40H _jj UNCLASSIFIED. AD AOY2 202 MASSACIWSETTS INST OF TECH LEX INSTON LINCOLN LAB fl5 ~ 2fl ASYeTOTICALLT OPTIMAL DETECTOR O ~ PENORY P FOR K Q ~

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1 AD AOY2 202 MASSACIWSETTS INST OF TECH LEX INSTON LINCOLN LAB fl5 2fl ASYeTOTICALLT OPTIMAL DETECTOR O PENORY P FOR K Q POG NT RAN CTCCU) MAY 79 L K JOttS fl9625 7e c 0002 UNCLASSIFIED 40H _jj TN l979 2 ESD TR Pt N a

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. -,- _ MASSACHUSETTS INSTITUTE OF TECHNOLO GY LINCOLN LABORA TORY ).SYMPTCYTICALLY 9 PTIMAL DETECTOR OF EMORY FOR k- EPENDENT RANDOM ffiignals J 92 (j L/, /.774J.t?

5 . -,- _ MASSACHUSETTS INSTITUTE OF TECHNOLO GY LINCOLN LABORA TORY ).SYMPTCYTICALLY 9 PTIMAL DETECTOR OF EMORY FOR k- EPENDENT RANDOM ffiignals J 92 (j L/, /.774J.t?? I / es iqn Fo? 2 JUS : L c...tiqatior pi t 0P6C1$1 D 0 C yi? I 7 UC Approved for public release; distribution unlimited. LEXINGTON MASSACHUSETTS C ( ( L;

6 - Abstract This note describes a general method for discriminating between two k-dependent stationary discrete random vanables using marginal statistics and their first order correlations. iii

7 Asymptotically Optimal Detector of Memory p for k - dependent Random Signals Lee K. Jones A stationary discrete time series {X ) is said to be 1 k-dependent if for every integer m, {X} < m is independent of {X } i i>k+m Suppose {X } 1 and {X } are two stationary k-dependent random signals occurring with prior probabilities a and 1-a respectively. If n pulses of a random signal are observed (n lar ge compared to k) how do we decide whether we observed or? A detector L of memory p is given by a function L(X) = L(x 1, x 2,..., x ) =1 g 1 (x1, x 1,.., x ) and a set of real numbers such that: for L(X) A we choose class 1 for L(X) c A C we choose class 2 In view of the stationarity we need only consider (for ii large compared to p) detectors for which g 1 g for all i. Suppo se bo th {X } and {X } are bounded with known (or estimates of) statistical properties (correlations, moments, etc.). In this note we use a stra ightforward ex tension of the method of minimal marginal moment variance 2 to determine the g which (1) For the cases we shall consider, A is the un ion of one or two intervals. (2) Introduced in [2] to solve for the case p O, k O.

8 rur k 3 minimizes the probability of error. This solves the problem addressed in (1J (p.0, 4-o+n1, x mn 1, n 1 k-dependent). Solution : Let 1, h 1, h 2,... he a complete set of continuous functions on For a h we find the coefficients 3 a which minim ize the probability of error. The constant function 1 need not be present in this expansion since i t trivial ly 3 has the same statistical properties under both hypotheses. As q becomes large the error probability using converges to the error probabili ty for the optimal g. Consider L(X) g (x., x.,..., x. ). It is a i p+l q 1 1 sum of bounded k+p - dependent random variables. Hence L is asyr1i)totically normal under each hypothesis. The set A will in general be described by 2 thresholds. (If a single threshold detector is desired the following analysis remains the same.) Let C (a ) probabili ty of error nun {ap 1, (l- o)p 2 } dx where p is the probability density of L under hypothesis i. By normali ty p 1 and p 2 are characterized by their means and vanances. We now restrict a such that F.2 L(X) -B ] L(X) - 1. C is then a function of the variances of L under each hypothesis: C.-C (v 1 (a 3 ), v 2 (a ). Taking the gradient wrt a 3 and using X as a Lagrange mul tiplier we have : + Vv 2 - X (V(E 2 L - E 1 L)) a 0 ( 1) r..-

