An Application of Fuzzy Hypotheses Testing in Radar Detection

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1 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS n pplcaton of Fuy Hypotheses estng n Radar Detecton.K.ELSHERIF, F.M.BBDY, G.M.BDELHMID Department of Mathematcs Mltary echncal Collage Caro, Egypt bstract method of testng of fuy hypotheses for fuy data s appled to radar detecton process, Imprecse parameters of the dstrbuton under each hypotheses are modeled as a fuy number n whch we consder that the receved sgnal data s fuy nstead we consder that the receved sgnal data s crsp, he advantages n sortng funess wll be stated.. Introducton Real observatons of contnuous quanttes are not precse numbers, such observaton are called fuy, he funess s dfferent from measurement probabltes, t s a feature of sngle observatons from contnuous quanttes as n reference [], probablty theory s related to random phenomena, relatve frequency and stochastc process; and, on the other hand, fuy set theory s related to non precse data, vague statements.. In secton we ntroduce the densty functon of both radar sgnal (targetand radar nose (no target. hrough secton 3, we analye and formulate the fuy test problem for fuy data as n reference []. In secton 4, we wll use the analyss and formulaton that we ntroduce n secton 3 to be appled for radar detecton.. Radar Receved Sgnal here are two basc operatons performed by radar are:. Detecton of the presence of the reflectng objects.. Extracton of nformaton from the receved waveform to obtan the target data as poston, velocty, angular poston. he detecton and the extracton depends on each other because radar that s good detecton devce s usually a good radar for extractng nformaton, and vce versa. But the problem s n detectng a sgnal n the presence of nose. Nose ultmately lmts the capablty of any radar. We ntroduce here how to obtan the probablty densty functon of radar sgnal and radar nose, so the jont probablty densty functon of two random varable ( t and φ( t accordng to reference[] s gven by: + cosϕ f (, ϕ = exp exp ( π Where ( t and φ( t represent the modulus and the phase of the receved sgnal and s the ISSN: ISBN:

2 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS mean of the random varable ( t, represent the standard devaton. he pdf for alone s obtaned by ntegratng Eq.( over ϕ π + f ( = f (, ϕ dϕ = exp π cosϕ exp d ϕ π ( Where the ntegral nsde equaton ( s known as the modfed Bessel functon of ero order π βcosθ I o ( β = e dθ π (3 π + f ( = f (, ϕ dϕ = I exp (4 o Whch s a rce probablty densty functon. If = (nose alone, hen Eq.(4becomes the Raylegh probablty densty functon f ( r = exp (5 lso, when s very large, Eq. (4 can be approxmately a Gaussan probablty densty functon of mean and varance : ( f ( = exp π (6 Probablty Densty functon Raylegh.5.4 Gaussan Fgure ( Gaussan and Raylegh probablty denstes. 3. estng of Fuy Hypotheses For Fuy Receved sgnal It has been notced that almostly measurable quanttes are mprecse by nature. For example, the mean power nterference s not generally known but must be estmated by samplng. Even the power of a transmtted sgnal can be known only to certan degree of precson and may fluctuate slghtly. We wll model such mprecse quanttes as fuy numbers. In ths secton we analye and formulate the fuy hypotheses test for fuy data (receved sgnal as ntroduced n reference [], then we wll make a comparson wth fuy threshold and crsp threshold. he hypothess model s gven by: H o : Interference (nose or no target H : arget + Interference (radar sgnal or t s a target ISSN: ISBN:

3 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS Under hypothess H, the probablty densty functon f ( ; θ k,where s the nput receved sgnal, whle θk s a vector ofθ k mprecse parameters n whch the probablty densty functon depends. Under hypothess H o, the probablty densty functon s gven by. f (, θ = f ( ;, f k ( ;, exp = (7 Wth mean ero and standard devaton (5.5 Under hypothess H, the probablty densty functon s gven by. f (, θ = f ( ;, f k ( ( ;, = exp π (8 Wth mean and standard devaton 4. pplcaton We wll use the analyss and formulaton n secton 3 to make fuy hypotheses test for fuy data Gven a set of N observaton = (,..., n n whch we want to test the hypothess. H o : H : > Now assume that the sample sgnals are symmetrc trangular fuy number wth support has an nterval of se equal. wth the values: =., =.4, 3 =.6, 4 =.8, 5 = 6 =., 7 =.4, 8 =.6, 9 =.8, = We wll test the hypothess n the case when = and =.5. In whch = and =.5 are symmetrc trangular fuy wth support has an nterval of se equal to and respectvely as shown n fgure below. y, ( (, y.5.5 Fgure(Membershp functon of symmetrc (FN and. Steps of analyss of ths model:. abulate and Plot the densty functon f ( of the sgnal for the gven values of.. By usng the extenson prncple plot η (, where η( s the membershp functon of the test-statstc of normal dstrbuton, whch s gven by: Where, = n = n = n (9 ( ISSN: ISBN:

