detection of random signals in gaussian mixture noise

1788 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 6, NOVEMBER 1995

etection of andom Signals in Gaussian Mixture Noise

David W. J. Stein, Member, IEEE

Abstruct- A locally optimal detection algorithm for random signals in dependent noise is derived and applied to independent identically distributed complex-valued Gaussian mixture noise. The resulting detector is essentially a weighted sum of power detectors-the power detector is the locally optimal detector for random signals in Gaussian noise. The weighting functions are modified to enhance the detection performance for small sample sizes. An implementation of the mixture detector, using the expectation-maximization algorithm, is described. Moments of these detectors are calculated from piecewise-polynomial approximations of the weighting functions. The sum of sufficiently many independent identically distributed detector outputs is then approximated by a normal distribution. Probability distributions are also derived for the power detector in Gaussian mixture noise. For a particular set of noise parameters, the theoretical distributions are compared with those obtained from Monte Carlo simulation and seen to be quite close. The theoretical distributions are then used to compare the performance of the mixture and power detectors in Gaussian mixture noise over a range of parameters and to assess the impact of parameter error on detection performance. In this study, the signal gain of the mixture detectors varies from 15 to 38 dB, and the degradation of the probability of detection due to parameter estimation error is relatively minor.

Index Terms- Locally optimal detection, random signal, Gaussian mixture noise, EM algorithm, receiver operating characteristics.

I. INTRODUCTION OISE in which individual sources are identifiable is likely to be non-Gaussian as the data may be the response

of a sensor or an array of sensors to an insufficient number of identical sources for the Central Limit Theorem to apply. Middleton 1211-[23], and Middleton and Spaulding [24] show that electromagnetic interference from a variety of sources is non-Gaussian. Evans and Griffiths [ l l ] show that extremely low-frequency electromagnetic noise in the atmosphere (3-300 Hz) may be non-Gaussian due to impulses of lightning. Bouvet and Schwartz [7] have shown that certain ocean acoustic data sets containing noise from snapping shrimp or merchant shipping are not Gaussian. Powell and Wilson [32] show that ocean acoustic data sets containing noise from snapping shrimp or seismic explorers may not have normal densities.

Manuscript received April 19, 1994; revised May 5, 1995. This work was supported by ONR Project SW17, and ONR program element 0601 152N. This work was also supported by Dr. R. Doolittle of ONR under Project SW17 and Dr. A. Gordon of NCCOSC RDT&E Division under the OCNR IR program. The material in this paper was presented in part at the SIAM Meeting on Simulation and Modeling, San Francisco, CA, 1993, and at the 28th Asilomar Conference on Signals, Systems, and Computers, Asilomar, CA, 1994.

The author is with NRaD, Code 761, San Diego, CA 92152-5000 USA. IEEE Log Number 9414741.

Machell et al. [16j, [17] reject the Gaussian assumption for ocean acoustic data sets containing the response to local shipping, seismic explorers, or seals and whales, and they accept the Gaussian hypothesis for data collected from a quiet area without nearby shipping. Stein et al. [35], [36] have shown that shipping noise in ocean acoustic data may be non- Gaussian. If the number of sources contributing to the noise fluctuates, then the noise may be non-Gaussian even as the expected number of noise sources goes to infinity, [33] and references therein.

Gaussian mixture densities (GMD) have been used to model a variety of non-Gaussian noise environments. If the random variable X = AU where A and U are independent, A is positive and discrete with p(A = a,) = p z , and U has a Gaussian density with mean ,U and covariance matrix C, then X is a spherically invariant random variable, and it has the as4D [21

m

Px(Zc> = Cp2N(a, ,U, .PC)(4. (1) 2 = 1

Thus the GMD applies to data sets for which the variance may vary from sample to sample and for which the data conditioned on the variance are normal. Other applications of mixture densities are discussed in [39], and continuous Gaussian mixtures are found in [33]. Middleton’s Class A noise density is a uni-variate Gaussian mixture distribution with m = 00, but it has only three independent parameters due to relations among them [3]. Gaussian mixture densities have been applied to impulsive noise, to various acoustic noise sets, and to sources moving in multimodal propagation environments [71, 1161, [171, 1211, [221, 1321, 1351, [361. Kassam [14] describes other parametric families of densities used to model noise. Noise has also been modeled by using nonparametric techniques [ 161, [ 171, [ 3 SI.

There are various techniques to detect signals in non- Gaussian noise. If the noise-only and signal-plus-noise densities are known, then a likelihood ratio test, using the Ney- man-Pearson criterion, provides the greatest probability of detection for a given false alarm rate [26], [31]. If these densities are not known then other approaches may be used. The minimax approach [13] defines classes of noise or signal- plus-noise density functions and then uses the detector which minimizes the maximum expected error over the classes. For example, while the matched filter is equivalent to a likelihood ratio for known signals in stationary independent Gaussian noise [12,] [26], [40], the correlator-limiter is the minimax detector for a known signal in stationary independent noise

0018-9448/95$04.00 0 1995 IEEE

STEIN: DETECTION OF RANDOM SIGNALS IN GAUSSIAN IMI[X'IURE NOISE 1789

with a distribution belonging to the class of +contaminated mixture distributions with a nominal Gaussian distribution [ 191. Nonparametric techniques are designed to work for more general classes of probability densities than is the minimax approach. For example, if the noise density is in the class of densities with median value zero and the signal-plus- noise density is not in this class, then the sign test provides a nonparametric means of detecting the signal [31]. If the signal-to-noise ratio is small, and if the noise density can be approximated by a member of a parametric family of densities for which the parameters can be estimated, or the noise density can be estimated nonparametrically , then locally optimal detection algorithms, which are based on Taylor approximations of the likelihood ratio, may Ibe employed

