4(a), and Case#4(b), respectively. It is observed that L_bs is constant in both Case#1 and Case#4(a) where the beam lattices are fixed. Specifically, L_bs = 0.89 dB at az_t = 0.20° in both Case#1 and Case#4(a). At az_t = 2.15°, L_bs = 2.26 dB and 5.39 dB in Case#1 and Case#4(a), respectively. Unlike Case#1 or Case#4(a), L_bs changes continuously with a period of N_s,azN_s,el in Case#4(b). Note that the average values of L_bs in Case#4(b) are higher or lower than those in Case#4(a) at az_t = 0.20° or 2.15°. As a result, the fluctuation observed in Case#4(a) is mitigated in Case#4(b). Note that R₉₀ of Case#4(b) is superior to that of Case#3 for the entire range of az_t, which shows that increasing beam spacing with MIS minimizes detection range performance degradation.

Although the results in Fig. 6 provides insights on why tuning beam spacing with MIS has advantages over controlling the other two parameters, the scenario of a target approaching to the radar platform at a constant azimuth and aspect angles are not a practical scenario. Thus, in the rest of this section, further analysis on detection range performance is presented for various realistic scenarios. Specifically, the following four scenarios with different target trajectories are considered for variety of the analysis, as shown in Fig. 8.

In the 1st scenario (Sc#1), the target starts from is an azimuth angle of +5° and moves southward (−x direction in Fig. 8(a)) with a constant heading of −15° referenced to north, as shown in Fig. 8(a). Thus, the target azimuth angle decreases as the target gets closer to the radar, as shown in Fig. 8(b). The target altitude is the same as that of the radar platform.

In the 2nd scenario (Sc#2), starting from an azimuth angle of +45°, the target moves with a constant heading of −80° referenced to north. The target altitude is 4,000 ft lower than that of the radar platform.

In the 3rd scenario (Sc#3), starting from an azimuth angle of +5°, the target moves with a constant heading of +15° referenced to north. The target altitude is 4,000 ft higher than that of the radar platform.

In the 4th scenario (Sc#4), the target moves southward while maneuvering with S-turn. The target altitude is the same as that of the radar platform.

The cumulative probabilities of detection from the four scenarios are shown in Fig. 9. The values of R₉₀ from each scenario are listed in Table 3. For all of the scenarios, it is observed that the detection range performance degradation is most severe for Case#2 due to the highest SNR loss, as analyzed in the previous section (Note that P_c does not reaches 90% in Sc#2). The second-worst performance is observed for Case#3, and the degradation is minimized in Case#4. In Sc#1, #2, and #4, Case#4(b) outperforms Case#4(a), while Case#4(a) shows slightly better performance in Sc#3, thus highlighting the advantages of applying MIS.

Among the three parameters (beamwidth, dwell time, and beam spacing), it has been shown that increasing beam spacing has the advantage over controlling the other two parameters for reducing search frame time in terms of detection range performance. However, increasing beam spacing may suffer large beamshape loss when a target is near the mid-point between two adjacent search beams. Applying MIS has an effect of averaging beamshape loss, and therefore increase in beamshape loss corresponding to increased beam spacing is mitigated. On the other hand, as shown with the SNR loss analysis in Section II, broadening beamwidth and shortening dwell time suffer larger SNR loss which degrades the detection range performance. With various scenarios, the detection range performance, which is defined as the cumulative probability of detection, has been analyzed for the four cases (Case#1, #2, #3, and #4), respectively, in Section III. The results verify that increasing beam spacing provides the superior detection range performance compared to controlling the other two parameters, and that applying MIS has an advantage of mitigating the large fluctuation of detection range performance depending on target angle.