Locally optimum detector for correlated random signals in a weakly dependent noise model

Signal Processing 74 (1999) 317—322

Locally optimum detector for correlated random signals in a weakly dependent noise model Kwang Soon Kim , Sun Yong Kim
, Iickho Song *, So Ryoung Park Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea
Department of Electronics Engineering, Hallym University, 1 Okcheon Dong, Chuncheon, Kangwon Do, 200-702, South Korea Received 24 April 1998; received in revised form 18 August 1998

Abstract In this paper, we consider a discrete-time random signal detection problem under the presence of additive noise exhibiting weak dependence. We derive the test statistic of the locally optimum detector under a weakly dependent noise model. The performance characteristic of the locally optimum detector is analyzed and compared with that of the square-law detector in term of asymptotic relative efficiency.  1999 Elsevier Science B.V. All rights reserved. Zusmmenfassung In dieser Arbeit wird ein Problem der Detektion eines zeitdiskreten Zufallssignals in additivem Rauschen mit schwacher Abha¨ngigkeit betrachtet. Die Teststatistik des lokal optimalen Detektors wird fu¨r ein Rauschmodell mit schwacher Abha¨ngigkeit abgeleitet. Die Leistungsfa¨higkeit des lokal optimalen Detektors wird analysiert und mit jener des quadratischen Detektors in Hinblick auf asymptotische relative Effizienz verglichen.  1999 Elsevier Science B.V. All rights reserved. Re´sume´ Dans cet article, nous conside´rons un proble`me de de´tection de signaux ale´atoires en temps discret, en pre´sence de bruit additif pre´sentant une de´pendance faible. Nous de´rivons une statistique de test du de´tecteur localement optimal sous un mode`le de bruit faiblement de´pendant. La performance du de´tecteur localement optimum est analyse´e et compare´e avec celle du de´tecteur a` loi quadratique, en termes d’efficacite´ relative asymptotique.  1999 Elsevier Science B.V. All rights reserved. Keywords: Locally optimum detector; Weakly dependent noise; Random signals

* Corresponding author. Tel.: #82 42 869 3445; fax: #82 42 869 3410; e-mail: [email protected] 0165-1684/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 8 ) 0 0 2 1 9 - 9

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K.S. Kim et al. / Signal Processing 74 (1999) 317—322

1. Introduction The signal detection problem in noisy observations has been considered in many previous studies. Among the various signal detection problems, weak signal detection has been of much interest in detection theory and applications. Among the typical investigations on locally optimum (LO) detectors are those considered in [1,4,5,11]. It has been commonly assumed that the additive noise samples are statistically independent. In practice, however, this assumption is often violated, and the optimum detectors designed under this assumption are no longer optimum in practice. Such a situation becomes more realistic as the sampling rate gets higher. Thus, investigations on signal detections in dependent noise should be considered. Among the investigations on the general optimum detection problem under various dependent noise models, including the -mixing noise model, mdependent noise model, and transformation noise model, are those in [2,3,7,9]. When the dependence of noise is weak, we can use a simpler model. In [6,8], the first order moving average (MA) of an i.i.d. random process is considered as a weakly dependent noise model. In these studies, however, detection schemes only for known signals were considered. In this paper, we will investigate the LO detection for random signals under a weakly-dependent noise model. The weakly dependent noise will be modeled as the first order MA of an i.i.d. random process.

2. Observation model In this paper, we will consider the detection of discrete-time random signals in weakly dependent noise environment. Let H be the null hypothesis  and H be the alternative hypothesis. Then, the  observation model can be written as H : X "¼ , i"1,2,2,n,  G G (1) H : X "hs #¼ , i"1,2,2,n,  G G G where +X , are the observations, +¼ , are the G G weakly-dependent noise components, h is the signal

strength parameter, +s , are the random signal G components with mean zero and variance +p,. G Then, the detection problem becomes a problem of the hypothesis decision based on the n observations, +X ,. G Weakly dependent noise can generally be modeled by the Volterra expansion with Volterra kernels and independent random processes [10]. This model, however, is almost intractable to handle because of the infinitely many terms of the expansion. In [6,8], simple first-order bilateral and unilateral MAs of an i.i.d. random process are used to model the weakly dependent noise, respectively. These two MA models are simple and good approximations to a weakly dependent noise. In addition, it was shown that the LO detector designed under one MA noise model can be applied with slight changes in the other one with almost the same performance [6]. In this paper, we will assume that the weakly dependent noise ¼ , G i"1,2,2,n, are the unilateral MA of i.i.d. random variables: ¼ "e #oe u , G G G\ G\