Detection algorithms for a particular application may be selected on the basis of comparative performance analyses and analyses of the sensitivity of the performance tcl modeling or parameter estimation error. The performance of locdly optimal detection algorithms is often studied by using asymptotic relative efficiency (ARE) [14], [24], [31]. ARE is most useful if the signal is very weak, and a very large number of samples is available. When these conditions are not met, receiver operating characteristics (ROC), which plot the probability of false alarm versus the probability of deteclion for fixed values of other parameters, are more revealing. These curves are calculated from probability distributions of the detection statistic conditioned on noise only or signal-plus-noise, and these distributions may be derived or obtained from Monte Carlo simulations. In most studies of mixture detectors performance analyses have relied upon ARE E141 and simulation VI, Wl.

In this paper we derive a locally optimal detection algorithm for discrete random signals in correlated noise, arid specialize it to the case of complex Gaussian mixture noise. 'fie algorithm is then modified to detect signals that are not vanishingly small. This algorithm is implemented by using the expectation-maximization (EM) algorithm to estimate the noise parameters. Noise-only and signal-plus-noise distributions of the power detector and approximate noise-only and signal- plus-noise distributions of the mixture detector are obtained for independent Gaussian signals in Gaussian mixture noise. These distributions are compared with the distributions obtained from Monte Carlo simulations, and they are used to study the relative performance of the power and mixture detectors over a wide range of noise parameters and to assess the impact of parameter error on detection performance.

[41-[61, [201, [241-[311.

LOBD for deterministic and random signals in ergodic Markov noise.

The LOBD statistic for a discrete zero-mean stochastic signal in correlated noise is derived under the assumptions that the signal and noise are independent, the probability density of the noise has continuous second-order partial derivatives, and there exists a likelihood ratio. The detection statistic is used to distinguish between the two hypotheses

H(3 : X; =N, and

H I : X, = &, +N;

1 5 i 5 n, where n is the length of the data segment, s is the unit variance signal, a is the unknown variance of the received signal, and Na is the ith noise sample. Let p s denote the probability density function of s, then

P&(Yl,...,T/n) = u - " ' 2 P s ( Y l / ~ , . . . , Y " / ~ ) .

Since the signal and noise processes are independent of each other, the probability density function (pdf) of the signal-plus- noise process, p f i + N , is the n-fold convolution of the noise pdf and the signal pdf. Thus for 3 E R", the locally optimal detection statistic is

(employing a chan,ge of variables)

Interchanging integration and differentiation operators leads to

To evaluate this limit, define

Then 11. LOCALLY OlTIh4AL DETECTION OF RANDOM SIGNALS

The locally optimal Bayes detection (LOBD) statistic is g(o,.'> = 1 !ER" (2 i=l %(3) * g a ) p s ( g ) dG equivalent (differs by a monotone transformation) to the first nonvanishing term of the Taylor approxirnation, at zero signal strength, of the likelihood ratio [20], [:241-[291, [31]. Middleton [20], [25], [26], [29] derives LOBD statistics for deterministic and certain random signals. Poosr and Thomas

= 5 gcs [ i=l yERn

= (2 ($1) E(ya)

Y i P m dii

[30] derive the LOBD for random signals in independent noise having a twice differentiable density. Maras [ 181 derives the

z = 1 = o


as E(y i ) = 0 by hypothesis. The limit in (2) has the form

Iim g(a, Z)/& a+O

which can be evaluated by using L'Hospital's rule.

(3)

where uzJ = E,(y,y,). The detection statistic can be estimated by using either a parametric model of the probability density [14], [24], [30] of the noise or nonparametric [41-[61, [381 techniques.

Equation (3) generalizes the detection statistics for discrete random signals utilized by Thomas and Poor, Middleton, Spaulding, and Maras in that it allows for quite general correlated noise and signal samples. It differs from the statistics employed by Middleton, Spaulding, and Maras in that they have an added bias term, &(a), which depends upon n and a. As this term is independent of the data, Z, adding a bias term to T produces a statistic which is equivalent to T up to monotone transformation. However, in their work the bias term facilitates the computation of the distributions of the statistics.

The bias term is selected so that the distributions of the detection statistics under HO and H I , {PO,,} and PI,^}, are contiguous [15], [18]. Since they are also asymptotically normal, for large n the densities under HO and HI are approximated by the normal densities N( -015a;, 02) and N(0.502, p i ) , respectively. Furthermore, as a -+ 0 and n --f 00, the distributions of the LOBD approach the distributions of the log of the likelihood ratio, so that the LOBD are asymptotically optimal (AO) [18], (251. In the present work, for the case of independent identically distributed (i.i.d.) Gaussian mixture noise and independent Gaussian signal samples, the moments of T are calculated under the HO and iY1 hypotheses, and thus approximate distributions are obtained without appealing to contiguity. Therefore, the bias term is not necessary.

