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IEEE New Hampshire Section
Radar Systems Course 1
Detection 1/1/2010 IEEE AES Society
Radar Systems Engineering
Lecture 6
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Radar Systems Course 2
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Block Diagram of Radar System
Trans...
Radar Systems Course 3
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Outline
• Basic concepts
– Probabil...
Radar Systems Course 4
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Radar Detection – “The Big Picture”...
Radar Systems Course 5
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Range-Azimuth-Doppler Cells to Be
T...
Radar Systems Course 6
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Target Detection in Noise
• Receive...
Radar Systems Course 7
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The Radar Detection Problem
Radar
R...
Radar Systems Course 8
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Maximize subject to no greater than...
Radar Systems Course 9
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Basic Target Detection Test
0 2 4 6...
Radar Systems Course 10
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Basic Target Detection Test
0 2 4 ...
Radar Systems Course 11
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Signal Plus Noise
SNR = 10 dB
PD =...
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Non-Fluctuating Target Distributio...
Radar Systems Course 13
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Probability of Detection vs. SNRPr...
Radar Systems Course 14
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Probability of Detection vs. SNR
S...
Radar Systems Course 15
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Tree of Detection Issues
Detection...
Radar Systems Course 16
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Detection Calculation Methodology
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Radar Systems Course 17
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Outline
• Basic concepts
• Integra...
Radar Systems Course 18
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Integration of Radar Pulses
Detect...
Radar Systems Course 19
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Integration of Pulses
Signal to No...
Radar Systems Course 20
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Different Types of Non-Coherent In...
Radar Systems Course 21
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Calculate
Binary (M-of-N) Integrat...
Radar Systems Course 22
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Detection Statistics for Binary In...
Radar Systems Course 23
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Optimum M for Binary Integration
O...
Radar Systems Course 24
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Optimum M for Binary Integration
•...
Radar Systems Course 25
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Detection Statistics for Different...
Radar Systems Course 26
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Detection Statistics for Different...
Radar Systems Course 27
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Detection Statistics for Different...
Radar Systems Course 28
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Signal to Noise Gain / Loss vs. # ...
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Effect of Pulse to Pulse Correlati...
Radar Systems Course 30
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Effect of Pulse to Pulse Correlati...
Radar Systems Course 31
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Albersheim Empirical Formula for S...
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Outline
• Basic concepts
• Integra...
Radar Systems Course 33
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Fluctuating Target Models
• For ma...
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Swerling Target Models
=σ Average ...
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Swerling Target Models
Amplitude
M...
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Other Fluctuation Models
• Detecti...
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RCS Variability for Different
Targ...
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Fluctuating Target Single Pulse
De...
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Fluctuating Target Single Pulse De...
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Fluctuating Target Multiple Pulse ...
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Detection Statistics for Different...
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Shnidman Empirical Formulae for SN...
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Shnidman Empirical Formulae for SN...
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Shnidman’s Equation
• Error in SNR...
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Outline
• Basic concepts
• Integra...
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Practical Setting of Thresholds
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Constant False Alarm Rate (CFAR)
T...
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Effect of Rain on CFAR Thresholdin...
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Effect of Rain on CFAR Thresholdin...
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Greatest-of Mean Level CFAR
• Find...
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Censored Greatest-of CFAR
• Comput...
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Mean Level CFAR Performance
PFA=10...
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CFAR Loss vs. Number of Reference ...
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CFAR Loss vs Number of Reference C...
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Summary
• Both target properties a...
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References
1. Skolnik, M., Introdu...
Radar Systems Course 57
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Homework Problems
• From Skolnik, ...