4 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS wll be calculated as dscussed n reference[5], whch s also a symmetrc trangular fuy number wth support equal to.. he membershp functon of the test-statstc s gven by: ( η( = sup{mn (, ( } t= 3. Plot the characterstc functon η ( whch s also a symmetrc trangular fuy number over the densty functon f ( of the normal dstrbuton η f ( Fgure (3 Densty f ( of standard normally dstrbuted test statstc and characterng functon η of the test statstc t. 4. s shown n the fgure we take δ =.5 for example to calculate the δ cut of characterng functon η of the test statstc t Cδ (.5 = [ t (.5, t (.5] ( C δ (. 5 = [.6,.6] ( 5. Calculate the δ cut of p under hypotheses of rght sded test Cδ ( p = [ p(.5, p (.5] (3 ( ( C ( p = [ P t (.5, P t (.5 ] (4 δ he value of p (.5 s the dark area as shown n fgure (3 p(.5 = P( t(.5 = P (.6 (5 p (.5 = P <.47 =.945 =.548 ( he value of p (.5 s the lght area + dark area as shown n fgure (3 p(.5 = P( t(.5 = P (.6 (6 p (.5 = P <.6 =.757 =.743 hen, ( C ( p [.548,.743] (7 δ = 6. Compare between the value of α (type one error whch here s the probablty of false alarm α =P(accept H / H o s true=p(accept sgnal (target / nose(no target s true So, If α = P fa =.5 he decson s accept H o (no target. If α = P fa =. ISSN: ISBN:

5 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS he decson s nether accept nor reject H o (no target. If α = P fa =.5 he decson s reject H o (no target.. = and =.5. = and =.5 3. = and =.75 decson Remark: hs can be done usng matlab for calculaton and wll gve..5 Crsp threshold = =,.5 = =,.5 = =, observaton Fgure (5 Fuy threshold for varous values of, wth the Crsp threshold. Fgure (4 Densty f ( of standard normally dstrbuted test statstc and characterng functon η( of the test statstc by usng matlab. Calculaton of crsp and fuy threshold he crsp threshold s calculated accordng to reference [3]. v = ln (8 P If α = = P fa fa and =, =.5 ( v =.5 ln.57 = (9 So we can compare the fuy threshold wth the crsp threshold for the followng cases: Calculaton of membershp functon of lkelhood rato he lkelhood rato functon s defned as the rato of the probablty densty functons under each hypothess f (, θ λ( ; Px; Py = < v ( f o(, θo ccordng to the extenson prncple, the lkelhood raton functon can be wrtten n terms of ts membershp functon accordng to reference []. λ λ ( θ ( θ,,,..., θ ( θ,, k k o o o o ( t = sup{mn } ( t=λ ( θ ( θ,,,..., (,, k θ θ k = ( π ( ;, exp ISSN: ISBN:

6 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS o get membershp functon for the lkelhood rato we wll use the extenson prncple. ( sup{mn (, (, ( } λ( t =,, o, t= λ( (3 Snce; ( = ( (4 o,, hen, equaton (7 Becomes λ( ( t = sup{mn, (, ( }, t= λ( λ ( λ ( λ ( ( λ( Fgure (6 Membershp functon of the lkelhood over the densty functon of the normal dstrbuton. he advantages for consderng fuy data rather than crsp data:. We enhance the decson problem as here we have three regon for decson crtera rather than two regon of decson crtera.. he probablty of detecton ncrease as the threshold level n case we consder funess to data became a functon rather than a crsp value. 3. he uncertanty n the sgnal s consdered due to mprecson but not due to randomness whch s more realstc, as the probablty theory s useful when we deal wth random data. 5. Concludng Remark We have demonstrated a method of sgnal detecton based on fuy hypothess testng for fuy receved data, later we wll drve:. he membershp functons of probablty of detecton and probablty of false alarm.. est of hypotheses for dfferent types of recever lke matched flter recever, correlaton recever, the nverse probablty recever, delay lne ntegrator, the bnary ntegraton recever. References [] P. Flmoser and R. vertl, estng hypotheses wth fuy data: he Fuy p-value, Metrca: sprnger-verlag, 4. [] M. Skolnk, Introducton to Radar heory, McGraw-Hll Scence/ Engneerng / Math; 3 edton (December,, 77 pages. [3] S. W. leung, J.W. Mnett, M. K. Lee, n pplcaton of Fuy Hypotheses estng to Sgnal Integraton, Hong Kong, Chna. [4] B. Moller and M. Beer, Fuy Randomness oward a new modelng of certanty, Ffth World Congress on computatonal mechancs, July 7-,, Venna, ustra [5]. Kaufmann, M. M. Gupta Introducton to Fuy rthmetc heory and pplcaton, n ISSN: ISBN:

7 Proceedngs of the th WSES Internatonal Conference on FUZZY SYSEMS Fuy Handbook, van Nostrand Renhold 985, 35 pages. ISSN: ISBN:

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)

ANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U) Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of