The detection statistic provided by (3) is evaluated for independent signals and i.i.d. complex Gaussian mixture noise, i.e., z E C and

(4)

where 0: < a; if i < j . The detection statistic is applied to a sequence of n complex numbers, Z' = ( z1 . . . zn) . These define 271 real numbers, z k = ( X 2 k - l 1 x 2 k ) , in (3). Independence of the signal implies that u , ~ = 0 if i # j , and unit signal variance implies that vll = 1 for 1 5 I 5 2 n . A direct calculation shows that for the noise model of (4) and 1 5 i 5 2 n , i an

odd number

x; + x;+l *exp (- 2 4 ) '

Similady, if 1 5 i 5 2 n , i an even number, then

1 1 P ( G - 1 , G)

Finally, substitution into (3) yields

where p ( z , I k , Ho) is the probability (assuming noise only) of zi given state k and

The power detector is the locally optimal detector of random signals in Gaussian noise [26], [31], and it is equivalent to

(7)

where m2 is an estimate of the noise variance. The detection algorithm, which compares (5) io a threshold, is thus a multistate power detector. It requires estimates of the probabilities of the states p k , the variances associated with each state U:,

and the conditional probability of each state for each data

If the signal variance af is not sufficiently small relative to c;, the performance of (5) may be enhanced by replacing p ( k I z,, Ho) with p ( k I z,, a:) defined by (let e =

point P(L I Z Z , B o ) .

( P I , d , . . . ,Pm, 4-J and U = llzl12$

%(U,, 6 03 = P(k I U,, 6 02)

STEIN DETECTION OF RANDOM SIGNALS IN GAUSSIAN MIX'NRE NOISE 1791

m = I ( 2 -k +) [w k (t , e', - wk (t , e', 0)] P n ( t ) d t

- - taining ZO. Define \

P

B$ G 0.5 11z3 \ I 2 .

0% is an estimate of the data variance. If oh < as, ~2 0, otherwise. B," is an estimate of the signal strength.

3=1

cr& - cr$, and B,"

The detection statistic studied below is

Equation (9), unlike a likelihood ratio, is independent of the signal pdf, thus it may be beneficial when tlhe pdf is not known. We now show that the ue correction is immaterial if n is sufficiently large and gf is sufficiently small. However, Fig. 14 shows the importance of this correction for n = 60 and a range of signal strengths.

A s s y e that the complex noise z has a GMD wiih parameter vector 0, and that the signal has a Gaussian dlensity with variance gin, where { B ~ , } ~ ' , is a monotone decreasing sequence that converges to 0. Let U G 1 1 ~ 1 1 ~ and Mo(u) be (9) with 0: = 0, and let M l ( u ) be (9) with crz = o&, i.e., the signal variance estimator is assumed to be perfect. Denote the signal-plus-noise density by

m

and the expected value of U under p , by E,(u). Note that { & ( U ) } is a monotone decreasing sequence. Assuming a perfect signal variance estimator and noise-only data, Mo(u) = M1 ( U ) , thus equivalent performance of the detixiors for large n and small B, follows from

00

i L t l IMo(t) - Ml(t)lp,(t)dt = 0 (10)

which we now argue. The noise density is an m state mixture, thus

and if a 5 b

Let M I , be M wiih a2 = 02,. For all 2, limnem MI,(%) = Mo(z). Thus M I , converges to M O uniformly on closed intervals. Thus for any fixed T ,

Equation (10) follows from (11) and (12).

detector that is actually implemented is As the noise parameters must be estimated from data, the

where now 32 and p k are the estimated variance and probability, respectively., of the kth state, e = (1;1,8;, . . * , Ijm , e%), and wk(ui , 4, 6e) is calculated from (8) by replacing the actual parameter viilues with the corresponding estimates. The technique used to estimate these parameters is discussed in the next section.

111. IMPLEMENTATION OF THE MIXTURE DETECTOR USING TKE EM ALGORITHM

To implement (13), the parameters of the mixture model, m, p k , and f f k , must be estimated. Once the number of terms has been fxed, there are four major techniques for estimating the other parameters: the method of moments (MOM) [71, 1321, 1421, the minimum distance (MD) [7], [32], the maximuim likelihood (ML) [32], [42], [43], or the expectation and maximization (EM) [lo], [39], 1411, [441 algorithms. Zabin and Poor [42]-[44] have shown that the EM algorithm can provide useful estimates of the parameters of Middleton's CliiSS A density by using on the order of 100 samples, whereas the other methods may require more than a tenfold increase in the number of points to achieve the same accuracy. Thus the EM algorithm is used here to estimate the parameters of two- and three-state mixture densities.

The EM algonthm is an indirect method of finding the maxima of the loglikelihood function [lo], and its application to Gaussian mixture probability models is described. The Gaussian mixture model is a multistate model, but the observer does not know wlhich state is operative at a given time. Thus


we can define the incomplete and the complete log-likelihood functions. The incomplete log-likelihood function is

n

2=1

The comple5e log-likelihood function+CL is a function of the parameters 6' and the vector of states $ = ($1, . . . , $n), where & is an integer between 1 and m denoting the operative state for sample i

n

C W , dI $1 = In (P(% $a I s',). (15) a=1

Given data vector z' and a current estimate of the parameters e'c'),, the conditional probabilities of the states are defined by

These conditional probabilities are then used to define the expected value of the complete log-likelihood function over t_he space of states, which is then a function of the parameter 0.

&(GI @'I) = E(CL(x, ~ I G) I z , 8'))

(17)

The EM algorithm selects the_value of the parameter e' that maximizes (17) f:r the given O('). This maximizing parameter value is denoted O('+'), and the process isjterated. +Thus from a given starting point G(O), a sequence, ~ ( o ) , . . . , @), . . . , is defined by choosing e'c'++') to be a value that maximizes &(e( 8')).

Wu [41] shows that under certain conditions the sequence {e'c')} converges to a saddle point or a local maximum of the incomplete log-likelihood function, L(e'). When these conditions occur, the maximum of L($) can be found by starting EM searches at a number of points and choosing the limit point of the EM searches that has the largest value of the incomplete log-likelihood function.

