adar Detection in Non-Gaussian Environment : Optimum Detectors Evaluation of their Performance Seminar at RUTGERS University Monday, the 2th June 2 JAY Emmanuelle Ph.D.Student in Statistical Signal Processing dvisors : Pr. DUVAUT Patrick (ETIS) and Dr. OVARLEZ Jean-Philippe (ONERA), 2th June 2 - Seminar at Rutgers - Emmanuelle JAY
Presentation Plan - Introduction Π Context, detection theory, classical procedure - The Padé Approximation Method Π Description Π First application : Performances evaluation for one pulse Π Further application : Coherent detection - Performances evaluation - Padé Estimated Optimum Detector (PEOD) Π Modelling non-gaussian clutter with compound processes : SIRP representation Π Resulting expression of the Optimum Detector Π Contributions of Padé approximation : PEOD expression - Simulations - Conclusions 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 2
- Introduction : Context ontext : High Resolution Radar - Low Grazing Angle Environment =) Decreasing of the number of reflectors - Preponderance of few of them =) The clutter is spiky + The clutter statistic is no more a gaussian one + A gaussian hypothesis is no more valid : It is necessary to model the non-gaussianity of the clutter 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 3
H > H - Introduction : Detection Theory TATISTICAL HYPOTHESIS TEST : y c Clutter only H y = s + c Target + Clutter py(y=h ) = pc(y=h ) ) = py(y s=h ) py(y=h he target signal s is a modified version of the transmitted signal p : s = T (A; ) p ETECTION TEST : Likelihood Ratio Test (LRT) = p y(y=h ) Λ(y) ) py(y=h < ERFORMANCES EVALUATION : False Alarm Probability (P fa ) and Detection Probability (P d ) Z Z P fa = IP(Λ(y) > H ) = py(y=h P ) d = IP(Λ(y) H > ) = dy and py(y=h ) dy D D 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 4
- Introduction : Classical Procedure Analytical determination of the noise + target density (H ) Target with constant envelope A Knowledge of the noise probability density (H ) Knowledge of the target fluctuations law Detection threshold derived for a fixed Pfa Theoritical expression of the detection probability P d 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 5
- Introduction : Classical Procedure - Gaussian Clutter AUSSIAN CLUTTER UNDER H HYPOTHESIS : py(y=h ) = 2 m jmj exp M y yy 2 ff 2 ) ßff (2 AXIMUM LIKELIHOOD (ML) ESTIMATION A OF : ML = argmax A ^A M y Λ(y) = p M p py y PTIMUM GAUSSIAN DETECTOR (OGD) : Output of the matched filter jp M yj 2H > y < 2ff log( )p M p y 2 H ETECTION PERFORMANCES : OGD statistics under H and H =) Padé approximation method to evaluate the statistics 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 6
LX a n u n b n u n Z MX μ n ( u) n k e ff k z k k ) Re(ff > k ff + u 2-Padé Approximation Method : Description ETHOD : PDF Estimation from Moment Generating Function (MGF) approximation MGF of a positive random variable Z with PDF p(z) and n-order moments μ n : Φ(u) = p(z) e uz dz = X = X c n u n Approximation P [L=M] (u) (L < M)ofΦ(u) : n= n= n! X L+M n= MX P [L=M] (u) = = c n u n + O(u L+M+ ) = MX n= k= n= Inverse Laplace Transform of P [L=M] (u) : ~p(z) = k= 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 7
