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Radar Systems Engineering Lecture 6 Detection of Signals in Noise Dr. Robert M. O’Donnell IEEE New Hampshire Section Guest Lecturer IEEE New Hampshire Section Radar Systems Course 1 Detection 1/1/2010 IEEE AES Society Block Diagram of Radar System Transmitter Propagation Medium Target Radar Cross Section Power Amplifier Waveform Generation T/R Switch Antenna Receiver Signal Processor Computer A/D Converter Pulse Compression Clutter Rejection (Doppler Filtering) User Displays and Radar Control General Purpose Computer Tracking Parameter Estimation Thresholding Detection Data Recording This Lecture Photo Image Courtesy of US Air Force Used with permission. Radar Systems Course Detection 11/1/2010 2 IEEE New Hampshire Section IEEE AES Society Outline • Basic concepts – Probabilities of detection and false alarm – Signal-to-noise ratio Radar Systems Course Detection 11/1/2010 3 • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary IEEE New Hampshire Section IEEE AES Society 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 ~ 1250 Hz Rotation Rate 12. rpm • • Mission – Detect and track all aircraft within 60 nmi of radar S-band λ ~ 10 cm Radar Systems Course Detection 11/1/2010 4 IEEE New Hampshire Section IEEE AES Society Range-Azimuth-Doppler Cells to Be Thresholded Range 60 nmi. Example – Typical Aircraft Surveillance Radar ASR-9 Magnitude (dB) 10 Pulses / Half BW Processed into 10 Doppler Filters Doppler Filter Bank Range Resolution 1/16 nmi 0 -25 -50 0 Rotation Rate 12.7 rpm 50 100 Radial Velocity (kts) As Antenna Rotates ~22 pulses / Beamwidth 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. Radar Systems Course Detection 11/1/2010 5 Az Beamwidth ~1.2 ° Radar Transmits Pulses at ~ 1250 Hz Is There a Target Present in Each Cell? IEEE New Hampshire Section IEEE AES Society Received Backscatter Power (dB) Target Detection in Noise • • • • 40 30 False Alarm Detected Strong Target 20 Threshold Weak target (not detected) Noise 10 0 0 25 50 100 Range (nmi.) 125 150 175 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 Detection 11/1/2010 6 IEEE New Hampshire Section IEEE AES Society The Radar Detection Problem Radar Receiver Measurement x For each measurement There are two possibilities: H0 or H1 Measurement Probability Density Target absent hypothesis, H 0 Noise only x=n p( x H 0 ) Target present hypothesis, H 1 Signal plus noise x = a+n p( x H 1 ) Decision H0 H1 For each measurement There are four decisions: H0 Truth H1 Radar Systems Course Detection 11/1/2010 Decision Detection Processing 7 Don’t Report False Alarm Missed Detection Detection IEEE New Hampshire Section IEEE AES Society Threshold Test is Optimum Decision H0 Truth H0 H1 Don’t Report False Alarm Missed Probability of Detection: PD H1 Detection Detection Objective: Neyman-Pearson criterion Likelihood Ratio The probability we choose H1 when H1 is true Probability of False Alarm: The probability we choose H1 when H 0 is true PFA Maximize PD subject to PFA no greater than specified (PFA ≤ α ) Likelihood Ratio Test p( x H 1 ) L( x ) = p( x H 0 ) Threshold H1 > <H η 0 Radar Systems Course Detection 11/1/2010 8 IEEE New Hampshire Section IEEE AES Society Basic Target Detection Test 0.7 Noise Probability Density 0.6 Probability Density