ETM 607 Application of Monte Carlo Simulation: Scheduling Radar Warning Receivers (RWRs)

1. ETM 607Application of Monte Carlo Simulation:Scheduling Radar Warning Receivers (RWRs) Scott R. Schultz Mercer University

2. Problem Statement • Develop an RWR scheduler that minimizes the time to detect multiple threats across multiple frequency bands.

3. RWR Scheduling Definitions Definitions: Revisit Time (RT) – time to rotate 360 degrees (rotating radar) Illumination Time (IT) – function of RT and BW Pulse Width (PW) – length of time while target is energized Pulse Repetition Interval (PRI) – time between pulses Beam Width (BW) Revisit Time (RT) Pulse Repetition Interval (PRI) Illumination Time (IT) Pulse Width (PW) Time

4. Example RWR Schedule RWR Schedule – a series of dwells on different frequency bands: sequence and length

5. RWR Scheduling Problem Objective – detect all threats as fast as possible (protect the pilot) How to sequence dwells? How to determine dwell length? How to evaluate / score schedules? Meta-Heuristics Simulation

6. Need for Simulation Given that the offset for each threat pulse train is unknown. Determine: MTDAT - expected time to detect all threats, MaxDAT - maximum time to detect all threats Threat detected in cycle 1 Note different offsets Threat detected in cycle 2

7. Simulation Algorithm n = 1 Objective: Evaluate / Score a single RWR schedule. N – number of iterations I – number of threats i = 1 Generate offset for threat i ~ U(0,RTi) Update MTDAT, MaxDAT Determine time when RWR schedule coincides with threat i n = n + 1 i = i + 1 Yes n < N i < I Yes No No Done

8. Simulation iterations - N When does the MTDAT running average begin to converge? MTDAT running average: 3 threats MTDAT running average: 5 threats MTDAT running average: 10 threats

9. Simulation Run-Time How long does simulation run to evaluate a single schedule?

10. Empirical Density Functions Can we take advantage of the distribution function of MTDAT to avoid costly simulation?

11. POI Theory – 2 pulse trains Enter: Kelly, Noone, and Perkins (1996) • The probability of intercept can be divided into four regions, associated with the pulse count, n, of the shorter periodic pulse train. • where P1 = (t1+ t2 - 2d + 1)/T2, and assumes T1 < T2. • * Note, Kelly Noone and Perkins did not add the 1, we believe trailing edge triggered.

12. MTD - Mean Time to Detect Our contribution: Knowing that, MTD = E[n] = , where t(n) is the intercept time for pulse n. What is t(n) for all n?

13. MTD - Mean Time to Detect Observation: When threat starts in positions -1,0,1, or 2, intercept occurs on pulse 1 of RWR. When threat starts in position 3, 4 or 5 intercept occurs on pulse 4, 7 and 10 respectively. Intercept time occurs at T1(n-1) + d + i, where n is the RWR pulse count and i is 0 if threat starts before start of cycle, else i is amount of time elapsed between start of RWR pulse and start of threat.

14. MTD - Mean Time to Detect Expected times t(n) per cycle n: where e is an indeterminate error bounded by: and, MTD = where E is the total error bounded by:

15. MTD - Mean Time to Detect Is there error, E, a concern? Note: calculated E[n] is better and faster than Simulated value.

16. Summary and Limitations • Summary: • An innovative closed form approach for determining the mean time for coincidence of periodic pulse trains has been developed using POI theory and insight on the coincidence of periodic pulse trains. • The approach is computationally faster and more accurate than a previous presented Monte Carlo simulation approach. • Limitations: • This method is limited to threats which exhibit strictly periodic pulse train behavior (e.g. rotating beacons). • Still need method to determine MaxDAT • Future: • An enumerative approach is being evaluated for non-periodic pulse trains.