: H jy(t)j = jb(t)j! p H (r) Z + MX PX Z + Z + MX k e ff k k ff 2 - First Application : Performances Evaluation for one pulse SOGD OR ENVELOPE DETECTOR : 8 < ADÉ APPROXIMATIONS : H jy(t)j = js(t) +b(t)j! p H (r) Clutter envelope ) p H (r) = MX k e ff k r k= Target Amplitude Fluctuations ) p(a; A ) = k fl e ffi k A=A A INK BETWEEN p H AND ESTIMATED p HO : k= PX i k rρj (ρr) fl 2 2 + ffi2 i )(ρ2 + ff 2 k ) dρ A p(ρ p H (r; A ) = i= k= ERFOMANCES EVALUATION : P fa = IP(jy(t)j > H ) = p H (r) dr = k= p H (r) dr d = IP(jy(t)j H > ) = P 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 8
RSOGD 3 2:63 increased :63: 2 2 - Performances Evaluation for one pulse : Results XPERIMENTAL DATA : Ground clutter (forest) Spatial PDF of the ground clutter (forest).5.9 Comparison between PDF.5 2 3 4 5 6 7 8 9 CDF.8.7.6 Data Rayleigh.5 Pd.5 5 2 3 4 5 6 7 8 9 Estimated threshold Rayleigh threshold.4.3.2. 5 5 2 Range bins 2 4 6 8 2 4 6 8 2 SNR (db) Π PERFORMANCES EVALUATION OF A VIRTUAL EMBEDDED TARGET IN THIS CLUTTER Initial Threshold Observed True value Padé true 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 9
H > D(Data) = V k;i e ff k;i V XM XM 2 - Further Application : Coherent detection OHERENT DETECTION PERFORMANCES EVALUATION : e output of the Detector (D(:), threshold T ) is a postive random variable V (Data depending) < H T =) Possible to estimate the Detection Test statistics using a Padé approximation i XM p Hi (V ) = k= P fa = IP(V > H T ) = k; e ff k; T k; ff k= d = IP(V H > T ) = P k; e ff k; T k; ff k= 2th June 2 - Seminar at Rutgers - Emmanuelle JAY
3-Padé Estimated Optimum Detector (PEOD) - SIRP ODELLING NON-GAUSSIAN CLUTTER : Spherically Invariant Random Process (SIRP) Compound processes : c = x p fi x is a complex circular gaussian vector with covariance matrix 2M, fi is a positive random variable, independent of x with a so-called characteristic function p(fi ), Conditionnally to fi, c (size m) is a complex circular gaussian vector with covariance matrix 2fi M, p(c=fi ) = m exp M c cy jmj ßfi) (2 2 fi, jmj where is the determinant of M, Z + p(c=fi ) f(fi ) dfi, p(c) = 2th June 2 - Seminar at Rutgers - Emmanuelle JAY
2 exp ( bjxj) b2n (b p p) N K N (b p p) sb 2 exp ( b 2 s 2 =2) N b (N +=2) 2 p (b 2 + p) N+=2 ß 2 3 @ b 2 exp s b b 2 3 - SIRV Examples SIRV with known characteristic PDF Characteristic PDF Marginal PDF fx (x) h 2N (p) f S (s) E(s2 ) Gauss p exp ( x 2 =2) exp ( p=2) ffi(s ) 2ß Laplace b 2 Cauchy b ß(b 2 + x 2 ) A 2 2s K-distribution 2b ψ! ν bx ν (bx) K (ν) b 2N 2 (ν) (b p p) ν N 2ν K N ν (b p p) exp 2 s 2 =2) 2b ( b (bs) 2ν 2ν (ν) 2ν b 2 Student-t (ν +=2) b p ß (ν) x ν =2 2 + A 2 N b 2ν (ν + N) @ 2 b (ν)(b 2 + p) N+ν exp b2 2b 2 2s (ν)2 ν b 2ν s 2ν b 2 2(ν ) Table : List of SIRV processes for which the characteristic PDF is known in closed form 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 2
Z + exp < H 3 - PEOD - SIRP Resulting Optimum Detector ESULTING EXPRESSIONS : ßfi) m jmj exp q (y) (2 py(y=h ) = p(fi ) dfi ; py(y=h ) = pc(y s=h ) DETECTION TEST : Z + 2 fi» q (y) exp q (y) log( ) H ) p(fi > dfi m fi 2 fi 2 fi M (y) y M y jpy yj 2 q = p M p y y if the target A amplitude is estimated in ML sense. q (y) = y y M y ; q (y) = q (y s) if s is known, 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 3
3 - SIRP Resulting Optimum Detector : Drawback and Solution RAWBACK : The expression of p(fi ) has to be known ) Dependence on the a priori clutter statistics ) Limitation to the PDF having a SIRP characteristic function ROPOSED SOLUTION : Estimation of p(fi ) directly from the clutter data Estimation of the variance : ^fi ) Padé approximation of p(^fi ) ) ) No a priori on the clutter statistics 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 4