(2)

where e , i"1,2,2,n, are the i.i.d. random variG ables with common p.d.f. f . The p.d.f. f is even   symmetric with bounded continuous derivatives and satisfies the regularity condition [4]. This model is not only an analytically tractable model but also a good representation of practical dependent noise when the dependence is weak. Here, o is called the dependence parameter determining the correlation coefficient of ¼ , and u is the unit step G G sequence, i.e., u "0 when i(0 and u "1 when G G i*0. Let X, w, e and s be the n-tuple vectors representing (x ,x ,2,x ), (¼ ,¼ ,2,¼ ), (e ,e ,2,e )   L   L   L and (s ,s ,2,s ), respectively, and fw(w), fe(e)"   L “ L f (e ) and fs(s) be the p.d.f.s of w, e and s, G  G Then, under H we have respectively.  fw(w)"f (X !hs ) f (X !hs !o(X !hs ))         2f (X !oX #2#(!o)L\X  L L\  ! h(s !os #2#(!o)L\s ) L L\ 

K.S. Kim et al. / Signal Processing 74 (1999) 317—322

L " “ f (½ !hc )  G G G "fe(Y!hc),

(3)

where Y"(½ ,½ ,2,½ ), ½ " G\ (!o)IX ,   L G I G\I c"(c ,c ,2,c ) and c " G\ (!o)Is .   L G I G\I 3. The locally optimum detector Let us define



(X"h)"

1L

fw(X!hs) fs(s) ds,

(4)

where 1L is the set of all n-tuples of real numbers. Then, the LO test statistic can be calculated by [4] dJ (X"h)/dhJ" F, ¹ (X)" * (X"0)

(5)

where l is the order of the first nonzero derivative of (X "h) at h"0. From Eqs. (3) and (4), it is easily seen that



d (X "h) dh

 



dfe(Y!hc) dh 1L

fs(s) ds F L L " fe(Y ) fs (s) c c g (½ ) G H *- G 1L G HH$G L ;g (½ )# ch (½ ) ds (6) *- H G *- G G

" F

and



(X "0)"

1L



fe(Y ) fs(s) ds"fe(Y),



(7)

where g (x)"!f (x)/f (x) and h (x)"f (x)/f (x). *  *  Then, the LO test statistic can be obtained as L L ¹ (Y)" Es+c c ,g (½ )g (½ ) *G H *- G *- H G HH$G L # Es+c,h (½ ), (8) G *- G G where Es+ ) , is the expectation over s. It is easily seen that the test statistic (8) is the same as that obtained for independent noise except

319

that we use weighted averages ½ of the output G samples and correlation coefficients of the weighted averages c of the signal components. It is also seen G that n!1 memories are required to implement the LO detector. Thus, it is clear that the implementation of the LO detector becomes more inefficient as the sample size gets larger. We can, however, obtain finite memory approximations, which are easy to implement and have less memory requirement, to the exact LO detector from (8) by ignoring higherorder terms of o. The performance of the exact LO detector is then the upper bound of that of those finite memory detectors. Due to the fact that "o" is small, the performance of the finite memory detectors is expected to be acceptable, which was shown in [6] for the known signal case.

4. Performance analysis In this section, we will analyze the performance characteristics of the LO detector under the weakly dependent noise model. The performance of the LO detector for known signals in weakly dependent noise was studied and shown to be better than those of the linear correlator and the sign correlator in [6]. In this paper, the performance of the LO detector will be compared with that of the squarelaw (SQ) detector whose test statistic is L ¹ " X. 1/ G G

(9)

In comparing the asymptotic performance of two detectors, the asymptotic relative efficiency (ARE) is generally employed. Under some regularity conditions [4] the ARE of detector D with respect   to detector D can be expressed as ARE "m /m ,     where m is the efficacy of D calculated as G G m " lim G L

[dJE+¹ "H ,/dhJ" ] G  F , i"1,2. n»+¹ "H , G 

(10)

In Eq. (10), ¹ denotes the test statistic of the G detector D , E+¹ "H , denotes the expected value of G G  ¹ under the alternative hypothesis, and »+¹ "H , G G  denotes the variance of ¹ under the null hypoG thesis.

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Theorem 1. The efficacy of the LO detector is m "2I( f )[1Es (c,c)2!1Es (c)2] *  # I ( f )1Es (c)2,   where

(11)

1 L L 1Es (c,c)2" lim Es +c c , G H n L G H 1 L L L L " lim Q (!o) GI n L G H I J ;Q (!o)r (i, j)r (k, l), (12) HJ Q Q 1 L 1Es (c)2" lim Es +c, G n L G 1 L L L L " lim Q (!o) GHIJ n L G H I J ;r (i, j)r (k, l), (13) Q Q f (y)  f (y) dy, (14) I ( f )"  f (y)


 


f (y)  f (y) dy, f (y)

(15)

x GH\G\H!xL\G\H> , Q (x)" GH 1!x

(16)

I ( f )" 

Q (x) GHIJ x GHIJ\G\H\I\J!xL\G\H\I\J> " 1!x

(17)

and r (i,j)"Es+s s ,. Q GH

1 L 1Es(c)2" lim Es+c, G n L G 1 L L " lim Q (!o)r (i, j), GH Q n L G H



m " xf (x) dx  

(21)

(22)

and




p" 



xf (x) dx 

 .