If the data have a Gaussian mixture probability !ens$ with a finite number of states, then the maximum over 0 = (pk , a,") of (17) can be obtained in closed form as

and

Furthermore, Wu's Theorem 2 [4 11 applies to Gaussian mixture densities, and it implies that a limit point of the EM algorithm is a local maximum or a saddle point of the incomplete log-likelihood equation (14).

I 0 200 400 600 800 1000

0.01 ' Sample size

(a)

0.0551

200 400 600 800 1 0.02;

Sample size

(b)

30

Fig. I . The mean-square error of the estimate of p l (a) and U: (b) as a function of sample size, with p l = 0.4 and = 6.25.

We have evaluated the mean-square error of EM estimates of the parameters using computer simulation of circular Gaussian mixture noise. The parameters of the two-state mixture model, m = 2 5 = 1,2, are determined by the probability of the low- variance state p l , the ratio of the high variance to the low variance (ag/a?), and the average noise variance.

The complex data are normalized so that the mean of the norm squared is 2. This reduces the problem to fitting p l and a?, as the following relations hold for the normalized data:

P I + p z = 1 and p l a t + ~ 2 0 2 1. (20)

For each value o f p l E (0.01, 0.05, 0.1, 0.2, 0.4, 0.8, 0.9, 0.95, 0.99}, az/o? E {1.6, 6.25, 25, loo}, and n E (16, 31, 62, 125, 250, 500, 1000) 100 sets of data are generated, and the mean-square error in p l and a? is calculated.

Fig. l(a) and (b) shows the mean-square error in p1 and 012, respectively, as a function of sample size. For this figure p1 = 0.4 and az/af = 6.25. Fig. 2(a) and (b) shows the mean-square error in the estimated parameters as a function of p l for az/o? = 6.25 and n = 125. The mean-square error increases rapidly for p l < 0.2 or p1 2 0.8. This is expected since as the probability of one of the states diminishes, the expected number of data points from that state diminishes in data sets of a fixed size.

Fig. 3(a) and (b) shows the mean-square error in the estimated parameters as a function of a:/a?, for p1 = 0.4 and

STEIN: DETECTION OF RANDOM SIGNALS IN GAUSSIAN MIXTURE NOISE

0.12-

0.1 * 4 E 0.08 2 &

0.06

6 0.04

0.02

m = U

2

1193

'

0.5

ll

0 0.2 0.4 0.6 0.8 1

P i

(a)

0.351 I

Fig. 2. The mean-square error of the estimate of p l (a) and 0: (b) as a function of p l , with uz/o: = 6.25 and n = 125.

Fig. 3. The mean-square error of the estimate of p l (a) and uf (b) as a function of u;/u? with p l = 0.4 and n = 125.

n = 125. The mean-square error diminishes up to a certain point as the ratio of o$/of increases for fixed p l and n. Again, this is expected: as this ratio increases, the conditional probabilities of the states given the data will tend toward the extreme values of zero and one.

Fig. 4 shows the cumulative distributions of the number of iterations required for the EM algorithm to converge. The solid curve (-) is the cumulative distribution for starting points and samples with n = 125, p1 = 0.4, and oz/u: = 6.25. The dashed curve (- -) is the cumulative distribution for starting points and samples with n = 125 and the oi.her parameters varying over their range, and the dash-dot curve (-.-) is the cumulative distribution using all starting points for all the samples and parameters. For example, the median number of iterations required for the EM algorithm to converge is approximately 12, and 90% of the time fewer than 50 iterations are required. The close correspondence of these distributions suggests that the number of iterations required For convergence is for the range of parameters evaluated relatively independent of the parameter values.

Effective initialization schemes are necessary to use the EM algorithm to estimate the parameters of Gaussian mixture densities having more than two states. The following procedure has been used to fit three-state Gaussian mixture densities to sets of ocean acoustic data. Let yk and pJl , denote the variance and probability, respectively, of the kth state for a

Number of iterations

Fig. 4. Cumulative distributions of the number of iterations required for the EM algorithm to comerge using all samples (- . -), all samples with n = 125 (- -), and all samples with n = 125, p l = 0.4, and uz/uf = 6.25 (-).

j-state mixture model, with q k < 4, if k < C. A data set 2 = ( z 1 . . + . z,} is normalized as described above, and the parameters, p121 and V z l , of the two-state mixture density are estimated. p22 and V 2 2 are then calculated from (20). For

1794 IEEE TRANSACTIONS ON EVJ?ORMATION THEORY, VOL 41, NO 6, NOVEMBER 1995

select p E [O, 13 and V E [VZl, V22] a partition of 2 is defined as follows. Let 7x2 be p . n rounded to the nearest integer. Arrange the elements of 2 in ascending order using the relation z1 51 22, which is defined by z1 < I zz if 10.51z112 - VI 5 10.5 /~21~ - VI, and let 22 be the n2 smallest elements of the Sl-ordered set. Let LB2 and UB2 be the greatest lower and least upper bounds of the norm squared of

and Z3 = (z E 2 : 1zI2 2 UB2). Then {21,22,23} is a

The characteristic function for n > 1 is the nth power of (25), which for m = 2 is

@G (w> e-zwkcll e-zw(n-k)czl

= 2 (z)pf(l -pl)n-k (1 - 2iWC12)k (1 - 2iwcz2)n-k ' k=O

the elements of 22, and define 21 = { z E 2 : 1 . ~ 1 ~ < LB2) (26)

partition of 2. For a finite set S, let n(S) be the number of ne density of p g for arbitrary (26), i.e.

