1. ECEN5633 Radar Theory Lecture #17 10 March 2015 Dr. George Scheets www.okstate.edu/elec-eng/scheets/ecen5633 n Read 12.2 n Problems 11.5, 8, & 12.5 n Corrected quizzes due 1 week after return u Live: 12 March n Exam #2, 31 March 2014 (< 4 April DL)

2. ECEN5633 Radar Theory Lecture #18 12 March 2015 Dr. George Scheets www.okstate.edu/elec-eng/scheets/ecen5633 n Read 13.1 & 2 n Problems 12.7, 8, & Web 3 n Corrected quizzes due 1 week after return u Live: 12 March n Exam #2, 31 March 2014 (< 4 April DL)

3. Coherent Detection (PLL), Single Pulse, Fixed P r Noise PDF Gaussian Mean = 0 Variance = kTº sys W n Echo PDF Gaussian Mean = P r 0.5 Variance = kTº sys W n r (volts) Matched Filter Output at Optimum Time γ

4. Coherent Detection (PLL) M Pulse Integration Fixed P r Noise PDF Gaussian Mean = 0 Variance = MkTº sys W n Signal PDF Gaussian Mean = MP r 0.5 Variance = MkTº sys W n r (volts) Matched Filter Output at Optimum Time γ

5. Coherent Detection, Single Pulse RCS Exponential PDF Noise PDF Gaussian Mean = 0 Variance = kTº sys W n Echo PDF Gaussian☺Rayleigh Mean = P r 0.5 Variance = Var(sig) + Var(noise) = 0.2734P r + kTº sys W n r (volts) Matched Filter Output at Optimum Time γ

6. Coherent Detection M Pulse Integration RCS Exponential PDF Noise PDF Gaussian Mean = 0 Variance = MkTº sys W n Signal PDF Gaussian Mean = MP r 0.5 Variance = MkTº sys W n + MP r 0.2734 r (volts) Matched Filter Output at Optimum Time γ Variance of MFD voltage (Rayleigh) PDF

7. Integral Result

8. Stephen O. Rice n Born 1907 n Died 1986 n Bell Labs 1930 – 1972 n IEEE Fellow n Paper "Mathematical Analysis of Random Noise" discusses Rice PDF Source: http://www.ieeeghn.org/wiki/index.php/Stephen_Rice

9. Friedrich Bessel n Born 1784 n Died 1846 n German Mathematician n In 1820's, while studying "many body" gravitational systems, generalized solutions for

10. Rice PDF x Starts to look somewhat Gaussian when v/σ 2 > 2

11. Coherent Detection n Previous Equations are Ideal u Require instantaneous phase lock to echo u Won't happen in reality F Will effectively lose part of echo pulse… … Till PLL or Phase-Frequency detector locks… Till PLL or Phase-Frequency detector locks u Lock can be obtained on Doppler Shifted echoes F Could use bank of PLL's, free running at different freqs n Coherent Detection not used a lot u But equations give feel as to process Have somewhat easily digestible derivations

12. Non Coherent Radar Detection n Fixed P r & Random Noise Single Range Bin u Noise has Rayleigh Distribution F Mean = 1.253 σ n F Variance = 0.4292 σ n 2 F σ n 2 = kTº sys W n (if calculations off front end) u Signal + Noise has Ricean Distribution ≈ Gaussian if α/σ n 2 = P r 0.5 /σ n 2 > 5 F Mean = P r 0.5 F Variance = kTº sys W n

13. Noncoherent (Quadrature) Detection, Single Pulse, Fixed Pr Noise PDF Rayleigh Mean = 1.253(kTº sys W n ) 0.5 Variance = 0.4292kTº sys W n Echo PDF ≈ Gaussian Mean = P r 0.5 Variance = kTº sys W n r (volts) Matched Filter Output at Optimum Time γ Ex) P(Hit | Coherent) = 0.3253 & P(Hit | Noncoherent) = 0.1692