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Radar 2009 a 6 detection of signals in noise

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Radar Systems Engineering Lect 6

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Radar 2009 a 6 detection of signals in noise

  1. 1. IEEE New Hampshire Section Radar Systems Course 1 Detection 1/1/2010 IEEE AES Society Radar Systems Engineering Lecture 6 Detection of Signals in Noise Dr. Robert M. O’Donnell IEEE New Hampshire Section Guest Lecturer
  2. 2. Radar Systems Course 2 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Block Diagram of Radar System Transmitter Waveform Generation Power Amplifier T / R Switch Antenna Propagation Medium Target Radar Cross Section Photo Image Courtesy of US Air Force Used with permission. Pulse Compression Receiver Clutter Rejection (Doppler Filtering) A / D Converter General Purpose Computer Tracking Data Recording Parameter Estimation Detection Signal Processor Computer Thresholding User Displays and Radar Control This Lecture
  3. 3. Radar Systems Course 3 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  4. 4. Radar Systems Course 4 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Radar Detection – “The Big Picture” • Mission – Detect and track all aircraft within 60 nmi of radar • S-band λ ~ 10 cm Example – Typical Aircraft Surveillance Radar ASR-9 Courtesy of MIT Lincoln Laboratory Used with permission Rotation Rate 12. rpm Range 60 nmi. Transmits Pulses at ~ 1250 Hz
  5. 5. Radar Systems Course 5 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Range-Azimuth-Doppler Cells to Be Thresholded Rotation Rate 12.7 rpm 10 Pulses / Half BW Processed into 10 Doppler Filters Example – Typical Aircraft Surveillance Radar ASR-9 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. 0 50 100 -25 -50 0 Radial Velocity (kts) Magnitude(dB) Doppler Filter Bank Az Beamwidth ~1.2 ° As Antenna Rotates ~22 pulses / Beamwidth Is There a Target Present in Each Cell? Range Resolution 1/16 nmi Radar Transmits Pulses at ~ 1250 Hz Range 60 nmi.
  6. 6. Radar Systems Course 6 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Target Detection in 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 ReceivedBackscatterPower(dB) 20 40 30 10 Noise Threshold False Alarm Detected Strong Target Weak target (not detected) 0 0 25 50 100 125 150 175 Range (nmi.)
  7. 7. Radar Systems Course 7 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society The Radar Detection Problem Radar Receiver Detection Processing Measurement Decision x 10 HH or nx = Measurement Probability Density Target absent hypothesis, Noise only Target present hypothesis, Signal plus noise nax += )Hx(p 0 1H )Hx(p 1 For each measurement There are two possibilities: For each measurement There are four decisions: 0H 0H 1H 1H Don’t Report False Alarm Detection Missed Detection Truth Decision 0H
  8. 8. Radar Systems Course 8 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Maximize subject to no greater than specifiedFAPDP ( )α≤FAP Threshold Test is Optimum 0H 0H 1H 1H Don’t Report False Alarm Detection Missed Detection Truth Decision Probability of Detection: Probability of False Alarm: DP FAP The probability we choose when is true The probability we choose when is true 1H 1H 1H 0H Likelihood Ratio Test > η= )Hx(p )Hx(p )x(L 0 1 < 1H 0H Likelihood Ratio Threshold Objective: Neyman-Pearson criterion
  9. 9. Radar Systems Course 9 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Basic Target Detection Test 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Voltage ProbabilityDensity Detection Threshold Noise Probability Density Noise Only Target Absent Probability of False Alarm ( PFA ) PFA = Prob{ threshold exceeded given target absent } i.e. the chance that noise is called a (false) target We want PFA to be very, very low! Probability of False Alarm ( PFA ) PFA = Prob{ threshold exceeded given target absent } i.e. the chance that noise is called a (false) target We want PFA to be very, very low! Courtesy of MIT Lincoln Laboratory Used with permission