is the Fourier of elements in S, and definep!& = n(Z,) /n(Z) and, if n(Zz) f 0

V$ = 0.5. (l/n(Zz)) 1 ~ 1 ~ . X € Z *

EM searches for the best fitting three-state exponential mixture k=O density are initialized with {&, V . : 1 5 i 5 3 ) . e--iwkCll -zw(n-k)czl --zwt

dw. (27) . s, (1 - 2fwcg)k(l - 2iw:22)'"-k IV. PERFORMANCE ANALYSIS

Probability distributions of the power detector 8, approximate formulas for the moments of M , and approximate distributions of M are derived below assuming Gaussian mixture noise and Gaussian signals. These formulas are used in the next section to compare the performance of M and 8.

Assume that the complex-valued noise z has a Gaussian

This integral is calculated by summing the residues at the poles, - i / (2cj2) [9]. Define 01 2 2 q 2 , ,O E 2cZ2, and, for 0 5 k 5 n, Tk(t) t + kcIl + (n - k ) c z l . Then

TO (4 2c22

" To(t)"-' exp ( - -) 6(t) = 2nr(n)c;2 mixture density given by (3). Then, if U = 1 1 ~ 1 1 ~ , U has an exponential mixture density [40] viz.

otherwise.

The density of in Gaussian mixture noise may be obtained as follows. For 1 5 j 5 m define c31 = 2/a2 and cj2 = (a: + 04)/04 (the noise-only density is calculated with c ~ f = 0). Presently cJ1 = Ckl, however, expressing the densities in terms of cZg is convenient for one of the approximations of (13) defined below. Let 4 I j be the random variable 8 assuming that state j occurs. The probability density of 4 is

m

and for n = 1

otherwise.

The density of 8 for arbitrary n may be calculated by using characteristic functions. The characteristic function of the exponential density (21) is [40]

-1 ZW2G - 1. @(w) = .

Thus the characteristic function of (22) is

The moments of 4 for arbitrary m may be calculated from (7) or from derivatives of (25) at 0. Using (7) we obtain

STEIN: DETECTION OF RANDOM SIGNALS IN GAUSSIAN MIXTURE NOISE 1795

Two approximations of the mixture detector ,U are devel- oped and used to construct approximate densities of M in GM noise. The first approach assumes that the detector knows

and assuming m = 2 and n = 1, we get

~ ( d 2 I j ) = - 1- 3 4W:j 8 ~ 1 . ~ 2 , dw;, + - + 2 ( 4 + 0:) which state occurs at each observation. Define the weights 8: I?;&; 8;

( W k 3 ) such that 0 5 Wk, 5 1 and

m

C W k 3 = 1. 2 2 [z 2 w a 3 ~ 2 3 4 3 1

k = l + 8 [ ( a , ” + ~ , ) ] ‘‘+r+- . 1 2 U2

(34) If state j occurs, the “clairvoyant” detector is

m Alternatively, &f’ may be approximated by a piecewise-

polynomial function obtained from piecewise-polynomial approximations of the: state weighting functions

(30) c^(u) = (2 + $) Wk3. k = l

C differs from M in that the weighting functions used to define C, but not M , are constant for each state. C could not be realized because the state from which each datum arises is generally unknown; however, assuming that the state is known, one might choose Wk, = 1 if k = j and 0, otherwise.

The density of c^ in GMN has the same form as the density of 9. Let m = 2, denote 2011 by w1, and 21122 by w 2 , and define

c11 (2w1)/6? + 2(1- w1)/6; c 1 2 = (a? + a:) [ W l / @ ; ) + (1 - w1)/(6$)] c 2 1 = (1 - w 2 ) ( 2 / 6 3 + w 2 ( 2 / 6 3

c 2 2 = (a; + a:) [(I - w 2 ) / @ 3 + w 2 / ( & 3 ] . (31)

Then, conditioned on the occurrence of state j

E A u - B .

Thus

G,(u) == p(j I U, 6, U,”), 1 5 j 5 m.

A procedure for obtaining the piecewise-polynomial approximation of Gji is described. For an m-state mixture limu+m&m(u) = 1. Select K so that if U 2 K then Gm(u) M 1. Select the degree d and number p of the polynomials to be used in the approximation. Define a partition (0 = u0 < . . . < U,, = K } of [0, K ] by U, = ( r .K) / (d .p ) for 0 5 T 5 p - d . Then for each 1 5 j 5 m-1 and each 1 5 f2 5 p let Pej be the Lagrange interpolating polynomial [37] of degree d defined by [ u ( e - l ) d , . . . , u e d ] and [ G j ( y e - l ) d , . . Gj(Ued>1. Other conditions, such as continuity of the derivatives up to certain orders at the points (U,) or vanishing of certain derivatives at 0 and K , may also be imposed [37]. Also, define

0, i f O < j < m P p + I j ( U ) = { 1, otherwise.

For a set A, let ;CA be the characteristic function of A, i.e., XA(U) = 1 if U E A and 0, otherwise. Define the piecewise- polynomial approximation of the weighting functions by

t + B P

PGj(’LL) = x [ . ( e - l ) d , U e d ) ( U ) P e 3 ( u ) e= i

(32) + X I U p d , ..)Pp+l, (U) (35) otherwise.

and the approxima,te detector by With these definitions of tag, the conditional densities of C are m given by (22), and thus the density for n = 1, characteristic function for n = 1, characteristic function for arbitrary n, and density for arbitrary n are given by (23), (25), 1(26), and (28), respectively.

