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the sup P.r(eluo=l) for the AND rule decreases as n increases. Considering U, = 0

M ( n ) = ll{l- [ P ( X , > t18)]n}~o(8) d8

X ( n ) = h{l- [ P ( X , > tl8)]”}n0(8) d8 H ( n ) = 1 - [ P ( X , > t l O , ) ] ”

where 8 2 E C is the value of 8 for which the sup P(U, = 010) is attained. The factor multiplying (1 - E) in (A2) is positive, since

n-1 \

(‘44)

and the quantity inside the curly bracket is positive. Similarly, the factor multiplying E in (A2) is positive. Therefore, the left-hand side of (A2) is positive. That is, the sup P”(elUo=O) for the AND rule decreases as n increases. Based on similar steps, we can prove that both inf P.r(eluo=l) and inf PT(eluo=o) decrease with increasing n.

Similarly, for the OR rule, we can prove that sup p“(Wo=1) sup p”(Wo=0) inf p “ ( @ l U o = 1 )

3 3

and inf P ” ( ~ I U O = O )

all increase with n.

REFERENCES

R. R. Tenney and N. R. Sandell, Jr., “Detection with distributed sensors,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-17, pp. 501-510, July 1981. I. Y. Hoballah and P. K. Varshney, “Distributed Bayesian signal detection,” IEEE Trans. Inform. Theory, vol. 35, pp. 995-1000, Sept. 1989. H. R. Hashemi and I. B. Rhodes, “Decentralized sequential detection,” IEEE Trans. Inform. Theory, vol. 35, pp. 509-520, May 1989. V. V. Veeravalli, “Comments on decentralized sequential detection,” IEEE Trans. Inform. Theoly, vol. 38, pp. 1428-1429, July 1992. J. N. Tsitsiklis, “ Decentralized detection,” in Advances in Statistical Processing, vol. 2, Signal Detection, H. V. Poor and J. B. Thomas, Eds. Greenwhich, CT: JAI Press, 1990. R. S . Blum and S. A. Kassam, “Optimum distributed detection of weak signals in dependent sensors,” IEEE Trans. Inform. Theory, vol. 38, pp. 1066-1079, May 1992. V. V. Veeravalli, T. Basar, and H. V. Poor, “Minimax robust decentral- ized detection,” IEEE Trans. Inform. Theory, vol. 40, pp. 35-40, Jan. 1994. E. Geraniotis and Y. A. Chau, “Robust data fusion for multi-sensor detection systems,” IEEE Trans. Inform. Theory, vol. 36, pp. 1265-1279, Nov. 1990. J. 0. Berger, Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag, 1985. C. H. Gowda, “Robustness of decentralized tests with t-contamination prior,” M.S. Thesis, Dept. elec. Eng.,, Southern Illinois Univ., Carbon- dale, IL 62901, Aug. 1994. R. Viswanathan and V. Aalo, “On counting rules in distributed detec- tion,” IEEE Transa. Acoust., Speech, Signal Process., pp. 772-715, May 1989.

Asymptotically Optimum Detection of a Weak Signal Sequence with Random Time Delays

Igor M. Arbekov

Abstract-The problem of designing asymptotically optimum detectors for a weak signal sequence with random time delays in the presence of a white Gaussian noise is considered. The multidimensional probability distribution of the time delays is assumed to be known. As a result of asymptotic analysis of the log-likelihood ratio, the asymptotically optimum linear or quadratic detectors and their probability distributions and efficiencies are found.

Zndex Term- Detection of dependent random weak signals, log-likelihood ratio, limiting probability distribution.

“The cognitive essence of the theory of probability is opened only by the limiting theorems.”

B. V. Gnedenko and A. N. Kolmogorov [7]

I. INTRODUCTION

The problem of detecting a signal with unknown time delay is one of the most important problems of statistical radioengineering and has applications in the construction of broadcasting systems, radiolocation, etc. When the signal power is much less than the noise power, the signal can be repeated a few times to improve signal reception. At the receiver, the detection of the signal can be complicated by several factors. One of them is the random time delay of each transmitted signal.

In this correspondence, we will be concemed with detection of a weak signal sequence with random dependent time delays in additive white Gaussian noise. In Section 11, we will prove the theorems establishing the limiting distribution of the log-likelihood ratio for the corresponding statistical hypotheses when m (the “depth” of the dependence of the time delays) is much less than n (the length of the signal sequence, m << n). Here we will assume that the signal-to- noise ratio p decreases to zero and the length of the signal sequence n grows to infinity, so that the following asymptotic representation of the log-likelihood ratio A(X) is true:

where the variance of Ln(X) is constant and + 0 in probability under both hypotheses. In this case the contiguity of the sequences of the probability measures corresponding to the statistical hypotheses takes place [5 ] .