MAP = argmax ^fi fi m 2 = argmax p(fi=y) fi MX MX k e ff k ^fi m q k ) 2 K m B k (y) (ff k 3 - PEOD : Expression DEA : Estimate p(fi ) thanks to Padé Approximation AXIMUM APOSTERIORI ESTIMATION : M y ) = d yy +2 p(y=fi ) g(fi a +) + d(m 2 p(y=fi ) is a gaussian pdf (likelihood of the clutter conditionnally to fi), g(fi ) is a conjugate prior, Inverse Gamma pdf, IG(a,d). ADÉ APPROXIMATION OF p(fi ) : ~p(^fi ) = EOD RESULTING EXPRESSION : k= MX q (y) k= q (y) m q k ) 2 K m B k (ff k H > < (y) H k= 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 5
3 - Needed quantities computation for OGD and PEOD PEOD p M -/2. -2 y M -/2. 2 - q (y ). 2 + q (y ) p OGD M -/2. -2 y M -/2. 2 q (y ) - q (y ) OGD 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 6
3 - PEOD : Block diagram for computation DATA N Reference.. cells MAP Estimation of the variance of the clutter (N estimates) Padé Approximation of the PDF of the variance α λ k k Cell under test Needed quantities calculation q (y ) q (y ) PEOD H H 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 7
4 - Comparison - OGD / OKD / PEOD ERFORMANCES COMPARISON BETWEEN OGD, OKD (OPTIMUM KDETECTOR) AND PEOD K-distribution corresponds to a spiky clutter when the shape parameter ν is low and tends to a gaussian distribution when ν! +, OKD Expression is obtained when p(fi ) is a Gamma PDF, The KPadé curves do not include MAP Estimation of fi, The Signal-to-Noise-Ratio (SNR) is given for pulse, m = pulses coherently integrated (SNR! SNR + ), fa value is and the true P 3 fa value for OGD is given (corresponds to the increasing of false alarm P if used), ν = :5; ; 2; 2 ; when ν = 2, the Optima curves blend with OGD curve. 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 8
4 - Simulations : Comparison - OGD / OKD / PEOD Bruit K : ν =.5 ; N= ; P fa = 3 ; Pfa KG real = 2.44. 2 Bruit K : ν= ; N= ; P fa = 3 ; Pfa KG reelle =.72. 2.9.8.7 KG G KG KK K Pade.9.8.7 KG G KG KK K Pade.6.6 P d.5 P d.5 P d.4.3.2.9.8.7.6.5.4.3.2.. 5 5 5 5 input SNR Bruit K : ν =2; N= ; P fa = 3 KG G KG KK K Pade 5 5 5 5 input SNR P d.4.3.2. 5 5 5 5 input SNR.9.8.7.6.5.4.3.2. Bruit K : ν=2 ; m= ; P fa = 3 ; Pfa KG reelle =. 3 KG G KG KK K Pade 5 5 5 5 input SNR 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 9
4 - Comparison between PEOD with and without MAP Estimation 3 25 2 5 5 Histogram of the true 5 τ 2 3 4 5 6 35 3 25 2 5 5 Histogram of the 5 MAP Estimated τ 2 3 4 5 6 P d.9.8.7.6.5.4.3.2. PEOD - K clutter ν = 2 ; P fa = -3 ; m= ; τ et τ map with τ with τ map -5 - -5 5 5 One pulse SNR 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 2
4 - Weibull Clutter : p(fi ) is unknown 2 Weibull - Histogram of the MAP estimate of τ ; size : 5 Weibull clutter as SIRV - τ map estimation ; N = ; P fa = -3 8.9 6.8 4.7 2.6 P d.5 8.4 6.3 4.2 2..2.4.6.8.2.4.6.8 2-5 - -5 5 5 One pulse SNR 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 2
5 - Conclusions epadé Approximation Method associated with a SIRP representation of the clutter ows to derive the Optimum Detector Structure matched to the clutter statistics No a priori on the clutter statistics, Estimation directly from the data of the characteristic function, Detector structure depends nevertheless only on Padé coefficients, Optimum in this sense, Low Computational cost, Moment estimation errors have no influence on performances, Possible to take into account the clutter correlation (SIRP), Show the real mismatch of the OGD. 2th June 2 - Seminar at Rutgers - Emmanuelle JAY 22