The conditional moments of c ̂ are also readily calculated in as much as Ck is a polynomial in U. The moments of C are then calculated from

(36)

Since PM is piiecewise-polynomial, its moments are easily calculated for Gaussian mixture noise and a Gaussian Signal having c:. Other signal models require appropriate modifications. For 1 5 l! 5 p + 1 define

m

j = 1

For example and define U ( ~ + ~ ) , I 00. Then

e= I

1196 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO 6, NOVEMBER 1995

and

Since &e is a truncated polynomial, the integrals in (37) may be evaluated using the incomplete Gamma function.

For m = 2, piecewise-linear approximations of ?&(U) may be defined by choosing (1 = CO L. c1 2 . . . L c, 2 0), g = wl(0)(c~,...,cT), ana (uo, . . . ,uT) so that w1(u,) = 3, for all 0 5 i 5 r.

Lz(u) = y,-1 + (U - uz-1) x (Yz - Yz-l)/(% - %I).

The x-intercept of Lr is

K = - yr x [ (UT - ~T-I)/(YT - 97--1)]

and the piecewise-linear approximation of 61, studied below, is

If r = 2, then the piecewise-linear approximation is determined by &(O) and the slope of L1, denoted by A and B respectively, and it is defined by

The approximate mixture detection statistic using L(w) with r = 2 in place of w is

Define

Then A

u 5 - B k 2 U 2 + k1u - ko,

6-2" c72 (41) -2 U

-+q, otherwise.

Expressions (41) and (42) are equivalent in the sense that one is a monotone transformation of the other [3 11. Equation (42) is more convenient because its range is [0, CO). The densities of (42) conditioned on noise only 2r signal plus noise are derived below, and the densities of M , are translations of the corresponding densities of (42).

The probability distribution of t = L M ( u ) is obtained by inverting (41). Define

LMYl( t ) 8;(t - ko + 2/8-22).

Pu(fi;l(t))(fi;l)'(t) t E LO, CPl)

Then, letting p , be the exponential mixture density with m = 2, the density function o f t is approximately

p t ( t ) = c p u ( M ; l ( t ) ) ( M ; l ) ' ( t ) t E [CPl, CP2)

;:;A& (My1 y ( t ) t E [C%,m). (43)

V. NUMERICAL RESULTS The piecewise-linear and piecewise-polynomial approxima-

tions of the state 1 weighting function of the two-state mixture detector and the resulting approximation of the detection statistic are next calculated for particular signal and noise parameters. The probability density of the detection algorithm is then calculated by using a two-term piecewise-linear approximation of the weighting function, and the approach of a sum of independent random variables having this density to a normally distributed random variable is studied. The first two moments of the various approximation schemes, C, P M , and LM are compared. Receiver operating characteristics (ROC) arc calculated for these approximate detectors and for 6, and they are compared with ROC'S obtained from Monte Carlo simulation of &l and 4. The performance of the power and mixture detectors over a wide range of parameters and the effect of parameter error on the mixture detectors are then studied. Finally, the importance of the signal correction term c7," in (9) is demonstrated.

The two-segment piecewise-linear approximations (c2 = 0.5 noise-only, and c2 = 0.3 for signal-plus-noise) and the three-segment piecewise-cubic polynomial approximations ( K is the solution of wl(K) = O.OOlwl(0)) of the noise-only and signal-plus-noise state 1 weighting functions are calculated

i3


noise only

-I

1797

25

20 0 = U) ii

U) m '0 15-

2 .- U

m U

10-

' 0. i---

-

-

signal plus noise

- 0 1 2 3 4 5 6 7

-0.1 ' intensity

Fig. 5. 201 (-) is approximated by P(w1) (0) and L(wl ( ) The noise and signal parameters are p l = 0.5, U: = 0.182, U$ i - i .818 , and U," = 0.182.

using the following parameters: 0: = 0.182, p1 = 0.5, CT; = 1.818, and 0," = 0.182. These parameter values are fairly close to parameters estimated from acoustic (data [35], [36], and the resulting density differs significantly from a normal density. The weighting functions and the approximations are compared in Fig. 5. The corresponding piecewise-quadratic and piecewise-quartic detection statistics are also calculated, and they are compared with the mixture detection statistic in Fig. 6. Evidently, using three polynomials of degree three gives an excellent approximation to the mixture weighting function and detection statistic.

The two-term piecewise-linear approximation is used to calculate an approximate density function (43). The resulting noise-only and signal-plus-noise densities are plotted in Fig. 7. This figure compares these densities to normal densities of the

1.4-

1.2-

2 'E 1 - C

U

- .20.8- 3 n E 0.6 -

0.41

0.2 I h - *._._ - - , . . . . . . . . . . . , . ..... .... 1 .-:.-.-.-. ,r.:../.:.:I.::L: -;.&

0 5 10 15 20 25 30 35 Mixture detector output

Fig. 7. Noise-only (--) and signal-plus-noise (--) densities of LM are compared with normal densities with the same mean and variance (. . .).

1 0.025

.- E 0.021 -I

Mixture detector output

Fig. 8. The noise-onlly (-) and signal-plus-noise (--) densities of a sum of eight independent random variables each having the density shown in Fig. 7 are compared with noimal densities with the same mean and variance (. . .).

same mean and variance, and the L2 norm of the difference is 0.199 and 0.1997 for noise only and signal plus noise, respectively.