(23)

The proofs of Theorems 1 and 2 are shown in Appendix A. Now, let us consider some examples to show the asymptotic performance of the LO detector more explicitly. Example 1. Let r (i, j)"rG\H, where 0("r"(1 Q and f (x)"(1/(2p)e\V. Then, we have I ( f )"1,    I ( f )"2, m "3, p"1,     (1!or)K(o,r) 1Es (c,c)2" , (1#or)(1!o)(1!r)

(24)

(1!or) , 1Es (c)2" (1!o)(1#or)

(25)

1!or 1Es(c)2" (1!o)(1#or)

(26)

and (18) r 1Es(s,c)2" , 1#or

Theorem 2. The efficacy of the SQ detector is 4((1!o)1Es(c)2#2o1Es(s, c)2) m " , 1/ (1#o)m !(1!o)p   where

1 L 1Es(s,c)2" lim Es+s c , G G\ n L G 1 L G\ " lim (!o)G\H\r (i, j), Q n L G H

(27)

where (19) K(o,r)"(1#or)(1!r)(1!o) # 2(r!o)(1#or).

(28)

Then, from Theorems 1 and 2, the ARE is *-1/ (20)

(1!or)(1#4o#o)K(o,r) ARE " . *-1/ (1!o)(1#or)(1!r)

(29)

K.S. Kim et al. / Signal Processing 74 (1999) 317—322

321

Example 2. Let r (i, j)"rG\H, where 0("r"(1 Q and f (x)"e\V/(1#e\V). Then, we have I ( f )"    , I ( f )", m "  n and p" n. Then, from         Theorems 1 and 2, the ARE is *-1/ n(4#13o#4o) ARE " *-1/ 2025




;

10(1!or)K(o,r) (1#or)(1!o)(1!r)

!



(1!or) . (1!o)(1#or)

(30)

In Figs. 1 and 2 , the ARE derived in the two *-1/ examples are plotted for various values of o when the additive noise is the first-order MA of the i.i.d. Gaussian process and the i.i.d. symmetric logistic process, respectively.

Fig. 1. ARE for various values of o when the noise is the *-1/ first-order MA of the i.i.d. Gaussian process.

5. Concluding remark In this paper, we considered the LO detection of random signals in additive weakly dependent noise. The test statistic of the LO detector for random signals in weakly dependent noise was derived and shown to have the same structure as that for independent noise with additional weighted averaging of output samples. The asymptotic performance of the LO detector was analyzed and compared to that of the SQ detector in terms of ARE. It was shown that the LO detector outperformed the SQ detector more as the correlation coefficient of the signal differed more from that of the noise.

Fig. 2. ARE for various values of o when the noise is the *-1/ first-order MA of the i.i.d. symmetric logistic process.

Acknowledgements ily seen that This research was supported by a 1997 Grant from the Hallym Academy of Science, Hallym University, for which the authors would like to express their thanks.

Appendix A. Proof of Theorem 1. Using x "y #oy , y "0, G G G\  E+g (½ )"H ,"0 and E+g (½ )"H ,"0, it is eas*- G  *- G 



dE+¹ (Y)"H , * dh

F "»+¹ (Y)"H , * L L "E 2 Es +c c ,g (½ )g (½ ) G H *- G *- H G HH$G L # Es +c,h (½ )"H G *- G  G





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K.S. Kim et al. / Signal Processing 74 (1999) 317—322



L L L "2I( f ) Es +c c ,! Es +c,   G H G G H G



L # I ( f ) Es +c,.   G G Thus, the m m " lim *L

*-

and

(31)

is

»+¹ (Y), *n

"2I( f )[1Es (c,c)2!1Es (c)2]   #I ( f )1Es (c)2. )  

(32)

Proof of Theorem 2. Using c "s !oc , c "0, G G G\  E+yh (y ),"2 and E+y g (y ),"1, it is easily G *- G G *- G seen that



dE+¹ "H , 1/  dh

F

"E+¹ (X)¹ (Y)"H , 1/ *

 



L "E (y #oy )¹ (Y)"H G G\ * G

L L "E yEs+c,#4o yEs+c c , G G G G G\ G G



L #o y Es+c ,"H G\  G\ G L "2 Es +c,#4oEs+c c , G G G\ G L #2o Es +c , G\ G L "2 Es +c,#4oEs+s c , G G G\ G L !2o Es +c , G\ G

(33)

»+¹ "H ,"E+¹ "H ,!E+¹ "H , 1/  1/  1/  "n[(1#o)m !(1!o)p]   !o[(2#o)m !(2!o)p]. (34)   Thus, m is 1/ [dE+¹ "H ,/dh" ] 1/  F lim n»+¹ , 1/ L 4((1!o)1Es(c)2#2o1Es(s,c)2) " . ) (35) (1#o)m !(1!o)p   References [1] J. Bae, I. Song, Rank-based detection of weak random signals in a multiplicative noise model, Signal Processing 63 (December 1997) 121—131. [2] D.R. Halverson, G.L. Wise, Discrete-time detection in

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