Radar Systems Engineering Lecture 6 Detection of Signals in Noise

Transcription

1 Radar Systems Engineering Lecture 6 Detection of Signals in Noise Dr. Robert M. O Donnell Guest Lecturer Radar Systems Course 1 Detection 1/1/010

2 Block Diagram of Radar System Target Radar Cross Section Propagation Medium Antenna T / R Switch Power Amplifier Transmitter Waveform Generation Signal Processor Computer Receiver A / D Converter Pulse Compression Clutter Rejection (Doppler Filtering) User Displays and Radar Control General Purpose Computer Tracking Parameter Estimation Thresholding Detection Data Recording Photo Image Courtesy of US Air Force Used with permission. Radar Systems Course This Lecture

3 Outline Basic concepts Probabilities of detection and false alarm Signal-to-noise ratio Integration of pulses Fluctuating targets Constant false alarm rate (CFAR) thresholding Summary Radar Systems Course 3

4 Radar Detection The Big Picture Example Typical Aircraft Surveillance Radar ASR-9 Range 60 nmi. Courtesy of MIT Lincoln Laboratory Used with permission Transmits Pulses at ~ 150 Hz Rotation Rate 1. rpm Mission Detect and track all aircraft within 60 nmi of radar S-band λ ~ 10 cm Radar Systems Course 4

5 Magnitude (db) Radar Systems Course 5 Range-Azimuth-Doppler Cells to Be Thresholded Doppler Filter Bank Radial Velocity (kts) As Antenna Rotates ~ pulses / Beamwidth Example Typical Aircraft Surveillance Radar ASR-9 10 Pulses / Half BW Processed into 10 Doppler Filters Range - Azimuth - Doppler Cells ~1000 Range cells ~500 Azimuth cells ~8-10 Doppler cells 5,000,000 Range-Az-Doppler Cells to be threshold every 4.7 sec. Az Beamwidth ~1. Range Resolution 1/16 nmi Rotation Rate 1.7 rpm Radar Transmits Pulses at ~ 150 Hz Range 60 nmi. Is There a Target Present in Each Cell?

6 Received Backscatter Power (db) False Alarm Target Detection in Noise Detected Strong Target Weak target (not detected) Threshold Range (nmi.) Noise Received background noise fluctuates randomly up and down The target echo also fluctuates. Both are random variables! To decide if a target is present, at a given range, we need to set a threshold (constant or variable) Detection performance (Probability of Detection) depends of the strength of the target relative to that of the noise and the threshold setting Signal-To Noise Ratio and Probability of False Alarm Radar Systems Course 6

7 The Radar Detection Problem Radar Receiver Measurement x Detection Processing Decision H or 0 H 1 For each measurement There are two possibilities: Target absent hypothesis, Noise only H 0 Measurement x = n Probability Density p(x H0 ) Target present hypothesis, Signal plus noise H 1 x = a + n p(x H1) For each measurement There are four decisions: Truth H 0 H 1 Decision H0 H1 Don t Report Missed Detection False Alarm Detection Radar Systems Course 7

8 Threshold Test is Optimum Truth H 0 H 1 H0 H1 Don t Report Missed Detection Decision False Alarm Detection Probability of Detection: P D Probability of False Alarm: P FA The probability we choose when is true H H1 1 The probability we choose when is true H H1 0 Objective: Neyman-Pearson criterion P D P FA Maximize subject to no greater than specified ( α) P FA Likelihood Ratio Likelihood Ratio Test p(x H = 1) L(x) p(x H ) 0 H 1 > η < H 0 Threshold Radar Systems Course 8

9 Basic Target Detection Test Noise Probability Density Noise Only Target Absent Probability Density Detection Threshold Probability of False Alarm ( P FA ) P FA = Prob{ threshold exceeded given target absent } i.e. the chance that noise is called a (false) target We want P FA to be very, very low! Voltage Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course 9