  10. 10. Radar Systems Course 10 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Basic Target Detection Test 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Voltage ProbabilityDensity Detection Threshold Noise Probability Density Signal-Plus-Noise Probability Density Probability of False Alarm ( PFA ) Probability of False Alarm ( PFA ) Signal + Noise Target Present Probability of Detection ( PD ) PD = Prob{ threshold exceeded given target present } I.e. the chance that target is correctly detected We want PD to be near 1 (perfect)! Probability of Detection ( PD ) PD = Prob{ threshold exceeded given target present } I.e. the chance that target is correctly detected We want PD to be near 1 (perfect)! )Hx(p 1 Courtesy of MIT Lincoln Laboratory Used with permission
  11. 11. Radar Systems Course 11 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Signal Plus Noise SNR = 10 dB PD = 0.61 Signal Plus Noise SNR = 20 dB PD ~ 1 Detection Examples with Different SNR • PD increases with target SNR for a fixed threshold (PFA) • Raising threshold reduces false alarm rate and increases SNR required for a specified Probability of Detection • PD increases with target SNR for a fixed threshold (PFA) • Raising threshold reduces false alarm rate and increases SNR required for a specified Probability of Detection 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Voltage ProbabilityDensity Noise Only Detection Threshold PFA = 0.01 Courtesy of MIT Lincoln Laboratory Used with permission
  12. 12. Radar Systems Course 12 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Non-Fluctuating Target Distributions 0 2 4 6 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Voltage Probability Threshold Noise pdf Signal-Plus-Noise pdf Set threshold rT based on desired false-alarm probability Compute detection probability for given SNR and false-alarm probability Is Marcum’s Q-Function (and I0(x) is a modified Bessel function) where Rayleigh Rician ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −= 2 r exprHrp 2 o ( ) ( )RrI 2 Rr exprHrp o 2 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + −= 2 R SNR = FAeT Plog2r −= ( ) ( )( )FAe r 1D Plog2,SNR2QdrHrpP T −== ∫ ∞ ( ) ( )drraI 2 ar exprb,aQ o b 22 ∫ ∞ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + −= Courtesy of MIT Lincoln Laboratory Used with permission
  13. 13. Radar Systems Course 13 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Probability of Detection vs. SNRProbabilityofDetection(PD) Signal-to-Noise Ratio (dB) PFA=10-4 PFA=10-6 PFA=10-10 PFA=10-12 PFA=10-8 4 6 8 10 12 14 16 18 0.6 1.0 0.4 0.0 0.2 0.8 SNR = 13.2 dB needed for PD = 0.9 and PFA = 10-6 Steady Target Remember This!
  14. 14. Radar Systems Course 14 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Probability of Detection vs. SNR SNR = 13.2 dB needed for PD = 0.9 and PFA = 10-6 Steady Target Remember This! ProbabilityofDetection(PD) Signal-to-Noise Ratio (dB) 4 6 8 10 12 14 16 18 20 0.7 0.9999 0.1 0.3 0.5 0.05 0.9 0.2 0.999 0.99 0.8 0.95 0.98 PFA=10-4 PFA=10-6 PFA=10-10 PFA=10-12 PFA=10-8
  15. 15. Radar Systems Course 15 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Tree of Detection Issues Detection Non-Fluctuating Multiple pulses Swerling I Fluctuating Fluctuating Single Pulse Square Law Detector Coherent Integration Single Pulse Decision Binary Integration Non-Coherent Integration Linear Detector Non-Fluctuating Uncorrelated Swerling IISwerling III Swerling IV Fully Correlated Partially Correlated
  16. 16. Radar Systems Course 16 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Calculation Methodology Probability of Detection vs. Probability of False Alarm and Signal-to-Noise Ratio Determine PDF at Detector Output • Single pulse • Fixed S/N Integrate N fixed target pulses Scan to Scan Fluctuations (Swerling Case I and III) Pulse to Pulse Fluctuations (Swerling Case II and IV) Average over signal fluctuations Integrate over N pulses Average over target fluctuations Integrate from threshold, T, to ∞ PD vs. PFA, S/N, & Number of pulses, N
  17. 17. Radar Systems Course 17 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Outline • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary
  18. 18. Radar Systems Course 18 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Integration of Radar Pulses Detection performance can be improved by integrating multiple pulses Calculate Threshold Calculate Coherent Integration Noncoherent Integration • Adds ‘voltages’ , then square • Phase is preserved • pulse-to-pulse phase coherence required • SNR Improvement = 10 log10 N • Adds ‘powers’ not voltages • Phase neither preserved nor required • Easier to implement, not as efficient … Pulses Tx N 1 2N 1n n >∑= Tx N 1 N 1n 2 n >∑= Sum Calculate Threshold Pulses … 2N 1n nx∑= nx Nx 2x 1x 1x Nx 2 1x Calculate 2 2x Calculate 2 Nx 2x ∑= N 1n 2 nx Target Detection Declared if Target Detection Declared if
  19. 19. Radar Systems Course 19 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Integration of Pulses Signal to Noise Ratio per Pulse (dB) 0 5 10 15 ProbabilityofDetection 0.6 1.0 0.4 For Most Cases, Coherent Integration is More Efficient than Non-Coherent Integration 0.2 0.8 Steady Target PFA=10-6 One Pulse Ten Pulses Coherent Integration Ten Pulses Non-Coherent Integration Coherent Integration Gain Non-Coherent Integration Gain
  20. 20. Radar Systems Course 20 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  21. 21. Radar Systems Course 21 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Calculate Binary (M-of-N) Integration Threshold T Calculate Sum… Pulse 1 Pulse 2 Pulse Threshold T Threshold T 2nd Threshold M Individual pulse detectors: Target present if at least M detections in N pulses 1x 1i Nx 2x 1i,Tx n 2 n =≥ ∑= = N 1n nim N 2nd thresholding: 2i Ni 0i,Tx n 2 n =< Mm ≥ Mm < , target present , target absent At Least M of N Detections At Least 1 of N ( ) ( ) kNk N Mk N/M p1p !kN!k !N P − = − − = ∑ ( )N C p11P −−= 2 1x Calculate 2 2x Calculate 2 Nx Binary Integration Cumulative Detection
  22. 22. Radar Systems Course 22 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Binary Integration Signal to Noise Ratio per Pulse (dB) ProbabilityofDetection 2 4 6 8 10 12 14 0.01 0.10 0.50 0.90 0.99 0.999 1/4 2/4 3/4 4/4 “1/4” = 1 Detection Out of 4 Pulses Steady (Non-Fluctuating ) Target PFA=10-6
  23. 23. Radar Systems Course 23 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Optimum M for Binary Integration OptimumM 1 10 100 1 10 100 Number of Pulses Steady (Non-Fluctuating ) Target PD=0.95 PFA=10-6 For each binary Integrator, M/N, there exists an optimum M M (optimum) ≈ 0.9 N0.8
  24. 24. Radar Systems Course 24 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Optimum M for Binary Integration • Optimum M varies somewhat with target fluctuation model, PD and PFA • Parameters for Estimating MOPT = Na 10b Adapted from Shnidman in Richards, reference 7 Target Fluctuations a b Range of N No Fluctuations 0.8 - 0.02 5 – 700 Swerling I 0.8 - 0.02 6 – 500 Swerling II 0.91 - 0.38 9 – 700 Swerling III 0.8 - 0.02 6 – 700 Swerling IV 0.873 - 0.27 10 – 700
  25. 25. Radar Systems Course 25 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration PFA=10-6 Coherent and Non-Coherent Integration 2 4 6 8 10 12 14 16 0.10 0.50 0.90 0.95 0.99 0.999 0.01 Coherent Integration 4 Pulses Non-Coherent Integration 4 Pulses Binary Integration 3 of 4 Pulses Single Pulse Signal to Noise Ratio (dB) ProbabilityofDetection
  26. 26. Radar Systems Course 26 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration Coherent and Non-Coherent Integration 0.10 0.50 0.90 0.95 0.99 0.999 0.01 2 4 6 8 10 12 14 16 PFA=10-6 Non-Coherent Integration 4 Pulses Coherent Integration 4 Pulses Binary Integration 3 of 4 Pulses Single Pulse ProbabilityofDetection Signal to Noise Ratio (dB)
  27. 27. Radar Systems Course 27 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration Coherent and Non-Coherent Integration 0.10 0.50 0.90 0.95 0.99 0.999 0.01 2 4 6 8 10 12 14 16 PFA=10-6 Non-Coherent Integration 4 Pulses Coherent Integration 4 Pulses Binary Integration 3 of 4 Pulses Single Pulse ProbabilityofDetection Signal to Noise Ratio (dB)
  28. 28. Radar Systems Course 28 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Signal to Noise Gain / Loss vs. # of Pulses Number of Pulses Number of Pulses 1 10 100 1 10 100 SignaltoNoiseRatioGain(dB) SignaltoNoiseRatioLoss(dB) 0 5 15 10 20 2 4 10 6 0 8 Steady Target PD=0.95 PFA=10-6 • Coherent Integration yields the greatest gain • Non-Coherent Integration a small loss • Binary integration has a slightly larger loss than regular Non-coherent integration Coherent Binary (Optimum M) Binary N1/2 Non-Coherent Binary N1/2 Binary (Optimum M) Non-Coherent Relative To Coherent Integration