Noise Only Target Absent Detection Threshold 0.5 Probability ) Probability of of False False Alarm Alarm (( P PFA FA ) P = Prob{ threshold exceeded given target absent } PFA FA = Prob{ threshold exceeded given target absent } i.e. i.e. the the chance chance that that noise noise is is called called aa (false) (false) target target We to be very, very low! We want want P PFA FA to be very, very low! 0.4 0.3 0.2 0.1 0 0 2 4 Voltage 6 8 Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course Detection 11/1/2010 9 IEEE New Hampshire Section IEEE AES Society Basic Target Detection Test 0.7 Noise Probability Density Probability Density 0.6 Detection Threshold 0.5 Probability Probability of of Detection Detection (( P PDD )) P PDD == Prob{ Prob{ threshold threshold exceeded exceeded given given target target present present }} I.e. I.e. the the chance chance that that target target is is correctly correctly detected detected We to be be near near 11 (perfect)! (perfect)! We want want P PDD to 0.4 Signal-Plus-Noise Probability Density 0.3 0.2 p( x H 1 ) 0.1 0 0 Probability Probability of of False Alarm ) False Alarm (( P PFA FA ) Radar Systems Course Detection 11/1/2010 10 Signal + Noise Target Present 2 4 Voltage 6 8 Courtesy of MIT Lincoln Laboratory Used with permission IEEE New Hampshire Section IEEE AES Society Detection Examples with Different SNR 0.7 Probability Density 0.6 Noise Only Courtesy of MIT Lincoln Laboratory Used with permission Detection Threshold PFA = 0.01 0.5 0.4 Signal Plus Noise SNR = 10 dB PD = 0.61 0.3 Signal Plus Noise SNR = 20 dB PD ~ 1 0.2 0.1 0 0 5 10 15 Voltage • • PDD increases with target SNR for a fixed threshold (PFA ) FA Raising threshold reduces false alarm rate and increases SNR required for a specified Probability of Detection Radar Systems Course Detection 11/1/2010 11 IEEE New Hampshire Section IEEE AES Society Probability Non-Fluctuating Target Distributions Rayleigh 0.7 0.6 0.5 0.4 0.3 0.2 0.1 00 Noise pdf Threshold 2 Rician ⎛ r2 ⎞ p ( r H o ) = r exp ⎜⎜ − ⎟⎟ ⎝ 2⎠ Signal-Plus-Noise pdf 4 6 Voltage ( ⎛ r2 + R ⎞ ⎟⎟ I o r R p ( r H 1 ) = r exp ⎜⎜ − 2 ⎠ ⎝ Set threshold rT based on desired false-alarm probability rT = − 2 log e PFA ∞ ) 8 SNR = R 2 Courtesy of MIT Lincoln Laboratory Used with permission Compute detection probability for P = p ( r H 1 ) dr = Q given SNR and false-alarm probability D ∫ ( 2 ( SNR ), − 2 log e PFA ) rT ∞ 2 2 ⎞ ⎛ r a + where Q ( a, b ) = r exp ⎜ − ⎟⎟ I o ( a r ) dr ∫b ⎜ 2 ⎠ ⎝ Radar Systems Course Detection 11/1/2010 12 Is Marcum’s Q-Function (and I0(x) is a modified Bessel function) IEEE New Hampshire Section IEEE AES Society Probability of Detection vs. SNR Probability of Detection (PD) 1.0 Remember This! PFA=10-4 0.8 SNR = 13.2 dB needed for PD = 0.9 and PFA = 10-6 PFA=10-6 0.6 Steady Target 0.4 PFA=10-8 0.2 PFA=10-10 PFA=10-12 0.0 4 6 8 10 12 14 16 18 Signal-to-Noise Ratio (dB) Radar Systems Course Detection 11/1/2010 13 IEEE New Hampshire Section IEEE AES Society Probability of Detection vs. SNR 0.9999 Probability of Detection (PD) 0.999 PFA=10-4 0.99 0.98 PFA=10-6 0.95 0.9 Remember This! 0.8 0.7 0.5 PFA=10-10 0.3 0.2 0.1 0.05 4 PFA=10-12 6 8 10 12 14 Signal-to-Noise Ratio (dB) Radar Systems Course Detection 11/1/2010 14 SNR = 13.2 dB needed for PD = 0.9 and PFA = 10-6 PFA=10-8 16 18 Steady