NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON LABORATORY NEW LONDON, CONNECTICUT Technical Memorandum
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1 NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON LABORATORY NEW LONDON, CONNECTICUT Tehnial Memorandum A FORTRAN COMPUTER PROGRAM TO EVALUATE AND PLOT THE STATISTICS OF THE MAGNITUDE-SQUARED COHERENCE FUNCTION Date: 5 February 1986 Prepared by: ~.A.~ Helen A. Duli~--r Eletronis Engineer~ng and Computer Appliations Division Surfae Ship Sonar Department APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED R fer NC ONLY
2 Report Doumentation Page Form Approved OMB No Publi reporting burden for the olletion of information is estimated to average 1 hour per response, inluding the time for reviewing instrutions, searhing existing data soures, gathering and maintaining the data needed, and ompleting and reviewing the olletion of information. Send omments regarding this burden estimate or any other aspet of this olletion of information, inluding suggestions for reduing this burden, to Washington Headquarters Servies, Diretorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subjet to a penalty for failing to omply with a olletion of information if it does not display a urrently valid OMB ontrol number. 1. REPORT DATE 05 FEB REPORT TYPE Tehnial Memo 3. DATES COVERED to TITLE AND SUBTITLE A FORTRAN Computer Program to Evaluate and Plot the Statistis of the Magnitude-Squared Coherene Funtion 6. AUTHOR(S) Helen Duling 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Underwater Systems Center,New London,CT, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publi release; distribution unlimited 13. SUPPLEMENTARY NOTES NUWC SPONSOR/MONITOR S REPORT NUMBER(S) 14. ABSTRACT This doument ontains omputer listings for two FORTRAN omputer programs. The first alulates the probability density funtion assoiated with the estimated magnitude-squared oherene funtion and the seond alulates the umulative distribution funtion assoiated with the estimated magnitude-squared oherene funtion. 15. SUBJECT TERMS FORTRAN 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unlassified b. ABSTRACT unlassified. THIS PAGE unlassified Same as Report (SAR) 18. NUMBER OF PAGES 16 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Presribed by ANSI Std Z39-18
3 ABSTRACT This doument ontains omputer listings for two FORTRAN omputer programs. The first alulates the probability density funtion assoiated with the estimated magnitude-squared oherene funtion and the seond alulates the umulative distribution funtion assoiated with the estimated magnitudesquared oherene funtion. ADMIMISTRATIVE INFORMATION This memorandum was prepared by Helen A. Duling, ode 33142, systems Analysis and Appliations Group. The author of the memorandum is loated at the New London Laboratory, Naval Underwater Systems Center, New London, Connetiut ACKNOWLEDGEMENT Dr. G. Clifford Carter of ode 3314 is responsible for providing me with the neessary douments and information whih enabled me to investigate and alulate the umulative distribution and probability density funtions assoiated with the estimation of the magnitude-squared oherene funtion. 3/4 Reverse Blank
4 I. INTRODUCTION The purpose of this memorandum is to doument the FORTRAN ode used to alulate the probability density funtion and the umulative distribution funtion assoiated with the estimated mean-squared oherene funtion, and to define the variables used in alulation. II. CALCULATIONS The first order probability density funtion (PDF) for the estimate of the magnitude-squared oherene funtion (MSC) qiven the true value of the MSC and the number of independent segments proessed, n, an be written as (equation 4.4 in (1)) p (t i,.,f\.,i X I a) : ( n.. I ) [ ( I I r I ') (I.. I i I.. ) 1\. (1-,r,"~trl"')a. (,_,r,'rrl 1 ),,. &.) 'I.~ I ( 1- n J 1. n ) I) It I I Yl ll-lli&.)z. where es+:,... 4~1J m~. -= e-sidh a fu I ms'' : o,., Note F (-m,b;;z) an be expressed as (equation in (2)) F l - b. C. ) ~ (-m)" (b") ~n. 2. '...,.'J J J i. ;; G...=.,. "" l.. ) "'' 5