The convergenlce of the sum of independent random variables having density given by (43) is studied by numerically convolving the density with the above parameters for n = 1. Newton-Cotes integration with polynomials of degree 7 and 8 is used to calculate the convolutions [37]. The singularities of the density require careful treatment and necessitate using a fine grid size-0.001 is used in this study over much of the domain. Performance comparisons using convolved densities are computationally expensive for large n. The discontinuity of the density could be removed by transforming the random variable, but the transformed random variable may have a discontinuous derivative, and thus the problem may be equally difficult. Fig. 8 shows the resulting densities for a sum of


~~

Detector

TABLE I THE MEAN AND STANDARD DEVIATION OF VARIOUS DETECTION STATISTICS OBTAINED FROM SIMULATION (SJM) AND THEORY FOR VARIOUS SIGNAL LEVELS USING PARAMETERS m 2,

R = 65, pi = 0.5, f12 - 0.182, AND fl: = 1.818. 1 - (The standard deviations of the estimated means

and standard deviations are given in parentheses.) ~

Signal Variance 0 045 091 182 364

Moment

G (sim)

G (theory)

mean .3 (.2) 6.2 (.2) 12.5 (.2) 25.1 ( 2 ) 49.8 (.3) std 17.8 (4.4) 19.0 (5.0) 20.1 (5.6) 22.4 (7.0) 28 (11)

mean 0 5.9 11.8 23.6 47.2 std 24.7 25.1 25.6 26.7 28.8

M (sim) LM

mean .3 (.3) 101 (.7) 202 (2) 404 (4) 812 (8) std 31 (13) 69 (66) 114 (181) 211 (622) 444 (2755)

mean -7.7 63.6 143.6 306.8. 646.1 (theory) std

eight independent random variables each having the noise- only or signal-plus-noise densities shown in Fig. 7. As seen in this figure, the noise-only and signal-plus-noise densities of the sum are nearly normal-the L2 norms of the differences between the corresponding normal densities and the noise-only and signal-plus-noise densities are 0.0017 and 0.0024, respectively. For small n, depending upon available computational resources, these densities may be used to compare detector performance, and for large n the densities of the detection statistics may be effectively normal. Alternatively (see [I] and the references therein), one may use other approximations of the density functions obtained from the moments.

A Monte Carlo study of the performance of G and M is conducted to evaluate these detectors. For parameter values p l = 0.5, ag/aT = 10, and a,”/a; = 2, I, 0.5, and 0.25, 10 240 independent sets of 130 independent noise-only samples are generated. For each set of 130 samples, the noise parameters are estimated using the EM algorithm, and the sample variance of each set is calculated. The values of 5; and fi are calculated at each of the first 65 samples of each set. For each of the 10240 sets, these 65 values are summed to form 10240 values of these detection statistics under the noise-only hypothesis. Thus the parameters, pk, a:, and a:, are estimated on sets of 130 samples but only the first 65 samples are used to calculate the detection statistic. For each set of parameters, another 10 240 independent sets of 130 independent noise-only samples are generated, the noise parameters are estimated using the EM algorithm, and the sample variance is calculated. For each signal level, 10240 independent sets of 65 independent signal samples (having a spherical Gaussian distribution) are generated and added to the corresponding noise samples in order to form 10240 sets of 65 signal-plus-noise samples. Thus a: is estimated using a window of 130 data points, even though the signal is only present for 65 points. Thus a certain amount, of error is present in a,” which is an estimate of a:. G and M are calculated to obtain 10240 output values of each of the detection statistics under the signal-plus-noise hypothesis. ROC curves are then generated.

41.4 45.5 51.4 63.9 90.1

10“ 1 os 1 o4 l o 2 1 oo False alarm> probability

Fig. 9. Simulation generated ROC of 0 ( E ) and M (0 are compared with the theoretical ROC of B (F), PM (A), LM(B), and C (0) The parameters used are p l = 0.5, a: = 0.182, U; = 1.818, 02 = 0 182, m = 2, and R = 65.

The first two moments of M and G obtained from this simulation are compared with calculated values of the first two moments of the detectors C, P M , L M , and G in Table I. The close approximation of the piecewise-quartic detector, defined using three polynomials of degree three to approximate the state 1 weighting function of M , suggests that the moments calculated using this approximation are very close to the moments of M. As seen from Table I, the standard deviations of LM are close to those of P M for all of the signal levels; however, LM underestimates the mean of M . The clairvoyant detector gives good estimates of the means of M for all signal levels, but it overestimates the standard deviations. The theoretical means of 5; are slightly below the simulation means. However, the theoretical standard deviations of 4 are within two standard deviations of the simulation standard deviations for all signal levels.

Fig. 9 presents seceivcr operating characteristic (ROC) for 6 (e, PM(A), LM (B), and C (D), that are calculated using the assumption that the detection statistics have normal densities and the moments shown in Table I for signal levels os = 0, 0.182. In this figure these curves are compared with ROC of B Q and M (C) which are calculated from the simulation data. The ROC calculated using either the piecewise-linear or piecewise-polynomial approximations of the weighting _function provided an upper bound on the performance of M , and the performance of C is somewhat below that of &f despite the degradation in the performance of J\;z due to parameter error. The deviation between the simulation and theoretical performance curves for the power detector may be due to the variability of the estimated curve.