In addition it will be proved that the limiting distribution of L, (X) is Gaussian under both hypotheses. These results give the opportunity to determine the minimum value n under which confident (efficient) detection of a weak signal sequence with given detection errors takes place. This is considered in Section 111.

The observation s(t) of a deterministic signal s ( t ) with random time delay 17 in additive noise n(t) can be written as

X ( t ) = s( t - 17) + n(t) .

Manuscript received November IO, 1992; revised November 5, 1994. The author is with the Laboratory for Control Systems Theory, Moscow

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1170 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995

Let p(y) be the probability density of the random variable 7. The signal s ( t - 17) can be rewritten as the sum of two components

(1) s ( t - 17) = E, ( t - 17) + S(t , 17)

where

is the mathematical expectation of the signal with respect to 17; s ( t , 17) = s ( t - 17) - E,s(t - 7 ) is the "purely" random component of the signal with the expectation E,s( t , 7) = 0 for each t.

In this form the problem of detecting a signal with a random time delay is a particular case of the problem of detecting a signal in a generalized observation model (i.e., detection of composite signals) [l]. In [I] the mathematical expression for the locally optimum detector of a weak signal was found as the first nonzero term of the asymptotic Taylor series expansion of the log-likelihood ratio. We will further establish, in Section II, that the remaining term of the Taylor series expansion of the log-likelihood ratio for our case decreases to zero under special conditions involving the number of observations n, the signal-to-noise ratio p, and some other parameters of the probability scheme.

In this correspondence, we consider noise with a Gaussian prob- ability distribution, which restricts the possible applications of the received results in comparison with [l]. On the other hand, in contrast to the conditions of [l], we shall consider the case of dependent time delays (dependent from one channel to another). The case of independent time delays was considered in [2].

11. ASYMPT~TIC ANALYSIS OF LOGLIKELIHOOD RATIO

Let us consider the sequence of real random functions

X=(z1(t), zz(t),...,z,(t)), t E [O,T+V]

and formulate the following statistical hypotheses:

Hypothesis Hypothesis

H I : xl(t) = s ( t - 17%) + nz( t ) HO : z z ( t ) = n z ( t ) .

Here s ( t ) is a known continuous deterministic signal, s ( t ) 0,

is a sequence of stationary connected, m-dependent t # [O, TI;

d T ) ( Y ) ? Y = (YZI, Y22>...>Y2T),

p , i = random variables with marginal probability densities

1 5 T 5 n, p(')(yl) 0, y1 6 [o, VI.

Here

P(')(Yk, Ys) = p'"'(y)dyidyz ... dYk-idyk+i . . . dys-idys+i . . . dy,.

J n,( t ) , i = is a sequence of mutually independent and

independent of vz, i = Gn, white Gaussian noise random functions with spectral density NO and with correlation function R(t) = NoS(t), where 6 ( t ) is the Dirac delta function (n(t) is expressed as the symbolic derivative of the Wiener process).

The log-likelihood ratio for hypotheses ho and H I , after trivial reductions, is written as:

where E?) indicates n-dimensional expectation with respect to 17 and

denotes the signal-to-noise ratio. (We omit the limits of integration in (2) and hereafter for brevity.)

Let

b(y) = l / ( r z N o ) / s ( t ) 4 t - Y) dt,

be the autocorrelation function of the signal.

b(O) = 1

Theorem I: Let p -+ 0 and 1) n + CO so that n 1 I 2 p = AI, 0 < A1 < 00; m = ne,

2) 0 5 (Y < 1/2

O < K l < C c

3) there exist constants 0 < v 5 2, 0 < C < CO such that

b(y) = 1 - Clvl" + ~ ( l v l " ) , as Y -+ 0.

Then under hypothesis HO the random functional A(X) has an asymptotically Gaussian distribution N ( -tc:A:/2, .:A:) and under hypothesis H I it has an asymptotically Gaussian distribution N ( K : A ~ / ~ , K : A : ) .

Proofi Let

&(I/) = l / ( p N o ) nt(t)s(t - Y)dt. I It is not difficult to show that &(y) is a stationary Gaussian process with the expectation EE(y) = 0 and correlation function E&(u)&(u - y) = b(y) , i = [6]. Here and hereafter E (or D ) indicate n-dimensional expectation (or variance) with respect to nz( t ) , i = 1,.