10 Basic Target Detection Test Noise Probability Density Probability of Detection ( P D ) P D = Prob{ threshold exceeded given target present } Probability Density Detection Threshold I.e. the chance that target is correctly detected We want P D to be near 1 (perfect)! Signal-Plus-Noise Probability Density p(x H1) Voltage Probability of False Alarm ( P FA ) Signal + Noise Target Present Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course 10

11 Detection Examples with Different SNR 0.7 Probability Density Noise Only Detection Threshold P FA = 0.01 Signal Plus Noise SNR = 10 db P D = 0.61 Courtesy of MIT Lincoln Laboratory Used with permission Signal Plus Noise SNR = 0 db P D ~ Voltage P D increases with target SNR for a fixed threshold (P FA ) Raising threshold reduces false alarm rate and increases SNR required for a specified Probability of Detection Radar Systems Course 11

12 Non-Fluctuating Target Distributions Noise pdf 0.5 Threshold Signal-Plus-Noise pdf Rayleigh Rician 4 Voltage 6 8 Probability p o r + R p( r H1) r exp Io ( r R ) = r ( r H ) = r exp Set threshold r T based on desired false-alarm probability Compute detection probability for given SNR and false-alarm probability P r = log D T = r T p e P FA ( r H ) dr = Q ( SNR) 1 SNR = R Courtesy of MIT Lincoln Laboratory Used with permission (, log P ) e FA where Q b r + a ( a, b) = r exp I ( a r)dr o Is Marcum s Q-Function (and I 0 (x) is a modified Bessel function) Radar Systems Course 1

13 Probability of Detection vs. SNR 1.0 Probability of Detection (P D ) P FA =10-4 P FA =10-6 P FA =10-8 P FA =10-10 Remember This! SNR = 13. db needed for P D = 0.9 and P FA = 10-6 Steady Target P FA = Radar Systems Course 13 Signal-to-Noise Ratio (db)

14 Probability of Detection vs. SNR P FA =10-4 Probability of Detection (P D ) Radar Systems Course P FA = P FA = P FA = P FA = Signal-to-Noise Ratio (db) Remember This! SNR = 13. db needed for P D = 0.9 and P FA = 10-6 Steady Target

15 Tree of Detection Issues Detection Single Pulse Multiple pulses Fluctuating Non-Fluctuating Non-Coherent Integration Coherent Integration Fluctuating Non-Fluctuating Square Law Detector Linear Detector Fully Correlated Partially Correlated Uncorrelated Swerling I Swerling III Swerling II Swerling IV Radar Systems Course 15 Single Pulse Decision Binary Integration

16 Detection Calculation Methodology Probability of Detection vs. Probability of False Alarm and Signal-to-Noise Ratio Scan to Scan Fluctuations (Swerling Case I and III) Determine PDF at Detector Output Single pulse Fixed S/N Pulse to Pulse Fluctuations (Swerling Case II and IV) Integrate N fixed target pulses Average over target fluctuations Average over signal fluctuations Integrate over N pulses Integrate from threshold, T, to P D vs. P FA, S/N, & Number of pulses, N Radar Systems Course 16

17 Outline Basic concepts Integration of pulses Fluctuating targets Constant false alarm rate (CFAR) thresholding Summary Radar Systems Course 17

18 Integration of Radar Pulses Pulses Coherent Integration 1 N T Adds voltages, then square Phase is preserved pulse-to-pulse phase coherence required SNR Improvement = 10 log 10 N N n= 1 x n > Pulses x1 1 x Sum Calculate N x Threshold n x x N Target Detection Declared if x n n= 1 Calculate x 1 Noncoherent Integration Calculate Detection performance can be improved by integrating multiple pulses x x N Calculate x Calculate x N Target Detection Declared if N x n n= 1 1 N N n= 1 Threshold T Adds powers not voltages Phase neither preserved nor required Easier to implement, not as efficient x > n Radar Systems Course 18