5 The first-order umulative distribution funtion for the estimate of the MSC, qiven the true value of MSC and the number, n, of independent seqments proessed, an be written as (equation 4.6 in (1)), ~ 1 " z " z 1\. n.z. I 'Z l l"tl -1~ 1 <.] p (I r 1 ~ ~I ) ~ I r I { (, 1 tl.. 1 ; 1.. ) ~~J/-1 rt'lit ~) Note that the hyperqeometri funtion F (-1,1-N;l;z) an be expressed as (equation 8 in (3)),.J &f,l-jjt-,jjl;,~):! Tf( "--=- 0 where T~;,; (.,t)'<(/-n).. -r. t T. = - TJC. T~~:-1 (1}1( t~ -. (~-1-J)(k... f-ti-n). i 6
6 TM No III. EXAMPLE CALCULATION The following is an example alulation orresponding to the speifi values MSC = 0.3 n = 32 The outputs of the FORTRAN programs are tabulated in Tables 1 and 2; and shown graphially in Figures 1 and 2. 7
7 PROBABILITY DENSITY FUNCTION Verifiation file : PDF.FOR Helen A. Duling NUSC Sientifi and Engineering studies Coherene Estimation Purpose : The purpose of this program is to verify the Probability Density Funtion found on page 28., TR4343 Estimation of the Magnitude Squared Coherene Funtion (Spetrum) by G..arter The PDF is alulated given n = 32, magnitude squared oherene funtion = 0.3. The estimated value ( magnitude squared oherene funtion) varies from 0 to l. REAL PDF,TEMPl,TEMP2,COH REAL Z,HSUM,ESTCOH,TERM4 INTEGER N,I,TERMl,TERM2,L OPEN (UNIT-3, FILE= 1 VERIFY2 1,STATUS= 1 NEW 1 ) Initialization - set onstants - N COH ESTCOH o.o TEMPl o.o TEMP2 o.o PDF = o.o Format statements 1 FORMAT (lox,a) 2 FORMAT (lx) 3 FORMAT (20x,a,7x,a) 4 FORMAT (22x,F7.5,9x,F7.5) 5 FORMAT (20x,a) 6 FORMAT (20X,A,lX,I2,A,lX,F2.1) 8 WRITE (3 '1) 'Probability Density Funtion Calulation' WRITE (3,2) WRITE (3' 6) 'Given : n = I 'N' I and = ',COH WRITE (3,2) WRITE (3,3) estimate, 'PDF value WRITE (3,2) I WRITE I I (3,3) WRITE (3,2) '
8 ESTCOH is varied from 0 through 1 in inrements of DO I ESTCOH = (I*0.025) TEMP1 ((((1-COH)*(1-ESTCOH))/((1-(ESTCOH*COH))**2))**N) TEMP2 = ((1-(COH*ESTCOH))/((1-COH)**2)) TERM1 = N-1 TERM2 = 1-N TERM4 = COH*ESTCOH CALL HYPER (TERM11 TERM2 I TERM4 I HSUM) PDF= ((N-1)* TEMP1 * TEMP2 * HSUM) WRITE WRITE PDF TEMP1 TEMP2 END DO END (3 I 4). ESTCOHI PDF (312) o.o - o.o = o.o SUBROUTINE - omputes the hypergeometri funtion SUBROUTINE HYPER (L 1 INTEGER L1K1TERM2 REALT TERM2 1Z1HSUM) HSUM 0.0 T=O.O DO K ""' 0 1 L IF (K.EQ. 0) THEN T 1.0 ELSE T = (((K-1-L)*(K-1+TERM2)*Z)*T)/(K**2) END IF HSUM =- END DO HSUM + T END RETURN 9
9 CUMULATIVE DISTRIBUTION FUNCTION Verifiation file : CDF.FOR Helen A. Duling C NUSC sientifi and Engineering studies Coherene Estimation. Purpose : The purpose of this program is to verify the umulative Distribution Funtion found on page 33. of TR4343 Estimation of the Magnitude Squared Coherene Funtion (Spetrum) by G. Clifford arter. The PDF is alulated given n = 32, and magnitude squared oherene funtion = 0.3 The estimated value (magnitude squared oherene funtion) varies from o to 1. REAL CDF, TEMP, SUM, COH, ESTCOH,HSUM REAL PARTIAL, Z,TERM4 INTEGER N, K, KK, I,L,TERMl,TERM2 OPEN (UNIT 3,. FILE 'VERIFYl',status='new') Initialization - set onstants N. 32 COH 0.3 ESTCOH = 0.0 TEMP 0.0 PARTIAL = 0.0 CDF 0.0 Format statements 1 FORMAT(lOx,a) 2 FORMAT(lx) 3 FORMAT(20x,a,9x,a) 4 FORMAT(22x,F7.5,lx,Fl8.5) 5 FORMAT(20x,a) 6 FORMAT(20X,A,lX,I2,A,lX,F2.1) 10 WRITE(3,1) umulative Distribution Funtion Calulation' WRITE(3,2) WRITE(3,6) 'Given: n = ',N,' and = ',COH WRITE(3,2) WRITE(3,3) ' estimate, 'CDF value' WRITE (3,2) WRITE(3,3) ' ',' WRITE(3,2)