9 The partials }f - may be negative (even for a single threshold de- V j tector) but are not both zero. It is easy to see that a solution of (1) is a critical point of the objective function Bvl + (1-181)v2 - (E 2L-E 1L-l) where is a Lagrange multiplier and - l < B < + 1. This critical point may be determined for various S values of B and the B corresponding to minimum probability of error determined from normal tables. We now proceed with the calculations. Let L h ht ) j. xi, x _i., x i_p j = E [ch B 1 h i ) (h - E By differentiating the above objective function wrt a 3 and setting the result equal to zero we obtain : (n-p) i [8 A 1 + (1-lBI)A 2J 4- (2) where m = E h E 1 h k+p and (A ) = (n-p)p 35 + E (n-p-l) + Solving equa tion (2) - =.j2. [M 1 + (1-IBDA 2] ; - Then - C. q g -l in ) from the condi tion E s 2 L-B 1 L-l. 3..TT

10 -. -n ,, -. - * -., ,. -.w.. v 1 and v 2 may now be calculated and the error (as a function of 8) determined. The 8 corresponding to minimum error is then obtained by a one-parameter minimization. The preceding method will yield an optimal detector whenever the Pj occurring in the expression for C(a ) depend only on the means and variances of L. We need only estimate the error as a function of 8 from the performance of L on sample data. I

, 1. P i J LJJ References [1] H. V. Poor and J. B. Thomas, Time Detection of a Constant Signal in rn-dependent Noise, IEEE Trans. Inform. Theory IT 25, Pages 54 61, (1979).

11 , 1. P i J LJJ References [1] H. V. Poor and J. B. Thomas, Time Detection of a Constant Signal in rn-dependent Noise, IEEE Trans. Inform. Theory IT 25, Pages 54 61, (1979). [2) On Optimal Discriminants between two Classes of Random Variables in Terms of the Moments of their Distributions, submitted to SIAM Journal of Appi. Math. I r rn

UNC kssif!ed CLASSIFICATION OP TINS PAGE VA.s DsIs 1EPORT DOCUMBITATIOW PAGE t ort sumt ER ESD-TR-79-83 READ INSTRUC T1ONS WORE CORPLE11NG POll ) ODYT ACCESSION NO. 3.

12 UNC kssif!ed CLASSIFICATION OP TINS PAGE VA.s DsIs 1EPORT DOCUMBITATIOW PAGE t ort sumt ER ESD-TR READ INSTRUC T1ONS WORE CORPLE11NG POll ) ODYT ACCESSION NO. 3. RECIPIENt S CATALOG SUMtER + WTLI s.d $.M4) 1. TYPE OP REPORT A PERIOD COVERED Asymptodcafly Optimal Detector cg 4 t o y p TedmiCat Note for k -Dependent Random agnal. A. PtePORMING ORG. REPORT SUMtER T.ctmlcal Note AUThOR a) S. r. tee K. J ones F C-0002 P ORjN ORGANIZATION NAME AND ADD*EU 10. PROGRAM ELEMENT PROJECT. TASK ARIA I WORK UNIT IUMSER S Lincoln Laboratory, M.I.T. P.O. Box 73 Program Element No ? Lexington. MA Project No. 627A II. CONTROLLING OFFICE NAME AND ADORESS 1). REPO RT DATE Air Force System. Command, LISA? 8 May 1979 WashIngton. DC sumter OF PAGIS II. WONITORDIG AGENCY NAME I ADDftESS( fd ff.nu ft.- Cs.Ir.U q Off...) Is. SECURITY CLASS. (.f saa. rip..i ) Electronic Systems Division Unclassified Hanscom Afl 3. DECLASSIPICATION DOINORADINO Bedford, MA SOIIDULE IA. DISTRIIUTIOW STATEM ENT 1.! S a R,.n) Approved for public release; distribution milmired OI$TRI IUT ION STATEMEN T (.1 IA. M.u....I.,.d.. IM.A JO. q a ft.- Rq ae) 1 IS. SUPPLEMENTARY NOTES None IL KEY WORDS (C...s....a a 14i f.s. s.d d..ufr Sy 54.ek..-Aw) k-dependent random variable, stationary discriminating IS. ASSTRACT (C...M s. r,.i.a. Ud. d, s i u, s.d d..*dfr Sy AA..h s.-a.r) H Thi. note describs u a general method for discriminating b.twoen two k-dependent stationary discrete random variable. using marginal statistic. and their first ordsr correlation.. I JAN EDITION OP 1 NOV 53 IS OSteLI TE LJNCLASmFIED SECURITY CLAU $PI CAIION OF THIS PAGE (IA.. 0. ti.nr,d) L _ _

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A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N