Target 20 IEEE New Hampshire Section IEEE AES Society Tree of Detection Issues Detection Multiple pulses Single Pulse Fluctuating Non-Coherent Integration Non-Fluctuating Fluctuating Square Law Detector Fully Correlated Swerling I Partially Correlated Swerling III Single Pulse Decision Radar Systems Course Detection 11/1/2010 15 Coherent Integration Non-Fluctuating Linear Detector Uncorrelated Swerling II Swerling IV Binary Integration 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 Scan to Scan Fluctuations (Swerling Case I and III) Pulse to Pulse Fluctuations (Swerling Case II and IV) Integrate N fixed target pulses Average over signal fluctuations Average over target fluctuations Integrate over N pulses Integrate from threshold, T, to ∞ Radar Systems Course Detection 11/1/2010 16 PD vs. PFA, S/N, & Number of pulses, N IEEE New Hampshire Section IEEE AES Society Outline Radar Systems Course Detection 11/1/2010 17 • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary IEEE New Hampshire Section IEEE AES Society Integration of Radar Pulses Coherent Integration Pulses x1 x2 Sum … xn Calculate 2 N n =1 Threshold n xN Target Detection Declared if • • • • 1 N n =1 Calculate x2 Calculate xN 2 N ∑x x1 … ∑x n Noncoherent Integration Pulses >T Adds ‘voltages’ , then square Phase is preserved pulse-to-pulse phase coherence required SNR Improvement = 10 log10 N x12 x Calculate N ∑x 2 2 Calculate n =1 Threshold 2 n x 2N Target Detection Declared if 1 N 2 xn > T ∑ N n =1 • Adds ‘powers’ not voltages • Phase neither preserved nor required • Easier to implement, not as efficient Detection performance can be improved by integrating multiple pulses Radar Systems Course Detection 11/1/2010 18 IEEE New Hampshire Section IEEE AES Society Integration of Pulses 1.0 Probability of Detection 0.8 0.6 0.4 Ten Pulses Coherent Integration Steady Target Coherent Integration Gain PFA=10-6 Non-Coherent Integration Gain One Pulse Ten Pulses Non-Coherent Integration 0.2 0 5 10 15 Signal to Noise Ratio per Pulse (dB) For Most Cases, Coherent Integration is More Efficient than Non-Coherent Integration Radar Systems Course Detection 11/1/2010 19 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) Radar Systems Course Detection 11/1/2010 – Similar to Binary Integration – Require at least 1 threshold crossing for N pulses 20 IEEE New Hampshire Section IEEE AES Society Binary (M-of-N) Integration Pulse 1 Threshold T Calculate x1 x12 Pulse 2 Calculate x2 x Threshold T 2 2 i1 i2 m = ∑ in n =1 Threshold T Calculate xN 2nd Threshold M N … Pulse N Calculate Sum x 2N iN Individual pulse detectors: 2 x n ≥ T, i n = 1 2 x n < T, i n = 0 2nd thresholding: m ≥ M , target present m < M , target absent Target present if at least M detections in N pulses Binary Integration At Least M of N Detections Radar Systems Course Detection 11/1/2010 21 PM / N = N ∑ k =M N! N −k pk ( 1 − p) k ! (N − k ) ! Cumulative Detection At Least 1 of N PC = 1 − ( 1 − p ) N IEEE New Hampshire Section IEEE AES Society Detection Statistics for Binary Integration 0.999 1/4 Probability of Detection 0.99 “1/4” = 1 Detection Out of 4 Pulses 2/4 3/4 4/4 0.90 0.50 Steady (Non-Fluctuating ) Target PFA=10-6 0.10 0.01 Radar Systems Course Detection 11/1/2010 22 2 4 6 8 10 12 Signal to Noise Ratio per Pulse (dB) 14 IEEE New