10 ESTCOH is varied from o through 1 in inrements of DO 1:=1,40,1 ESTCOH = (1:*0.025) TEMP= ((( 1-COH)/(1-(ESTCOH * COH ))) ** N) DO KK=1,N-1 K=KK-1 SUM 1.0 l:f (ESTCOH.NE. 1.0) THEN SUM= ( ((1.0-ESTCOH)/(1.0 - (COH * ESTCOH))) ** K) ENDl:F TERM1 = K TERM2 = 1-N TERM4 = COH*ESTCOH CALL HYPER(TERM1,TERM2,TERM4,HSUM) PARTIAL = PARTIAL + (SUM * HSUM) END DO IF (ESTCOH.EQ. 1.0) THEN CDF 1.0 ELSE CDF = (ESTCOH * TEMP * PARTIAL) END IF WRITE (3,4) ESTCOH,CDF WRITE (3,2) CDF = 0.0 PARTIAL = TEMP, END DO END C SUBROUTINE - omputes the hypergeometri funtion SUBROUTINE HYPER (L, TERM2,Z,HSUM) INTEGER TERM2,K,L REAL T HSUM = 0.0 T = 0.0 DO K = O,L l:f (K.EQ. 0) THEN T = 1.0 ELSE T = (((K-1-L)*(K-1+TERM2)*Z)*T)/ (K**2) END IF HSUM = HSUM + T END DO RETURN END 11
11 Probability Density Funtion Calulation Given : n 32 and =.3 estimate PDF value Table 1 12
12 umulative Distribution Funtion Calulation Given : n = 32 and =.3 estimate CDF value ~ r r Table 2 13
13 TM NO PDF 0 0 ID ID Gl 00 " Gl 0 0 Gl ~----r-----r-----~----~----~----,-----,-----,-----,-----, n ~ ~ PDF versus estimated oherene 14 Figure 1
14 TM NO CDF ~ OOr-----~~~-----,N------~y ~r-----~~~-----,G ~r Gr------,G , CDF versus estimated oherene Figure 2 15
15 IV. CONCLUSION The magnitude-squared oherene funtion an be used for signal detetion and passive sonar parameter estimation. To exploit this funtion it is neessary to estimate the quantity. The statistis of the estimate have been well doumented, see referene (2) Carter, 1972; however, the omputer software is omplex and involves a number of subtleties beause the hypergeometri funtion onsists of ombinations of very large and very small numbers in ways that prelude a simple straight-forward programming of the -equations. Moreover, if you follow the proedure outlined in this memorandum, whih on the surfae is very straightforward and use the omputer software listed, then one an evaluate these omplex mumerial problems with little or no numerial overflow or underflow problem in the parameter range of interest. A omputer program was written and tested to evaluate the probability density funtion and umulative distribution funtion assoiated with the magnitude-squared oherene funtion. This memorandum presents the formulas and FORTRAN programs tested for the partiular ase when the magnitude squared oherene equals.3 and the number of independent segments proessed is 32. The results of the sample test ase are presented in tabular and graphial form in addition to the atual program listing. 16
16 REFERENCES 1. "Estimation of the Magnitude-Squared Coherene Funtion (Spetrum)," arter, G.., NUSC Tehnial Report, 19 May "Handbook of Mathematial Funtions," Abramowitz and Stegun, U.S. Department of Commere, National Bureau of standards, Applied Mathematis Series.55, "Reeiver Operating Charateristis for a Linearly Thresholded Coherene Estimation Detetor," Carter, G.C., IEEE Transations on Aoustis, Speeh, and Signal Proessing, February 1977.
17 A FORTRAN COMPUTER PROGRAM TO EVALUATE AND PLOT THE STATISTICS OF THE MAGNITUDE-SQUARED COHERENCE FUNCTION Helen A. Duling Systems Analysis and Appliations Division surfae Ship Sonar Department 13 January 1986 UNCLASSIFIED Internal DISTRIBUTION LIST Codes B Total 26 (4 opies) (K. Andronowitz, K. Briotti, A. Brown, E. Eby, A. Failone, v. Gonzalez, E. Martins, D. Lerro, w. Rahr, s. Sutherland) H.Duling (3 opies) (D. Klingbeil) (M. Cloutier) (R. Bernier) (NLON Library - (NPT Library) 3 opies)
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