19 Integration of Pulses 1.0 Probability of Detection Ten Pulses Coherent Integration Ten Pulses Non-Coherent Integration Coherent Integration Gain Non-Coherent Integration Gain One Pulse Signal to Noise Ratio per Pulse (db) Steady Target P FA =10-6 For Most Cases, Coherent Integration is More Efficient than Non-Coherent Integration Radar Systems Course 19

20 Different Types of Non-Coherent Integration Non-Coherent Integration Also called ( video integration ) Generate magnitude for each of N pulses Add magnitudes and then threshold Binary Integration (M-of-N Detection) Separately threshold each pulse 1 if signal > threshold; 0 otherwise Count number of threshold crossings (the # of 1s) Threshold this sum of threshold crossings Simpler to implement than coherent and non-coherent Cumulative Detection (1-of-N Detection) Similar to Binary Integration Require at least 1 threshold crossing for N pulses Radar Systems Course 0

21 Binary (M-of-N) Integration Pulse 1 x 1 Pulse x Pulse N x N Calculate x 1 Calculate x Calculate x N Threshold T Threshold T Threshold T Individual pulse detectors: xn T, in = 1 x < T, i 0 n n = i 1 i i N Calculate Sum m = N n= 1 i n m M m < M nd Threshold M nd thresholding:, target present, target absent Target present if at least M detections in N pulses Binary Integration Cumulative Detection At Least M of N Detections Radar Systems Course 1 P M / N N k N k At Least = p ( 1 p) 1 of N P ( ) k! ( N k) N C = 1 1 p! k= M N!

22 Detection Statistics for Binary Integration Probability of Detection /4 /4 3/4 4/4 Steady (Non-Fluctuating ) Target P FA = Signal to Noise Ratio per Pulse (db) 1/4 = 1 Detection Out of 4 Pulses Radar Systems Course

23 Optimum M for Binary Integration 100 Steady (Non-Fluctuating ) Target Optimum M 10 P D =0.95 P FA = Number of Pulses For each binary Integrator, M/N, there exists an optimum M M (optimum) 0.9 N 0.8 Radar Systems Course 3

24 Optimum M for Binary Integration Optimum M varies somewhat with target fluctuation model, P D and P FA Parameters for Estimating M OPT = N a 10 b Target Fluctuations a b Range of N No Fluctuations Swerling I Swerling II Swerling III Swerling IV Adapted from Shnidman in Richards, reference 7 Radar Systems Course 4

25 Detection Statistics for Different Types of Integration Probability of Detection Coherent and Non-Coherent Integration Non-Coherent Integration 4 Pulses Coherent Integration 4 Pulses Binary Integration 3 of 4 Pulses Single Pulse P FA = Signal to Noise Ratio (db) Radar Systems Course 5

28 Signal to Noise Gain / Loss vs. # of Pulses 0 10 Signal to Noise Ratio Gain (db) Signal to Noise Ratio Loss (db) Number of Pulses Number of Pulses Steady Target P D =0.95 P FA =10-6 Non-Coherent Binary (Optimum M) Coherent Binary N 1/ Coherent Integration yields the greatest gain Non-Coherent Integration a small loss Binary integration has a slightly larger loss than regular Non-coherent integration Binary (Optimum M) Non-Coherent Binary N 1/ Relative To Coherent Integration Radar Systems Course 8

29 Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain Equivalent Number of Independent Pulses Radar Systems Course Dependent Sampling Region N= σ / λ v f r Non-coherent Integration Can Be Very Inefficient in Correlated Clutter 10 3 Independent Sampling Region σ v Velocity Spread Of Clutter f r = 1 T Pulse Repetition Rate

30 Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain Equivalent Number of Independent Pulses Radar Systems Course Adapted from nathanson, Reference 8 5 ρ (T) Dependent Sampling Region N= σ / λ v f r Non-coherent Integration Can Be Very Inefficient in Correlated Clutter 10 3 Independent Sampling Region σ v Velocity Spread Of Clutter f r = 1 T Pulse Repetition Rate

