Array radar resource management Alexander Charlish* and Fotios Katsilieris*
3.2 Task management
3.2.1.2 Revisit interval time and dwell duration
one-scan cumulative detection probability drops significantly between the beams.
For closer range targets, shorter dwells can be used resulting in closely spaced beams without significant drops in the cumulative detection probability in angle.
However, it can be seen that the maxima are all broad and a choice between 0.8 and 1.0 performs reasonably well for all target ranges. The trade-off has been widely studied [7,8,9, Section 14.3], with similar conclusions of a very broad optimum beam separation at approximately 0.8 times the 3 dB beamwidth. The difference between a rectangular or triangular search lattice is not significant [10]; however, a triangular lattice has the benefit that the difference between the maximum and minimum one-scan cumulative detection probability in angle is smaller than for a rectangular lattice.
The cumulative detection probabilityPcof at least one detection is:
PcðrÞ ¼ EDs 1Y
n2N
1PsðrþðnþDsÞtrvrÞ
½
( )
(3.3) wherePsðrÞis the single look probability of detection for a target at range r,EDs denotes the expectation with respect to Ds;vr is the radial velocity of the target, N ¼ f0;. . .;bðrpurÞ=trvrcg is a set representing the number of looks until the target reaches ranger;andrpuis the target pop-up range. The offsetDs2½0;1is a uniformly distributed random variable that represents the possibility of the target arrival at any time during a revisit interval. In the following, the rangerand radial velocityvr are normalized with respect toR0, which is the range at which a unity signal-to-noise ratio (SNR) is achieved using a representative dwell lengthtd.
The single look probability of detection depends on the target radar cross- section fluctuation and the type of integration applied. For a Swerling 1 target and coherent integration, the single look probability of detection is:
PsðrÞ ¼P
1þSNRðrÞ1
FA (3.4)
where the expected SNR at range r can be calculated based on the radar range equation. If a fixed temporal loading ofls is available for the beam position, then the revisit interval also determines the dwell time. Assuming coherent integration over the dwell, the SNR is subsequently:
SNRðrÞ ¼ Dr Dre
1 r
4
(3.5) where Dre¼tdvr=ls is a representative closure range (which is the distance tra- velled by the target in one scan normalized by the temporal loading) that specifies the search problem. Once the cumulative detection probability as a function of range is calculated, it is possible to find the cumulative detection rangeR90.
This model for the cumulative detection probability results in a trade-off for selecting the revisit interval and hence dwell duration. Reducing the revisit interval time for the beam position increases the number of opportunities to detect the target in a given time frame. These multiple opportunities to observe the target give a
‘sampling gain’ [7], whereby the 90% cumulative detection range is greater than the 90% single look detection range. Conversely, increasing the revisit interval time for each beam position allows for more time in each beam position, which enables a gain through integration. Therefore, the trade-off when selecting the revisit interval can be thought of as a balance between sampling gain and integra- tion gain.
The trade-off in selecting the revisit interval is illustrated in Figure 3.6(a), where the multi-scan cumulative detection range is plotted against the distance that a target can close in one revisit interval time. In Figure 3.6(a), an inbound target approaching from infinite range is considered. It can be seen that the maximum in
0 5 10 15 20 0.25
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Closure distance in one scan (%/R0) (a)
Cumulative detection range R90
ΔRe = 2.5%
ΔRe = 5%
ΔRe = 10%
(b)
0 5 10 15 20
Closure distance in one scan (%/R0) 0.25
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Cumulative detection range R90
ΔRe = 2.5%
ΔRe = 5%
ΔRe = 10%
Figure 3.6 Multi-scan cumulative detection range as a function of the target closure range in one search revisit interval. (a) Infinite Range Inbound Target. (b) Pop-up target at 60% of R0
the cumulative detection range depends on the value of Dre. For the considered values of Dre, the maximums in the cumulative detection range occur when the target closes 5%–10% of the instrumental range R0 in one revisit interval period.
For example, ifDre¼0:05, the instrumental range is 150 km and the target has a 300 m/s radial velocity then the target should close 7.5% of R0 in one revisit interval time. This corresponds to a revisit interval of 37.5 s.
The 37.5 s revisit interval found in the previous example is much larger than commonly used. Shorter revisit intervals are commonly used, partly to accom- modate targets that pop-up at closer range. In Figure 3.6(b), the cumulative detection range is plotted for a target that pop-ups at 60% of the radar instrumental rangeR0. ForDre¼0:05, the ideal target closure range is now around 3% of the instrumental range, which corresponds to a revisit interval of 15 s. The trade-off for selecting the revisit interval has been studied many times [7,11,12], with similar conclusions to those presented here.
Limits on the coherent processing interval
Long revisit intervals and hence long coherent processing intervals may not be desirable due to range and Doppler cell migration. Range cell migration occurs when the target travels through multiple range cells leading to a spread in the signal energy. Consequently, to avoid additional processing that compensates for range migration, the maximum coherent processing interval can be limited by the target radial velocity and the range cell resolution, which is determined by the signal bandwidth. The maximum coherent processing intervalTCVfor a constant velocity target can be found by combining (2.7) and (2.8) from [13, Chapter 2]:
TCV< c
2Bvr (3.6)
whereBis the signal bandwidth,cis the speed of the electromagnetic propagation andvris the target radial velocity. If the target is accelerating with acceleration a, then the maximum coherent processing intervalTACCis:
TACC¼vrv2r þ2ac=2B
a (3.7)
The maximum coherent processing interval that prevents range cell migration is illustrated in Figure 3.7(a) for a constant velocity target.
In addition to limiting the coherent processing interval to avoid range cell migration, it should also be limited to avoid Doppler cell migration due to the target acceleration. The maximum allowable coherent integration time TDOP to prevent Doppler cell migration is:
TDOP<
ffiffiffiffiffi l 2a r
(3.8) wherelis the wavelength of the transmitted waveform. The maximum coherent pro- cessing interval times due to Doppler cell migration are illustrated in Figure 3.7(b).
Based on these models, it is possible to configure the search based on the range resolution required by the application and then the dwell duration and revisit interval to match the expected target motion. Consequently, it can be possible to specify multiple search modes that are suited to different target types, such as a
103
102
101
103
102
101 102
10–1 100 101
103 Radial velocity (m/s)
(a)
(b)
Maximum coherent processing interval (ms)
10 MHz 5 MHz 2.5 MHz 1 MHz
Radial acceleration (g)
Maximum coherent processing interval (ms)
2 GHz 4 GHz 1 GHz 0.5 GHz
Figure 3.7 Limitations on the coherent processing interval to avoid range and Doppler cell migration. (a) Limits on range cell migration for varying signal bandwidth. (b) Limit on Doppler cell migration for varying operating frequency
short-range search for manoeuvrable pop-up targets and a long-range search for less manoeuvrable targets approaching from greater range. The parameters of the search can be adapted online, to match the current requirements of the operator.