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 79 83 Pt N a
1.0 L a I s 1112.2 Ii L 11111 1 25 IIlII1 lmu 6 MICROC(1t V RI SOt Ii I ION I I liar I NAIIONAL AIRI Al l A l.iani \lll. l -J
I C,C
(..fls GR Ju L...t Lca i 1strib : io11abt Ava
. -,- _ 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 6 1919 Approved for public release; distribution unlimited. LEXINGTON MASSACHUSETTS C ( ( L; 79 8 2 001
- 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
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.
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..-
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 3 1 - 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 - 3 2 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
-. -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 4 4.. - -.. -
, 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. RECIPIENt S CATALOG SUMtER + WTLI s.d $.M4) 1. TYPE OP REPORT A PERIOD COVERED Asymptodcafly Optimal Detector cg 4 t o y p TedmiCat Note for k -Dependent Random agnal. A. PtePORMING ORG. REPORT SUMtER T.ctmlcal Note 1979-42 7. AUThOR a) S. r. tee K. J ones F19628-78-C-0002 P ORjN ORGANIZATION NAME AND ADD*EU 10. PROGRAM ELEMENT PROJECT. TASK ARIA I WORK UNIT IUMSER S Lincoln Laboratory, M.I.T. P.O. Box 73 Program Element No. 63311? Lexington. MA 02173 Project No. 627A II. CONTROLLING OFFICE NAME AND ADORESS 1). REPO RT DATE Air Force System. Command, LISA? 8 May 1979 WashIngton. DC 20331 sumter OF PAGIS II. WONITORDIG AGENCY NAME I ADDftESS( fd ff.nu ft.- Cs.Ir.U q Off...) Is. SECURITY CLASS. (.f saa. rip..i ) Electronic Systems Division Unclassified Hanscom Afl 3. DECLASSIPICATION DOINORADINO Bedford, MA 01 31 SOIIDULE IA. DISTRIIUTIOW STATEM ENT 1.! S a R,.n) Approved for public release; distribution milmired.... 7. OI$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 13 103 EDITION OP 1 NOV 53 IS OSteLI TE LJNCLASmFIED SECURITY CLAU $PI CAIION OF THIS PAGE (IA.. 0. ti.nr,d) L..... -..._..... - -..... _