Hampshire Section IEEE AES Society Optimum M for Binary Integration 100 Optimum M Steady (Non-Fluctuating ) Target PD=0.95 10 1 PFA=10-6 1 10 Number of Pulses For each binary Integrator, M/N, there exists an optimum M Radar Systems Course Detection 11/1/2010 23 100 M (optimum) ≈ 0.9 N0.8 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 Target Fluctuations No Fluctuations Swerling I Swerling II Swerling III Swerling IV a 0.8 0.8 0.91 0.8 0.873 b - 0.02 - 0.02 - 0.38 - 0.02 - 0.27 Range of N 5 – 700 6 – 500 9 – 700 6 – 700 10 – 700 Adapted from Shnidman in Richards, reference 7 Radar Systems Course Detection 11/1/2010 24 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration Coherent and Non-Coherent Integration 0.999 Coherent Integration 4 Pulses Probability of Detection 0.99 0.95 0.90 Binary Integration 3 of 4 Pulses Non-Coherent Integration 4 Pulses 0.50 0.10 Single Pulse PFA=10-6 0.01 Radar Systems Course Detection 11/1/2010 2 25 4 6 8 10 12 Signal to Noise Ratio (dB) 14 16 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration Coherent and Non-Coherent Integration 0.999 Coherent Integration 4 Pulses Probability of Detection 0.99 0.95 0.90 Binary Integration 3 of 4 Pulses Non-Coherent Integration 4 Pulses 0.50 0.10 Single Pulse PFA=10-6 0.01 Radar Systems Course Detection 11/1/2010 2 26 4 6 8 10 12 Signal to Noise Ratio (dB) 14 16 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Types of Integration Coherent and Non-Coherent Integration 0.999 Coherent Integration 4 Pulses Probability of Detection 0.99 0.95 0.90 Binary Integration 3 of 4 Pulses Non-Coherent Integration 4 Pulses 0.50 0.10 Single Pulse PFA=10-6 0.01 Radar Systems Course Detection 11/1/2010 2 27 4 6 8 10 12 Signal to Noise Ratio (dB) 14 16 IEEE New Hampshire Section IEEE AES Society Signal to Noise Gain / Loss vs. # of Pulses 10 Signal to Noise Ratio Loss (dB) Signal to Noise Ratio Gain (dB) 20 Coherent Non-Coherent 15 Binary (Optimum M) 10 5 Binary N1/2 0 1 10 Number of Pulses • • • Steady Target PD=0.95 PFA=10-6 Radar Systems Course Detection 11/1/2010 28 100 Binary N1/2 8 Binary (Optimum M) 6 Non-Coherent 4 Relative To Coherent Integration 2 0 1 10 Number of Pulses 100 Coherent Integration yields the greatest gain Non-Coherent Integration a small loss Binary integration has a slightly larger loss than regular Non-coherent integration IEEE New Hampshire Section IEEE AES Society 100 29 σv Velocity Spread Of Clutter =3 00 20 10 0 50 10 Independent Sampling Region 30 10 20 5 1 fr = T Pulse Repetition Rate 3 2 1 0.001 0.01 0.1 1 σ v / λ fr • Radar Systems Course Detection 11/1/2010 Dependent Sampling Region 50 N Equivalent Number of Independent Pulses Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain Non-coherent Integration Can Be Very Inefficient in Correlated Clutter IEEE New Hampshire Section IEEE AES Society Effect of Pulse to Pulse Correlation on Non-Coherent Integration Gain .99 0.9 0.1 0.02 100 20 Dependent Sampling Region σv Velocity Spread Of Clutter =3 00 50 50 10 10 0 N Independent Sampling Region 30 5 10 20 Equivalent Number of Independent Pulses ρ (T) Pulse Repetition Rate 3 2 1 0.001 1 fr = T 0.01 0.1 1 σ v / λ fr • Non-coherent Integration Can Be Very Inefficient in Correlated Clutter Adapted from nathanson, Reference 8 Radar Systems Course Detection 