  29. 29. Radar Systems Course 29 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain • Non-coherent Integration Can Be Very Inefficient in Correlated Clutter EquivalentNumberofIndependentPulses N =300 50 30 20 10 3 100 0.001 0.01 0.1 1 1 5 2 50 10 100 20 rv f/ λσ Independent Sampling Region Dependent Sampling Region vσ T 1 fr = Velocity Spread Of Clutter Pulse Repetition Rate
  30. 30. Radar Systems Course 30 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain • Non-coherent Integration Can Be Very Inefficient in Correlated Clutter )T(ρ .99 0.9 0.1 0.02 EquivalentNumberofIndependentPulses N =300 50 30 20 10 3 100 0.001 0.01 0.1 1 1 5 2 50 10 100 20 rv f/ λσ Independent Sampling Region Dependent Sampling Region vσ T 1 fr = Velocity Spread Of Clutter Pulse Repetition Rate Adapted from nathanson, Reference 8
  31. 31. Radar Systems Course 31 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Albersheim Empirical Formula for SNR (Steady Target - Good Method for Approximate Calculations) • Single pulse: – Where: – Less than .2 dB error for: – Target assumed to be non-fluctuating • For n independent integrated samples: – Less than .2 dB error for: – For more details, see References 1 or 5 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ++= D D e FA e P1 P logB P 62.0 logA B7.1BA12.0A)unitsnatural(SNR ( )B7.1BA12.0Alog 44.0n 54.4 2.6nlog5)dB(SNR 1010n ++⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ++−= 1.0P9.010P10 D 7 FA 3 >>>> −− 1.0P9.010P101n8096 D 7 FA 3 >>>>>> −− SNR Per Sample
  32. 32. Radar Systems Course 32 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Outline • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary
  33. 33. Radar Systems Course 33 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Fluctuating Target Models • 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 • Fluctuating target models are used to more accurately predict detection statistics (PD vs., PFA, and S/N) in the presence of target amplitude fluctuationsRCS versus Azimuth B-26, 3 GHz RCS vs. Azimuth for a Typical Complex Target
  34. 34. Radar Systems Course 34 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Swerling Target Models =σ Average RCS (m2) RCS Model Nature of Scattering Fast Fluctuation “Pulse-to-Pulse” Slow Fluctuation “Scan-to-Scan” Fluctuation Rate Swerling I Swerling IVSwerling III Swerling II Similar amplitudes One scatterer much Larger than others Exponential (Chi-Squared DOF=2) (Chi-Squared DOF=4) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ σ σ − σ =σ exp 1 p ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ σ σ − σ σ =σ 2 exp 4 p 2 Courtesy of MIT Lincoln Laboratory Used with permission
  35. 35. Radar Systems Course 35 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Swerling Target Models Amplitude Model Nature of Scattering Fast Fluctuation “Pulse-to-Pulse” Slow Fluctuation “Scan-to-Scan” Fluctuation Rate Swerling I Swerling IVSwerling III Swerling II Similar amplitudes One scatterer much Larger than others =σ Average RCS (m2) Central Rayleigh, DOF=4 Rayleigh ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ σ − σ = 2 a exp a2 ap ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ σ − σ = 2 2 3 a2 exp a8 ap Courtesy of MIT Lincoln Laboratory Used with permission
  36. 36. Radar Systems Course 36 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Other Fluctuation Models • Detection Statistics Calculations – Steady and Swerling 1,2,3,4 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. 37. Radar Systems Course 37 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society RCS Variability for Different Target Models 0 5 10 15 20 0 5 10 15 20 RCS(dBsm) 0 20 40 60 80 100 0 5 10 15 20 Sample # Non-fluctuating Target Swerling I/II Swerling III/IV Constant High Fluctuation Medium Fluctuation Courtesy of MIT Lincoln Laboratory Used with permission