31 Albersheim Empirical Formula for SNR (Steady Target - Good Method for Approximate Calculations) Single pulse: Where: SNR(natural units) A Less than. db error for: = log e 0.6 P FA = A A B B B = log e PD 1 P D > PFA > > PD > 0.1 SNR Per Sample Target assumed to be non-fluctuating For n independent integrated samples: Radar Systems Course SNR (db) = 5log10 n log10 n Less than. db error for: For more details, see References 1 or 5 ( A A B 1.7 B) n > n > 1 10 > PFA > > PD > 0.1

33 Fluctuating Target Models RCS vs. Azimuth for a Typical Complex Target For many types of targets, the received radar backscatter amplitude of the target will vary a lot from pulse-to-pulse: Different scattering centers on complex targets can interfere constructively and destructively Small aspect angle changes or frequency diversity of the radar s waveform can cause this effect B-6, 3 GHz RCS versus Azimuth Fluctuating target models are used to more accurately predict detection statistics (P D vs., P FA, and S/N) in the presence of target amplitude fluctuations Radar Systems Course 33

34 Swerling Target Models Nature of Scattering Similar amplitudes RCS Model Exponential (Chi-Squared DOF=) p 1 σ ( σ) = exp σ σ Fluctuation Rate Slow Fluctuation Scan-to-Scan Swerling I Fast Fluctuation Pulse-to-Pulse Swerling II One scatterer much Larger than others (Chi-Squared DOF=4) 4 σ σ ( σ) = exp p σ σ Swerling III Swerling IV σ = Average RCS (m ) Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course 34

35 Swerling Target Models Nature of Scattering Similar amplitudes Amplitude Model Rayleigh Fluctuation Rate Slow Fluctuation Scan-to-Scan Fast Fluctuation Pulse-to-Pulse p a ( a) = exp σ σ a Swerling I Swerling II One scatterer much Larger than others p Central Rayleigh, DOF=4 8 a 3 ( a) = exp σ σ a Swerling III Swerling IV σ = Average RCS (m ) Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course 35

36 Other Fluctuation Models Radar Systems Course 36 Detection Statistics Calculations Steady and Swerling 1,,3,4 Targets in Gaussian Noise Chi- Square Targets in Gaussian Noise Log Normal Targets in Gaussian Noise Steady Targets in Log Normal Noise Log Normal Targets in Log Normal Noise Weibel Targets in Gaussian Noise Chi Square, Log Normal and Weibel Distributions have long tails One more parameter to specify distribution Mean to median ratio for log normal distribution When used Ground clutter Weibel Sea Clutter Log Normal HF noise Log Normal Birds Log Normal Rotating Cylinder Log Normal

37 RCS Variability for Different Target Models Non-fluctuating Target Constant Swerling I/II Swerling III/IV RCS (dbsm) High Fluctuation Medium Fluctuation Radar Systems Course Sample # Courtesy of MIT Lincoln Laboratory Used with permission

38 Fluctuating Target Single Pulse Detection : Rayleigh Amplitude jφ H1 : x = ae + H 0 : x = n Detection Test z Probability of False Alarm (same as for non-fluctuating) Probability of Detection Test = n P x x FA > < T T H H 1 0 T exp σ = N = p(z > T H,a) p(a)dadz PD 1 Rayleigh amplitude model p(a) = Average SNR per pulse a σ exp ξ = σ a σ N σ 1 = 1+ξ P D P FA Radar Systems Course 38 T = exp σ PD N 1 1+ ξ

39 Fluctuating Target Single Pulse Detection 1.0 Probability of Detection (P D ) P FA = Signal-to-Noise Ratio (db) Radar Systems Course 39 Fluctuation Loss Non-Fluctuating Swerling Case 3,4 Swerling Case 1, For high detection probabilities, more signal-to-noise is required for fluctuating targets. The fluctuation loss depends on the target fluctuations, probability of detection, and probability of false alarm.