The performance of 4 and M is also compared over a range of values of p1 and a,”/af. Figs. 10 and 11 are the resulting probabilities of detection of 4 and M , respectively, at a probability of false alarm of 0.001, for n = 65, a,“ = 0.182, and various values of p1 and .,”/at. The overall noise variance is equal to 1 for all pairs (p1,a,”/$). Fig. 12 shows the

PM (theory)

mean -.4 89.9 180.3 360.8 721.8 std 41.8 47.3 53.1 64.8 88.1

C (theory)

mean 0 90.3 180.5 361.1 722.2 std 63.0 79.4 96.8 133.1 207.7


0.08, , I , , , , ,

0.07

0.06

5 0.05

0.04 .s 2 0.03 2

0.02 a

0.01

0 z

V

-

I -

-

-

-

-

-

-

d 0:l 0:2 0:3 0:4 0:5 0:6 017 10.6 0.9 1

P l

Fig. 10. The probability of detection of Q at PFA = 0.001 for p i ranging from 0.05 to 0.95, cr;/u: ranging from 2 to 20 (pluf + p2uz is held constant at l), n = 65, and U: = 0.182.

I--L---J L ,

0:l 0:2 0:3 0:4 0:5 0:6 0:7 0.8 0.9 1

P l

Fig. 11. The probability of detection of M using the same parameter values as in Fig. 10.

4

P u r

\ U 16-

12

10

2 .--%&16 0.1 0.2 0.3 '0.4 0.5 0.6 0.7 0.8 0.9

P I

Fig. 12. The signal gain (dB) of &t over 6 using the same parameter values as in Fig. 10.

theoretical signal gain (dB) from using the mixture detector rather than the power detector for the parameters studied in Figs. 10 and 11. If, for a fixed false-alarm probability, is the signal level which 8 requires to achieve the same probability of detection as M achieves for a signal level a:, then the signal gain of M relative to 8 is sg (dB) = 10log,o ( ~ ; / C T ? ) . Evidently, for these parameters M offers significant improvement over 8.

1799

Relative error in a:

Fig. 13. Contours of the probability of detection of M at PFA = 0.001, pi = 0.5, u?/u; = 10, n = 65, (T: = 0.182, ((T: - u ~ ) / u ~ = -0.2, for various relative errors i n p1 and uf-plu? + pzuz is held constant at 1.

I I

I I

I I

I

I

-15 -10 -5 0 5 10 15

io*log( a: /a: )

Fig. 14. The probability of detection of M with p1 = 0.5, u$/crf = 10, n = 65, and cr: = 0 (-) or U; = a: (--).

The effect of parameter estimation error on the performance of M is also assessed using the assumption that the detector outputs have Gaussian distributions. Fig. 13 is a contour plot of the probability of detection as a function of relative error in p l and 0; for a fixed relative deviation in cr," - 0,". This plot indicates thal for the parameters of the calculation M is rather insensitive to parameter errors of a magnitude that are achieved by the ]EM algorithm.

The significance of the 0," term in (8) is demonstrated in Fig. 14 which shows the probability of detection versus signal level for JMo (-), calculated using g," = 0, and M I (---), calculated using g," = 0: when processing signal- plus-noise data and .," = 0 when processing noise-only data. To compute these curves the parameters are set at cr; = 1, 02 = 10, p l = 0.5, n = 60, PFA = 0.0001, and various values of C:/CT~. The first two moments of M are calculated from (37). For these parameters, Fig. 14 indicates that M1 performs significantly better than M O . However, in Section 11, above, these detection statistics are shown to be equivalent for sufficiently large n and small a:.

VI. CONCLUSIONS The locally optimal detection algorithm (3 ) for random sig-

nals in correlated noise is derived and applied to independent circular Gaussian mixture noise. The algorithm is modified

1800 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 41, NO 6, NOVEMBER 1995

so that it can detect signals that are not “vanishingly small.” To use the algorithm, parameters of the probability density function of the noise must be estimated, and a method for doing this by using the EM algorithm is described. The error in the parameter estimates is shown to depend upon their values and the number of available samples. Probability density functions of the power detector (7) for Gaussian mixture noise and independent Gaussian signals are derived. The moments of the mixture detector (9) are very accurately approximated by using piecewise-polynomial approximations of the state-weighting functions, and piecewise-linear approximations of the state- weighting functions are used to obtain approximations of the density functions of the mixture detector.

These approximation techniques are next compared with the “clairvoyant” mixture detector (30). The clairvoyant detector is shown to overestimate the variance of the mixture detector, and thus to predict poorer performance than is predicted by either the piecewise-polynomial or piecewise-linear approximation techniques.

These approximate performance results are then compared with Monte Carlo simulation of the mixture detector (9). For certain parameter values, the performance of this algorithm, as implemented by using the EM algorithm, is shown to be very close to the performance predicted by the piecewise- polynomial and piecewise-linear approximation techniques and to exceed the performance predicted by the “clairvoyant” approximation.

The theoretical performance of the power and mixture detectors in the absence of parameter estimation error is then compared over a wide range of noise parameters, and the mixture detector is seen to offer signal gains of 15-37 dB. For certain noise parameter values, the mixture detector is shown to be fairly insensitive to parameter estimation errors.

The performance of the mixture detector (9) with and without the correction for larger signals (8) is then evaluated. This correction is shown to significantly improve the detection performance over a range of signal values even though the two algorithms have equivalent performance as the number of samples goes to infinity and the signal strength goes to zero. Future studies may compare (9) with the likelihood ratio for various signal pdf‘s, and analyze LOBD detection algorithms for broader classes of signals and noise.

ACKNOWLEDGMENT The author wishes to thank Dr. C. Baker, Dr. J. Bond, Dr.

J. Zeidler, and Dr. G. Dillard for helpful discussions, and Dr. Dillard for reviewing the manuscript.

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detection of random signals in gaussian mixture noise

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