Then under hypothesis HO we can rewrite A(X) as I "

Without loss of generality, we assume that n = (I + m ) ~ , where I, m, T are positive integers, I > m.

Let the set of indices N = { 1, 2,. . . n} be the union of Lk's and M k ' s

L1 = (1, 2,...Z} Ml = {l+1, I + 2 , . . . I + m }

Lz = {I + m + 1, Z + m + 2, . . . ,2I + m}

Mz = { 2 Z + m + l , 2 I + m + 2 ,..., 21+2m+1}, . . . ,

= I , IMk( = m , k =-

and let L be the union of Lk'S: L = L1 U LZ U ... U L , and M be the union of Mk's: M = M I U MZ U ... U M,. Let us divide the observations <1(~1), & ( q ~ ) , . . . ,En(qn) into two groups: { & ( q t ) , i E L } and { & ( q z ) , i E M}. Then using the property of m-dependence of the random variables 17% we can show that

1 A(X) = lnE(,")exp CLC(E~(T~~) - p / 2 ) { z:l

(3)

= ClnE:)exP CLC(Et(71t)-P/2) k=l { + l n ( l + en)

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995 1171

where

We prove that E E ~ --+ 0 when n + CO.

According to Schwarz’s inequality [3], we have with probability 1

I

According to (E<)-’ 5 E<-’ by convexity [3], we have with probability 1

Since { & ( q z ) , i E L } and {&(qZ), i E M } are independent, we have

Using the fact that E exp { E } = exp { a 2 / 2 } for a random variable E with Gaussian probability distribution N ( 0 , 0 2 ) we obtain

E

Let us determine r = no, where D is such that 0 < cy + p < 1. Then from the condition a) of Theorem 1 and from equalities (4) for some constant c we obtain the estimate

EE: 5 Cna+L1-l

which decreases to zero. From this estimate we can see that In (1 + G,) + 0 in probability.

Now we consider the random variable

Let 0, be the set defined by

Since the process ( y ) is determined on a limited segment of time, from condition c) of Theorem 1 and from the estimates, derived in {4], we can show that for some constant C > 0 and all n, the probability P(Q,) satisfies the inequality

2 1 - Cn(lnn)(2-”)/uexp{-ln2 n / ~ }

and P(0, ) + 1 under n + 03. Using the Taylor series expansion of exp {z} in the vicinity of zero and the expressions

I = (n - m r ) / r 5 nl-’ p = ~ln - ’ / ’

we can obtain on the set 0,

n - 3 d

+ 0 ( n(d-l) ( J ! : / Z L ) .

It is clear that for any a < 1/2 we can choose 3 > l / Z so that a + P < 1 (to make Eft + 0) and for sufficiently large d the inequality P - d /Z (d - 1) > 0 will be true. Then

n-19 Ind n n ( d - l ) ( o - d / Z ( d - 1)) = o(n-”‘

Since

under fixed qz has Gaussian distribution N ( 0 , I ) , 1 I n1-3, it is not difficult to see that

if s is odd and

E

1172 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41. NO. 4, JULY 1995

Using the Taylor series expansion of the function In (1 + z) in the vicinity of zero and remembering that T = no, we have k=l

where Cn -+ 0 under n -+ 03 in probability. Taking into account that

we can show that the expectations and the variances of the terms of the asymptotic expansion (5) satisfy the relations

Since In (1 + E " ) decreases to zero in probability, we finally notice that under hypothesis HO the difference

decreases to zero in probability. The asymptotic probability distri- bution of

is Gaussian with the expectation (-n:Xf/2) and the variance This easily follows from the central limiting theorem.

Thus we obtain that that log-likelihood ratio A(X) has an asymp- totically Gaussian distribution N ( -4X;/4, K ; X ; / ~ ) . By [5, Theo- rem 6.11 in this case the contiguity of the sequences of the probability measures corresponding to the statistical hypotheses HO and H I takes place. The direct use of [5, Theorem 7.1 and Corollary 7.21 establishes the convergence of the probability distribution of A(X) to N ( 4 X 2 / 2 , .:A2) under hypothesis H I . Thus Theorem 1 is proved.