  38. 38. Radar Systems Course 38 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Fluctuating Target Single Pulse Detection : Rayleigh Amplitude ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ σ − σ = 2 a exp a2 )a(p 2 Nσ σ =ξ0 1 HTx HTxz < >= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ σ −= 2 N 2 FA T expP ∫ >= dzda)a(p)a,HTz(pP 1D ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ξ+σ −= 1 1T expP 2 N 2 D naex:H j 1 += φ Detection Test Probability of False Alarm (same as for non-fluctuating) Average SNR per pulse Rayleigh amplitude model Probability of Detection Test ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ξ+= 1 1 FAD PP nx:H0 =
  39. 39. Radar Systems Course 39 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Fluctuating Target Single Pulse DetectionProbabilityofDetection(PD) 0.6 1.0 0.4 0.0 0.2 0.8 Signal-to-Noise Ratio (dB) 0 5 10 15 20 25 Non-Fluctuating Swerling Case 3,4 Swerling Case 1,2 Fluctuation Loss 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. PFA=10-6
  40. 40. Radar Systems Course 40 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Fluctuating Target Multiple Pulse Detection ProbabilityofDetection(PD) 0.6 1.0 0.4 0.0 0.2 0.8 Signal-to-Noise Ratio per Pulse (dB) 0 4 8 12 16 20 Steady Target Coherent Integration Swerling 2 Fluctuations Non-Coherent Integration, Frequency Diversity Swerling 1 Fluctuations Coherent Integration, Single Frequency PFA=10-6 N=4 Pulses • In some fluctuating target cases, non-coherent integration with frequency diversity (pulse to pulse) can outperform coherent integration
  41. 41. Radar Systems Course 41 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Target Fluctuation Models Steady Target Swerling Case I Swerling Case II Swerling Case III Swerling Case IV Signal-to-Noise Ratio (SNR) (dB) ProbabilityofDetection(PD) -2 0 2 4 6 8 10 12 14 0.0 0.4 0.6 0.8 1.0 0.2 10 pulses Non-coherently Integrated PFA=10-8 Adapted from Richards, Reference 7
  42. 42. Radar Systems Course 42 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets) • Analytical forms of SNR vs. PD, PFA, 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 1st order radar systems calculations: * Back of the Envelope ( )( ) ( ) ( )( )DDDFAFA P1P4ln8.05.0PsignP1P4ln8.0 −−−+−−=η =α 40N0 ≤ 40N 4 1 > N2 ,2 ,NK ,1 = ∞ Non-fluctuating target (“Swerling 0 / 5”) Swerling Case 1 Swerling Case 2 Swerling Case 3 Swerling Case 4 Adapted from Shnidman in Richards, Reference 7
  43. 43. Radar Systems Course 43 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets) ( )( )( ) ( ) ( ) ( ) )SNR(log10dBSNR N XC )unitsnatural(SNR 10CC 80 20N2 P 10 ln7.08.0Pe K 1 C K/525.3P5339.14P4496.18P7006.17C 4 1 2 N 2X 10 10 C dB FA 5 D )14.25P31.27 2 DDD1 dB D == == ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+= −+−= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −α++ηη= ∞ − − ∞ 99.0P872.0CC 872.0P1.0C D21 D1 ≤≤+ ≤≤ Adapted from Shnidman in Richards, Reference 7
  44. 44. Radar Systems Course 44 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Shnidman’s Equation • Error in SNR < 0.5 dB within these bounds – 0.1 ≤ PD ≤ 0.99 10-9 ≤ PFA ≤ 10-3 1 ≤ N ≤ 100 Steady Target Swerling I Swerling II Swerling III Swerling IV SNRCalculationError(dB) Probability of Detection 0.0 0.2 0.4 0.6 0.8 1.0 0.0 -0.5 1.0 0.5 -1.0 SNR Error vs. Probability of Detection N = 10 pulses PFA= 10-6 Adapted from Shnidman in Richards, Reference 7
  45. 45. Radar Systems Course 45 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Outline • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary
  46. 46. Radar Systems Course 46 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Practical Setting of Thresholds • 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 Display, NEXRAD Radar, of Rain Clouds Range (nmi) Rain Cloud Receiver Noise 0 1 2 S-Band Data 0 20 40 Rain Backscatter Data Ideal Case- Little variation in Noise Power Power Courtesy of NOAA
  47. 47. Radar Systems Course 47 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Constant False Alarm Rate (CFAR) Thresholding • 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 = CFAR Window – Range Cells CFAR Window – Range and Doppler Cells Range Range cells DopplerVelocitycells Cell to be Thresholded “Guard” Cells Data Cells Used to Determine Mean Level of Background ∑= N 1n nx N 1