40 Fluctuating Target Multiple Pulse Detection 1.0 Probability of Detection (P D ) Steady Target Coherent Integration Swerling Fluctuations Non-Coherent Integration, Frequency Diversity Swerling 1 Fluctuations Coherent Integration, Single Frequency P FA = Radar Systems Course 40 N=4 Pulses Signal-to-Noise Ratio per Pulse (db) In some fluctuating target cases, non-coherent integration with frequency diversity (pulse to pulse) can outperform coherent integration

41 Detection Statistics for Different Target Fluctuation Models Probability of Detection (P D ) pulses Non-coherently Integrated P FA =10-8 Steady Target Swerling Case I Swerling Case II Swerling Case III Swerling Case IV Signal-to-Noise Ratio (SNR) (db) Adapted from Richards, Reference 7 Radar Systems Course 41

42 Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets) Analytical forms of SNR vs. P D, P FA, and Number of pulses are quite complex and not amenable to BOTE* calculations Shnidman has developed a set of empirical formulae that are quite accurate for most 1 st order radar systems calculations: K = α = Non-fluctuating target ( Swerling 0 / 5 ) 1, N,, N 0 N 1 4 N > Swerling Case 1 Swerling Case Swerling Case 3 Swerling Case η = ( 4P ( 1 P )) + sign( P 0.5) 0.8 ln( 4P ( 1 P )) 0.8 ln FA FA D D D Adapted from Shnidman in Richards, Reference 7 Radar Systems Course 4 * Back of the Envelope

43 Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets) X C C 1 = η η + = = ((( PD ) PD ) PD 3.55) / K P 5.14) 10 ( ) ( N 0) D e P ln 1 K N + α 1 4 D P FA + 80 C db = C 1 C 1 + C 0.1 P D 0.87 P D C = 10 C 10 db SNR(natural units) = C X N SNR db ( ) = 10 log (SNR) 10 Adapted from Shnidman in Richards, Reference 7 Radar Systems Course 43

44 Shnidman s Equation Error in SNR < 0.5 db within these bounds 0.1 P D P FA N 100 SNR Calculation Error (db) SNR Error vs. Probability of Detection N = 10 pulses P FA = 10-6 Steady Target Swerling I Swerling II Swerling III Swerling IV Adapted from Shnidman in Richards, Reference 7 Radar Systems Course Probability of Detection

46 Practical Setting of Thresholds Display, NEXRAD Radar, of Rain Clouds Ideal Case- Little variation in Noise Power Courtesy of NOAA Power S-Band Data Rain Backscatter Data Rain Cloud Receiver Noise 0 1 Range (nmi) Need to develop a methodology to set target detection threshold that will adapt to: Temporal and spatial changes in the background noise Clutter residue from rain, other diffuse wind blown clutter, Sharp edges due to spatial transitions from one type of background (e.g. noise) to another (e.g. rain) can suppress targets Background estimation distortions due to nearby targets Radar Systems Course 46

47 Constant False Alarm Rate (CFAR) Thresholding CFAR Window Range Cells CFAR Window Range and Doppler Cells Guard Cells Range Cell to be Thresholded Doppler Velocity cells Range cells Data Cells Used to Determine Mean Level of Background Estimate background (noise, etc.) from data Use range, or range and Doppler filter data Set threshold as constant times the mean value of background Mean Background Estimate = 1 N N n= 1 x n Radar Systems Course 47

48 Effect of Rain on CFAR Thresholding Mean Level Threshold CFAR Radar Backscatter (Linear Units) Range Cells Rain Cloud. db Receiver Noise.6 Slant Range, nmi 9 db C Band 5500 MHz Receiver Noise 4.5 Window Slides Through Data Cell Under Test Courtesy of MIT Lincoln Laboratory Used with permission Guard Cells Data Cells for Mean Level Computation Radar Systems Course 48