Theorem2: Let p -+ 0 and 1) n -+ 03 so that n(l+Y)/'p2 = ~ 2 , o < XZ < 03; m = ne,

0 5 c u < 1 / 4 , 0 5 y < c u

3) . b(y - u ) d y d u

= ~ f X : ( l + 0(na+'-')) (6)

- p'lr -- 2 (7)

4) there exist constants 0 < Y 5 2, 0 < C < 03, such that 5 cP4lzr 5 cn-$ (8) b b ) = 1 - CIyI" + o(lyI"). as y -+ 0.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995 1173

Then under hypothesis HO the random functional A(X) has asymptotically Gaussian distribution N ( -tczX;/4, K;X;/~), and under hypothesis H1 -asymptotically Gaussian distribution N (4X; /4 , 4X;/2).

The proof of Theorem 2 is analogous to the proof of Theorem 1, taking into account the terms of the expansion (5) up to p5.

The first (linear) term of the expansion (5) is

II.2 / p ( ' ) ( y ) E C ( y ) d y

and its variance according to (6) and assumption b) of the Theorem 2 is equal to

k = l t E L k

* / s ( t - yl)s(t - u1) dt

* / s ( t - Yl+k)S(t - Ul+k) d t dy1 du1 dYl+k dUl+k 1 ll2 .

The choice of one detector or another depends on the relation between the values Ql(n) and Qz(n).

Let us suppose that the known signal ~ ( t ) is transmitted and p << 1 at the receiver. h t 61, 62 be the errors of the detection and we want, for example, to find nl-the length of the signals sequence that

51 = P(Ai(X) 2 R/Ho) 6 2 = P(Ai(X) < R/Hi)

for the test threshold R. p(l)(y)s(t - y)dy dt = o(1). Let z(&), ~ ( 6 2 ) be determined by

5, = (27r)- ' / 'C, , exp {-22/2) dz, j = 1 , 2 .

s (J l2 X Z T 1 n - ( l + Y ) / 2 1 No p2

That is why the second term in (5) is the main term. After some transformations we can rewrite it as

*Ez(Yl)Ec+k(yl+k) dYl d?/l+k) + 6,

where 6, + 0 in probability. This fact finally leads to the changes in the parameters of the limiting probability distribution.

III. ASYMPTOTICALLY OFTIMUM DETECTORS From the proofs of Theorems 1 and 2, it is not difficult to see that

the asymptotical optimum detector is either the linear detector

where

* (1 s ( t - y)s(t - U ) d t dy d u ) 2

Then we calculate Ql(nl) and find nl from the equation

~ ( 6 1 ) + ~ ( 6 2 ) = Qi(ni). (12)

It is clear that nl ensures the detection of the signal with detection errors 61 and 6 2 . After that we calculate Q~(n1) . If ~ ( 6 1 ) + ~ ( 6 2 ) > Q ~ ( n 1 ) then we use the linear detector hl(X). Otherwise, we use the quadratic detector A z ( X ) . In the last case we find nz-the new length of the signals sequence from the equation

It must be clear that only if nl (or nz) are sufficiently large we can correctly use the Gaussian approximation for the determination of the detection errors. In particular, it will be correct if n1 (or nz) is more than max{(Z(p1))6, ( ~ ( P z ) ) ~ ) [81.

Most probably conditions c) and d) of Theorem 1 or 2 will be correct for any practical situation.

For example, let 1)

Then

and b(y) satisfies conditions c) and d) with C = 1 and Y = 1. 2 )

sin (2rk t /T) , t E [0, TI, k-integer t 10, TI s ( t ) =

and b(y) satisfies conditions c) or d) with C = 27r2k2 and v = 2. In the limiting theorems we do not assume that the signal s ( t )

and other parameters of the probability scheme must change as a special function of n. The theorems' meaning is that if for some concrete practical situation, the calculations n1 and n 2 from (12), (13) give that n1 >> n2 and nl is sufficiently large, then we can be sure that the linear detector is the optimum detector and that we can correctly use the Gaussian approximation for the determination of the detection errors.