11/1/2010 30 IEEE New Hampshire Section IEEE AES Society Albersheim Empirical Formula for SNR (Steady Target - Good Method for Approximate Calculations) • Single pulse: SNR (natural units ) = A + 0.12 A B + 1.7 B ⎛ 0.62 ⎞ ⎟⎟ A = log e ⎜⎜ ⎝ PFA ⎠ – Where: ⎛ PD ⎞ ⎟⎟ B = log e ⎜⎜ ⎝ 1 − PD ⎠ – Less than .2 dB error for: 10 −3 > PFA > 10 −7 0.9 > PD > 0.1 – Target assumed to be non-fluctuating • For n independent integrated samples: 4.54 ⎞ ⎛ ⎟ log 10 (A + 0.12 A B + 1.7 B ) SNR n (dB ) = −5 log 10 n + ⎜ 6.2 + n + 0.44 ⎠ ⎝ SNR Per Sample – Less than .2 dB error for: 8096 > n > 1 10 −3 > PFA > 10 −7 0.9 > PD > 0.1 – For more details, see References 1 or 5 Radar Systems Course Detection 11/1/2010 31 IEEE New Hampshire Section IEEE AES Society Outline Radar Systems Course Detection 11/1/2010 32 • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary IEEE New Hampshire Section IEEE AES Society 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: – – • B-26, 3 GHz RCS versus Azimuth Radar Systems Course Detection 11/1/2010 33 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 fluctuations IEEE New Hampshire Section IEEE AES Society Swerling Target Models Nature of Scattering Similar amplitudes RCS Model Radar Systems Course Detection 11/1/2010 34 Fast Fluctuation “Pulse-to-Pulse” Swerling I Swerling II Swerling III Swerling IV (Chi-Squared DOF=4) 4σ ⎛ 2σ⎞ p (σ ) = 2 exp⎜ − ⎟ σ ⎝ σ ⎠ σ= Slow Fluctuation “Scan-to-Scan” Exponential (Chi-Squared DOF=2) 1 ⎛ σ⎞ p(σ ) = exp⎜ − ⎟ σ ⎝ σ⎠ One scatterer much Larger than others Fluctuation Rate Average RCS (m2) Courtesy of MIT Lincoln Laboratory Used with permission IEEE New Hampshire Section IEEE AES Society Swerling Target Models Nature of Scattering Amplitude Model Similar amplitudes Rayleigh ⎛ a2 ⎞ 2a p(a ) = exp⎜⎜ − ⎟⎟ σ ⎝ σ⎠ One scatterer much Larger than others ⎛ 2 a2 ⎞ 8a ⎟⎟ p (a ) = 2 exp⎜⎜ − σ σ ⎝ ⎠ Radar Systems Course Detection 11/1/2010 35 Slow Fluctuation “Scan-to-Scan” Average RCS (m2) Fast Fluctuation “Pulse-to-Pulse” Swerling I Swerling II Swerling III Swerling IV Central Rayleigh, DOF=4 3 σ= Fluctuation Rate Courtesy of MIT Lincoln Laboratory Used with permission 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 – – – – – Radar Systems Course Detection 11/1/2010 36 Ground clutter Sea Clutter HF noise Birds Rotating Cylinder Weibel Log Normal Log Normal Log Normal Log Normal IEEE New Hampshire Section IEEE AES Society RCS Variability for Different Target Models 20 Non-fluctuating Target 15 Constant 10 Swerling I/II RCS (dBsm) 5 0 20 15 High Fluctuation 10 5 0 20 Swerling III/IV 15 Medium Fluctuation 10 5 0 0 20 40 60 Sample # Radar Systems Course Detection 11/1/2010 37 80 100 Courtesy of MIT Lincoln Laboratory Used with permission IEEE New Hampshire Section IEEE AES Society Fluctuating Target Single Pulse Detection : Rayleigh Amplitude H 1 : x = ae jφ + n H0 : x = n Detection Test Rayleigh amplitude model z= x >T H1 x <T H0 Probability of ⎛ T2 False Alarm PFA = exp⎜⎜ − 2 (same as for ⎝ σN non-fluctuating) Probability of Detection Test Radar Systems Course Detection 11/1/2010 38 Average SNR per pulse σ ξ= 2 σN ⎞ ⎟⎟ ⎠ PD = ∫ p ( z > T H 1 , a) p(a) da dz ⎛ T2 PD = exp⎜⎜ − 2 ⎝ σN ⎛ a2 ⎞ 2a p(a ) = exp⎜⎜ − ⎟⎟ σ ⎝ σ⎠ ⎛ 1 ⎞ ⎜⎜ ⎟⎟ 1 + ξ ⎠ FA ⎝ PD = P ⎛ 1 ⎞⎞ ⎜⎜ ⎟⎟ ⎟⎟ ⎝ 1+ ξ ⎠⎠ IEEE New Hampshire Section IEEE