49 Effect of Rain on CFAR Thresholding Mean Level Threshold CFAR Radar Backscatter (Linear Units) Range Cells Rain Cloud. db Receiver Noise.6 Slant Range, nmi 9 db C Band 5500 MHz Receiver Noise 4.5 Window Slides Through Data Sharp Clutter or Interference Boundaries Can Lead to Excessive False Alarms Courtesy of MIT Lincoln Laboratory Used with permission Cell Under Test Guard Cells Data Cells for Mean Level Computation Radar Systems Course 49

50 Greatest-of Mean Level CFAR Find mean value of N/ cells before and after test cell separately Use larger noise estimate to determine threshold Window Slides Through Data Data Cells for Mean Level 1 Cell Under Test Data Cells for Mean Level Guard Cells Use Larger Value Helps reduce false alarms near sharp clutter or interference boundaries Nearby targets still raise threshold and suppress detection Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course 50

51 Censored Greatest-of CFAR Compute and use noise estimates as in Greatest-of, but remove the largest M samples before computing each average Window Slides Through Data Censored Data (Not Used in Computation of Average) Data Cells for Mean Level 1 Cell Under Test Data Cells for Mean Level Guard Cells Use Larger Value Up to M nearby targets can be in each window without affecting threshold Ordering the samples from each window is computationally expensive Radar Systems Course 51

52 Mean Level CFAR Performance Probability of Detection (P D ) Matched Filter Mean Level CFAR 10 Samples Mean Level CFAR 50 Samples Mean Level CFAR 100 Samples CFAR Loss Signal-to-Noise Ratio per Pulse (db) P FA = Pulse Radar Systems Course 5

53 CFAR Loss vs. Number of Reference Cells 6 5 P FA =10-8 P D = 0.9 CFAR Loss (db) Radar Systems Course Number of Reference Cells Adapted from Richards, Reference 7 The greater the number of reference cells in the CFAR, the better is the estimate of clutter or noise and the less will be the loss in detectability. (Signal to Noise Ratio)

54 CFAR Loss vs Number of Reference Cells CFAR Loss, db N=3 P FA N= For Single Pulse Detection Approximation CFAR Loss (db) = - (5/N) log P FA Dotted Curve P FA = 10-4 Dashed Curve P FA = 10-6 Solid Curve P FA = 10-8 N = 15 to 0 (typically) N=10 N=30 N= Number of Reference Cells, N Since a finite number of cells are used, the estimate of the clutter or noise is not precise. Adapted from Skolnik, Reference 1 Radar Systems Course 54

55 Summary Both target properties and radar design features affect the ability to detect signals in noise Fluctuating targets vs. non-fluctuating targets Allowable false alarm rate and integration scheme (if any) Integration of multiple pulses improves target detection Coherent integration is best when phase information is available Noncoherent integration and frequency diversity can improve detection performance, but usually not as efficient An adaptive detection threshold scheme is needed in real environments Many different CFAR (Constant False Alarm Rate) algorithms exist to solve various problems All CFARs algorithms introduce some loss and additional processing Radar Systems Course 55

56 References 1. Skolnik, M., Introduction to Radar Systems, McGraw-Hill, New York,3 rd Ed., Skolnik, M., Radar Handbook, McGraw-Hill, New York, 3 rd Ed., DiFranco, J. V. and Rubin, W. L., Radar Detection, Artech House, Norwood, MA, Whalen, A. D. and McDonough, R. N., Detection of Signals in Noise, Academic Press, New York, Levanon, N., Radar Principles, Wiley, New York, Van Trees, H., Detection, Estimation, and Modulation Theory, Vols. I and III, Wiley, New York, Richards, M., Fundamentals of Radar Signal Processing, McGraw-Hill, New York, Nathanson, F., Radar Design Principles, McGraw-Hill, New York, rd Ed., Radar Systems Course 56

57 Homework Problems From Skolnik, Reference 1 Problems.5,.6,.15,.17,.18,.8, and.9 Problems 5.13, 5.14, and 5-18 Radar Systems Course 57