14. Noncoherent Detection, M Pulse Integration (Envelope Detection, fixed P r ) n Sample envelope M times, sum results Make decision based on sum Noise and Signal PDF's approximately Gaussian n P(Hit) = Q[0.6551Q -1 [P(FA)] + 1.253M 0.5 – (M*SNR) 0.5 ]

15. Noncoherent (Quadrature) Detection M Pulse Integration Fixed P r Noise PDF ≈ Gaussian Mean = M1.253(kTº sys W n ) 0.5 Variance = M0.4292kTº sys W n Signal PDF Gaussian Mean = MP r 0.5 Variance = MkTº sys W n r (volts) Matched Filter Output at Optimum Time γ Ex) P(Hit | Coherent) = Q(-8.848) & P(Hit | Noncoherent) = Q(-6.523)

16. Comment n Noncoherent Integration Gain u Sometimes stated as M 0.5 u P(Hit) ≈ Q[ Q -1 [P(FA)] – (M 0.5 *SNR) 0.5 ] n "Noncoherent Integration Gain, and it's Approximation" u Mark Richards, GaTech, May 2013 u Has an example where gain is M 0.8333 u P(Hit) ≈ Q[ Q -1 [P(FA)] – (M 0.833 *SNR) 0.5 ] u EX) P(Hit) ≈ Q[4.753 – 10 0.833 *18.5) 0.5 = Q[4.753 – 11.22] = Q[-6.469] n Safer to say gain is M a ; 0.5 < a < 1.0

17. Radar P(Hit), Fixed P r n Single Pulse, Coherent P(Hit) = Q[ Q -1 [P(FA)] – SNR 0.5 ] u Equation 12.19 in text n M Pulse Integration, Coherent P(Hit) = Q[ Q -1 [P(FA)] – (M*SNR) 0.5 ] u See equation 13.3 in text

18. Radar P(Hit), Exponential P r n Single Pulse, Coherent Noise is Gaussian Signal (echo) Voltage is Rayleigh u Evaluate 2nd Order PDF f(n,s) or f(n)☺f(s) n M Pulse Integration, Coherent P(Hit) ≈ Q{[Q -1 [P(FA)]σ n – (M*P signal_1 ) 0.5 ]/σ sum } u where σ sum = (σ 2 n + σ 2 s ) 0.5 u σ 2 n = noise power u σ 2 s = variance of noise free signal (echo) voltage = 0.2734*M*P signal_1

19. Radar P(Hit), Fixed P r n Single Pulse, Noncoherent Noise is Rayleigh Distributed Signal is Ricean Distributed → Gaussian P(Hit) ≈ Q[γ/σ n – SNR 0.5 ] where γ = {ln[1/P(FA)]2σ n 2 } 0.5 u Equation 12.49 in Text n M Pulse Integration, Noncoherent P(Hit) ≈ Q[ Q -1 [P(FA)] – (M a SNR) 0.5 ] P(Hit) ≈ Q[ 0.655Q -1 [P(FA)] +1.253M 0.5 - (M*SNR) 0.5 ]

20. Noncoherent Detection Fluctuating P r n Will not be derived in class n Text has calculations for several cases n Below is PDF of Signal u Sum of S.I. Gaussian noise & Rayleigh echo u Need PDF of I sum 2 added to another SI Q sum 2, then take square root.

21. Peter Swerling n Born 1929 n Died 2000 n PhD in Math at UCLA n Worked at RAND Entrepreneur (founded 2 consulting companies) n Developed & analyzed Swerling Target Models in 1950's while at RAND

22. Swerling Model Performance M = 10 Noncoherent Integration P(FA) = 10 -9 Source: Merrill Skolnik's Introduction to Radar Systems, 3rd Edition

23. Receiver Phase Locked Loop X Active Low Pass Filter Voltage Controlled Oscillator cosω c t (from antenna) sin((ω vco t +θ) -sin((ω vco -ω c )t+θ) VCO set to free run at ≈ ω c VCO output frequency = ω c + K * input voltage LPF with negative gain. 2 sinα cosβ = sin(α-β) + sin(α+β)

24. PPI with clutter Source: www.radartutorial.eu