AES Society Fluctuating Target Single Pulse Detection Probability of Detection (PD) 1.0 Fluctuation Loss 0.8 0.6 Non-Fluctuating 0.4 Swerling Case 3,4 Swerling Case 1,2 0.2 PFA=10-6 0.0 0 5 10 15 Signal-to-Noise Ratio (dB) 20 25 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. Radar Systems Course Detection 11/1/2010 39 IEEE New Hampshire Section IEEE AES Society Fluctuating Target Multiple Pulse Detection Probability of Detection (PD) 1.0 0.8 Steady Target Coherent Integration Swerling 2 Fluctuations Non-Coherent Integration, Frequency Diversity Swerling 1 Fluctuations Coherent Integration, Single Frequency 0.6 0.4 0.2 PFA=10-6 N=4 Pulses 0.0 0 4 8 12 16 20 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 Radar Systems Course Detection 11/1/2010 40 IEEE New Hampshire Section IEEE AES Society Detection Statistics for Different Target Fluctuation Models Probability of Detection (PD) 1.0 10 pulses Non-coherently Integrated 0.8 PFA=10-8 Steady Target Swerling Case I 0.6 Swerling Case II Swerling Case III Swerling Case IV 0.4 0.2 0.0 -2 0 2 4 6 8 10 12 Signal-to-Noise Ratio (SNR) (dB) Radar Systems Course Detection 11/1/2010 41 14 Adapted from Richards, Reference 7 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: K= α= ∞ Non-fluctuating target (“Swerling 0 / 5”) 1, N, 2, Swerling Case 1 Swerling Case 3 2N Swerling Case 4 Swerling Case 2 0 N ≤ 40 1 4 N > 40 η = − 0.8 ln(4 PFA (1 − PFA )) + sign (PD − 0.5 ) − 0.8 ln (4 PD (1 − PD )) Adapted from Shnidman in Richards, Reference 7 Radar Systems Course Detection 11/1/2010 42 * Back of the Envelope IEEE New Hampshire Section IEEE AES Society Shnidman Empirical Formulae for SNR (for Steady and Swerling Targets) ⎛ N ⎛ 1 ⎞ ⎞⎟ ⎜ X ∞ = η⎜ η + 2 + ⎜ α − ⎟⎟ 2 ⎝ 4 ⎠⎠ ⎝ C1 = (((17.7006 PD − 18.4496 ) PD + 14.5339 ) PD − 3.525 ) / K ⎛ ⎛ 10 − 5 ⎞ (2 N − 20 ) ⎞ ⎞⎟ 1 ⎛⎜ 27.31 PD − 25.14 ) ⎟ ⎟⎟ + C2 = e + (PD − 0.8) ⎜⎜ 0.7 ln⎜⎜ ⎟⎟ ⎜ K⎝ 80 ⎝ PFA ⎠ ⎠⎠ ⎝ CdB = C1 0.1 ≤ PD ≤ 0.872 C1 + C 2 0.872 ≤ PD ≤ 0.99 C X∞ SNR (natural units ) = N C = 10 CdB 10 SNR (dB ) = 10 log 10 (SNR ) Adapted from Shnidman in Richards, Reference 7 Radar Systems Course Detection 11/1/2010 43 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 SNR Error vs. Probability of Detection SNR Calculation Error (dB) 1.0 0.5 0.0 -0.5 -1.0 0.0 Adapted from Shnidman in Richards, Reference 7 Radar Systems Course Detection 11/1/2010 44 Steady Target Swerling I Swerling II Swerling III Swerling IV N = 10 pulses PFA= 10-6 0.2 0.4 0.6 0.8 1.0 Probability of Detection IEEE New Hampshire Section IEEE AES Society Outline Radar Systems Course Detection 11/1/2010 45 • Basic concepts • Integration of pulses • Fluctuating targets • Constant false alarm rate (CFAR) thresholding • Summary IEEE New Hampshire Section IEEE AES Society Practical Setting of Thresholds Ideal Case- Little variation in Noise Power Display, NEXRAD Radar, of Rain Clouds Rain Backscatter Data Power 40 • Rain Cloud 20 0 Courtesy of NOAA S-Band Data 0 Receiver Noise 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 Detection 11/1/2010 46 IEEE New Hampshire Section IEEE AES Society 2 Constant False Alarm Rate (CFAR) Thresholding Range Î “Guard” Cells • Cell to be Thresholded CFAR Window – Range and Doppler Cells Doppler Velocity cells Î CFAR Window – Range Cells Estimate background (noise, etc.) from data Range cells Î Data Cells Used to Determine Mean Level of Background – Use range, or range and Doppler filter data – Set threshold as constant times the mean value of background 1 N • Mean Background Estimate = ∑ xn N n =1 Radar Systems Course Detection 11/1/2010 47 IEEE New Hampshire Section IEEE AES Society Effect of Rain on CFAR Thresholding Radar Backscatter (Linear Units) Mean Level Threshold CFAR Range Cells 2.2 dB C Band 5500 MHz Rain Cloud 9 dB Receiver Noise Receiver Noise 2.6 4.5 Slant Range, nmi Window Slides Through Data Cell Under Test Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course Detection 11/1/2010 48 “Guard” Cells Data Cells for Mean Level Computation IEEE New Hampshire Section IEEE AES Society Effect of Rain on CFAR Thresholding Radar Backscatter (Linear Units) Mean Level Threshold CFAR Range Cells 2.2 dB C Band 5500 MHz 9 dB Rain Cloud Receiver Noise Receiver Noise 2.6 4.5 Slant Range, nmi Window Slides Through Data Sharp Clutter or Interference Boundaries Can Lead to Excessive False Alarms Courtesy of MIT Lincoln Laboratory Used with permission Radar Systems Course Detection 11/1/2010 49 Cell Under Test “Guard” Cells Data Cells for Mean Level Computation 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 Window Slides Through Data Cell Under Test Data Cells for Mean Level 1 “Guard” Cells Data Cells for Mean Level 2 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 Detection 11/1/2010 50 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 Window Slides Through Data “Censored” Data (Not Used in Computation of Average) Data Cells for Mean Level 1 Cell Under Test “Guard” Cells Data Cells for Mean Level 2 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 Detection 11/1/2010 51 IEEE New Hampshire Section IEEE AES Society Mean Level CFAR Performance PFA=10-6 0.9999 Matched Filter Mean Level CFAR – 10 Samples Mean Level CFAR – 50 Samples Mean Level CFAR – 100 Samples Probability of Detection (PD) 0.999 0.99 1 Pulse 0.90 0.50 CFAR Loss 0.10 0.01 4 6 8 10 12 14 16 18 Signal-to-Noise Ratio per Pulse (dB) Radar Systems Course Detection 11/1/2010 52 IEEE New Hampshire Section IEEE AES Society CFAR Loss vs. Number of Reference Cells 6 PD= 0.9 CFAR Loss (dB) 5 PFA=10-8 4 10-6 3 10-4 2 1 0 0 10 20 30 40 Number of Reference Cells 50 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) Radar Systems Course Detection 11/1/2010 53 IEEE New Hampshire Section IEEE AES Society CFAR Loss vs Number of Reference Cells 5 PFA 4 10-4 10-6 10-8 3 CFAR Loss, dB For Single Pulse Detection Approximation CFAR Loss (dB) = - (5/N) log PFA 2 N=1 Dotted Curve Dashed Curve Solid Curve N=3 1 0.7 PFA = 10-4 PFA = 10-6 PFA = 10-8 N = 15 to 20 (typically) 0.5 N=10 0.4 Since a finite number of cells are used, the estimate of the clutter or noise is not precise. N=30 0.3 N=100 0.2 2 5 7 10 20 50 70 100 Number of Reference Cells, N Adapted from Skolnik, Reference 1 Radar Systems Course Detection 11/1/2010 54 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 Radar Systems Course Detection 11/1/2010 55 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. Radar Systems Course Detection 11/1/2010 56 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 Radar Systems Course Detection 11/1/2010 57 IEEE New Hampshire Section IEEE AES Society