  48. 48. Radar Systems Course 48 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Effect of Rain on CFAR Thresholding Cell Under Test “Guard” Cells Data Cells for Mean Level Computation Window Slides Through Data 2.2 dB 9 dB 4.52.6 Slant Range, nmi RadarBackscatter(LinearUnits) Rain Cloud C Band 5500 MHz Receiver Noise Receiver Noise Range Cells Mean Level Threshold CFAR Courtesy of MIT Lincoln Laboratory Used with permission
  49. 49. Radar Systems Course 49 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Effect of Rain on CFAR Thresholding 2.2 dB 9 dB 4.52.6 Slant Range, nmi Rain Cloud C Band 5500 MHz Range Cells Cell Under Test “Guard” Cells Data Cells for Mean Level Computation Window Slides Through Data Receiver Noise Receiver Noise Sharp Clutter or Interference Boundaries Can Lead to Excessive False Alarms RadarBackscatter(LinearUnits) Mean Level Threshold CFAR Courtesy of MIT Lincoln Laboratory Used with permission
  50. 50. Radar Systems Course 50 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Greatest-of Mean Level CFAR • Find mean value of N/2 cells before and after test cell separately • Use larger noise estimate to determine threshold • Helps reduce false alarms near sharp clutter or interference boundaries • Nearby targets still raise threshold and suppress detection Cell Under Test “Guard” Cells Data Cells for Mean Level 1 Window Slides Through Data Data Cells for Mean Level 2 Use Larger Value Courtesy of MIT Lincoln Laboratory Used with permission
  51. 51. Radar Systems Course 51 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Censored Greatest-of CFAR • Compute and use noise estimates as in Greatest-of, but remove the largest M samples before computing each average • Up to M nearby targets can be in each window without affecting threshold • Ordering the samples from each window is computationally expensive Cell Under Test “Guard” Cells Data Cells for Mean Level 1 Window Slides Through Data Data Cells for Mean Level 2 Use Larger Value “Censored” Data (Not Used in Computation of Average)
  52. 52. Radar Systems Course 52 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Mean Level CFAR Performance PFA=10-6 1 PulseMatched 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) ProbabilityofDetection(PD) 4 6 8 10 12 14 16 18 0.01 0.999 0.99 0.90 0.50 0.10 0.9999
  53. 53. Radar Systems Course 53 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society CFAR Loss vs. Number of Reference Cells PFA=10-8 10-6 10-4 Number of Reference Cells 0 10 20 30 40 50 CFARLoss(dB) 2 1 0 3 4 5 6 PD= 0.9 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) Adapted from Richards, Reference 7
  54. 54. Radar Systems Course 54 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society CFAR Loss vs Number of Reference Cells For Single Pulse Detection Approximation CFAR Loss (dB) = - (5/N) log PFA Dotted Curve PFA = 10-4 Dashed Curve PFA = 10-6 Solid Curve PFA = 10-8 N = 15 to 20 (typically) Since a finite number of cells are used, the estimate of the clutter or noise is not precise. Number of Reference Cells, N CFARLoss,dB N=100 N=3 N=10 N=30 N=1 PFA 10-8 10-6 10-4 2 5 7 10 20 50 70 100 4 0.4 1 0.5 0.2 0.3 0.7 3 2 5 Adapted from Skolnik, Reference 1
  55. 55. Radar Systems Course 55 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society 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
  56. 56. Radar Systems Course 56 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society References 1. Skolnik, M., Introduction to Radar Systems, McGraw-Hill, New York,3rd Ed., 2001. 2. Skolnik, M., Radar Handbook, McGraw-Hill, New York, 3rd Ed., 2008. 3. DiFranco, J. V. and Rubin, W. L., Radar Detection, Artech House, Norwood, MA, 1994. 4. Whalen, A. D. and McDonough, R. N., Detection of Signals in Noise, Academic Press, New York, 1995. 5. Levanon, N., Radar Principles, Wiley, New York, 1988 6. Van Trees, H., Detection, Estimation, and Modulation Theory, Vols. I and III, Wiley, New York, 2001 7. Richards, M., Fundamentals of Radar Signal Processing, McGraw-Hill, New York, 2005 8. Nathanson, F., Radar Design Principles, McGraw-Hill, New York, 2rd Ed., 1999.
  57. 57. Radar Systems Course 57 Detection 11/1/2010 IEEE New Hampshire Section IEEE AES Society Homework Problems • From Skolnik, Reference 1 – Problems 2.5, 2.6, 2.15, 2.17, 2.18, 2.28, and 2.29 – Problems 5.13 , 5.14, and 5-18

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