1. August 27, 2012 ETM 607 Slide 1 ETM 607 Application of Monte Carlo Simulation: Scheduling Radar Warning Receivers (RWRs) Scott R. Schultz Mercer University

2. August 27, 2012 ETM 607 Slide 2 Problem Statement  Develop an RWR scheduler that minimizes the time to detect multiple threats across multiple frequency bands.

3. August 27, 2012 ETM 607 Slide 3 RWR Scheduling Definitions Pulse Width (PW) Revisit Time (RT) Illumination Time (IT) Pulse Repetition Interval (PRI) Beam Width (BW) 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 Time

4. August 27, 2012 ETM 607 Slide 4 Example RWR Schedule RWR Schedule – a series of dwells on different frequency bands: sequence and length

5. August 27, 2012 ETM 607 Slide 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. August 27, 2012 ETM 607 Slide 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 Note different offsets Threat detected in cycle 1 Threat detected in cycle 2

7. August 27, 2012 ETM 607 Slide 7 Simulation Algorithm n = 1 i = 1 Generate offset for threat i ~ U(0,RT i ) Determine time when RWR schedule coincides with threat i i = i + 1 i < I Objective: Evaluate / Score a single RWR schedule. N – number of iterations I – number of threats n = n + 1 Update MTDAT, MaxDAT n < N Done Yes No

8. August 27, 2012 ETM 607 Slide 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. August 27, 2012 ETM 607 Slide 9 Simulation Run-Time How long does simulation run to evaluate a single schedule?

10. August 27, 2012 ETM 607 Slide 10 Empirical Density Functions Can we take advantage of the distribution function of MTDAT to avoid costly simulation?

11. August 27, 2012 ETM 607 Slide 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 P 1 = (  1 +  2 - 2d + 1)/T 2, and assumes T 1 < T 2. * Note, Kelly Noone and Perkins did not add the 1, we believe trailing edge triggered.

12. August 27, 2012 ETM 607 Slide 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. August 27, 2012 ETM 607 Slide 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 T 1 (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. d = 2 T 2 = 20T 1 = 7 Time of Intercept Start Time of Threat      ThreatRWR 2 200204060 21 1 32 4 243 7 454 10 665 96 97 98 2 109 5 3110 8 5211 7312 1613 1614 1615 3 1716 6 3817 9 5918 12 -19 -20

14. August 27, 2012 ETM 607 Slide 14 MTD - Mean Time to Detect Expected times t(n) per cycle n: where  is an indeterminate error bounded by: and, MTD = where E is the total error bounded by:

15. August 27, 2012 ETM 607 Slide 15 MTD - Mean Time to Detect Is there error, E, a concern? RWRThreatCoincidenceMTD T1tau1T2tau2dRPTEnumeration (1)Simulation (2)%Error (3) E[n]  t(n)p(n) %Error (4) 417307251400.4209.952141.93%209.950.00% 417307252428.0225.772270.54%225.770.00% 417307256591.0289.282900.25%289.280.00% 417307257653.3348.363490.18%348.360.00% 417321101818.2592.125970.82%590.470.28% 417321102935.1655.986651.37%654.670.20% 4173211031090.9728.047310.41%727.070.13% 417277401240.8115.571160.37%115.190.33% 417277402251.8118.221152.72%117.950.23% (1) Enumeration - by hand, correct value (2) Simulated for 10,000 iterations (3) %Error = ABS(Enumeration - Simulation)/Enumeration (4) %Error = (Enumeration -  t(n)p(n))/Enumeration Note: calculated E[n] is better and faster than Simulated value.

16. August 27, 2012 ETM 607 Slide 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.