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For example, let us have the observation of the signal Multiuser Signaling in the Symbol-Synchronous AWGN Channel

J. A. Fergus Ross, Member, IEEE, and Desmond P . Taylor, Fellow, IEEE

Abstruct- Multiuser signal set design for the energy-constrained linear real-additive symbol-synchronous additive white Gaussian noise channel is investigated in this correspondence. An argument is presented showing the suboptimality of binary alphabets which, in turn, uncovers a rule for the formation of multiuser signal sets. This simple rule leads to a theorem stating the dimensionality at which additional users can be added to a multiuser system without decreasing the minimum Euclidean distance. Specific vector sets with the desired property are then constructed as examples and some generalizations are discussed. The theorem may also be used as a platform for the design of more efficient multiuser codes incorporating redundant signal sets. Two such codes are presented. In particular, a code combining four-dimensional unit-energy signals with a rate-W3 convolutional code obtains a summed rate of 1.75 bitddimension in the two-user adder channel with received minimum Euclidean distance of 2.

with the random time delay 7 and its probability density

1/V> Y E 10, VI , v I T. p(y) = { 0, y [O, VI

Then for independent time delays we have

Qi(n) = n1/2p(1 - V/3T)’/’

& 2 ( n ) = 2-’/’n1/’pZ(1 - (2V/3T)(1 - V / 4 T ) ) 1 / 2 .

It is easy to see that Ql(n) >> Qz(n) under p << 1 for all n and in this case the linear detector is much more effective than the quadratic detector.

ACKNOWLEDGMENT

The author wishes to thank the Associate Editor and the anonymous reviewers for their very helpful comments and suggestions.

REFERENCES

[ l ] I. Song and S. A. Kassam, “Locally optimum detection of signals in a generalized observation model: The random signal case,” IEEE Trans. Inform. Theory, vol. 36, pp. 51CL530, 1990.

[2] I. M. Arbekov, “Detection of weak signals sequence in the conditions of the time delay fast fluctuation,” Radioeng. Electron., vol. 36, no. 4, pp. 737-745, 1991 (in Russian).

[3] M. Loeve, Probability Theory. New York: Van Nostrand, 1960. [4] M. R. Leadbetter and M. Rootzen, “Extrema1 theory for stochastic

processes,” Ann. Probab., vol. 16, pp. 431-478, 1988. [5] G. G. Roussas, Contiguity of Probability Measures. Cambridge, U K

Univ. of Cambridge Press, 1972. [6] B. R. Levin, Theoretical Bases of the Statistical Radioengineering. vol.

2. Moscow, USSR: Sov. Radio, 1975 (in Russian). [7] B. V. Gnedenko and A. N. Kolmogorov, Limiting Distributions for the

Sum of the Independent Random Variables. Moscow, USSR GITl’L, 1949 (in Russian).

[8] W. Feller, An Introduction to Probability Theory and its Applications, vol. 2. New York: Wiley, 1971.

I. INTRODUCTION: SUMMING SIGNAL SETS Consider the addition of two one-dimensional, equally-likely, an-

tipodal, binary symbols over one symbol period. The symbols are to be restricted in energy such that user 1 selects from { -a , a} and user 2 selects from { - b , b } where lal, Ibl 5 1 . Assuming la1 > Ibl, the summed, or global, signal set contains four points, Fig. 1. This four-point constellation is clearly uniquely decodable.

The channel under consideration is corrupted by additive white Gaussian noise (AWGN), implying the use of the Euclidean metric in a “soft” decision process. Thus we strive to maximize the minimum Euclidean distance between points in the signal set. In the present example, the distances among the four summed signals depend on a and b. For a given a, we may select b to maximize the minimum distance between the points. That is, choose b to maximize min (2a -2b , 2b) , a 2 b. This occurs at b = ;. Any greater value for b would force the inner points closer together than necessary while any lesser value of b would result in the outer points being closer together than necessary.

Thus a simple rule related to multiuser signal set design for the AWGN channel has been uncovered. Note that mappings from GF(2) onto a single set of binary symbols violate this rule, resulting in a minimum distance of zero in the summed set.

Previous work in the symbol-synchronous multiuser channel has had an algebraic viewpoint, considering superposed codes in both noiseless and noisy channels, where the transmitted symbols are typically restricted to 0 , l [1]-[4]. Chang and Weldon [ l ] do discuss multilevel codes but they are formed by grouping sets of binary codes together. In any case, Hamming distance and L-distance are the design criteria of the cited works.

The Hamming distance is the number of positions in which two code vectors are different while the L-distance is the sum of the

Manuscript received January 10, 1994; revised October 26, 1994.This work was presented in part at the 1993 Canadian Workshop on Information Theory, Rockland, Ontario, and is based on the Ph.D. dissertation of the first author.

J. A. Fergus Ross was with the Communications Research Laboratory, McMaster University, Hamilton, Ont., Canada. He is now with the Electronic Systems Laboratory, GE Corporate Research and Development, Schenectady, NY USA.

D. P. Taylor is with the Department of Electrical and Electronic Engineer- ing, University of Canterbury, Christchurch, New Zealand.

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