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Sampling design for weak signal detection in SIRP noise
ARTICLE IN PRESS
Signal Processing 85 (2005) 205–214 www.elsevier.com/locate/sigpro
Sampling design for weak signal detection in SIRP noise$ Chao-Tang Yu Department of Electronic Engineering, Southern Taiwan University of Technology, No. 1, Nan-Tai Street, Yung-Kang City, Tainan, Taiwan Received 28 February 2002; received in revised form 27 September 2004
Abstract In this paper, we present a sampling design for weak signal detection in additive spherically invariant random process (SIRP) noise. Our approach for the design of sampling schemes is based on the class of Ali–Silvey distance measures. A design objective function for this problem is developed. This objective function is maximized to obtain the sampling scheme for the detection problem under consideration. Two examples are presented to illustrate our sampling design. Numerical results show that our sampling design outperforms the uniform sampling. It is also numerically shown that the sampling design is insensitive to the statistics (the characteristic probability density function) of the SIRP noise samples. r 2004 Elsevier B.V. All rights reserved. Keywords: Sampling design; SIRP; Detection
1. Introduction Sampling design for the Gaussian detection problem has been studied in [2,4,22], where the received signals under hypotheses H 1 and H 0 were Gaussian random processes. Both random and deterministic sampling schemes for the detection of known signals in Gaussian noise were considered in [4]. Using the decision statistic based on the $ Research sponsored by National Science Council, Taiwan, R.O.C., under Grant No. NSC 88-2213-E-218-010. Tel.: +886 6 2533131x3137; fax: +886 6 2426911, +886 6 2537461. E-mail addresses: [email protected], [email protected] (C.-T. Yu).
sampled data, instead of the continuous-time detector (matched filter or correlation receiver), naturally results in a degradation of performance. Deterministic and random sampling schemes were designed so as to minimize the degradation in the performance of the detector. In [2], an asymptotically optimal periodic sampling design for the Gaussian hypothesis testing problem was presented. Under very general conditions, the probabilities of error decrease exponentially with increasing sample size. For this problem, an asymptotically optimal periodic sampling scheme was designed by maximizing the rate of exponential decrease. In [22], a universal approach based on the Ali–Silvey distance measure (ASDM) for
0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.09.012
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the design of sampling schemes for Gaussian hypothesis testing problems was presented. In the strong signal case, sampling points were determined that maximize the distance measures between the conditional densities. In the weak signal case, a new member of the class of ASDMs suitable for the weak Gaussian signal detection problem was developed. The sampling scheme was designed by maximizing the distance measure. In the weak signal detection problem, the assumption of Gaussian noise is not suitable for many situations. Here, we are interested in sampling design for weak signal detection in nonGaussian noise characterized as a spherically invariant random process (SIRP). When we sample a SIRP, the probability density function (PDF) of the sampled data vector is uniquely determined by the specification of a mean vector, a covariance matrix and a characteristic first order PDF. SIRP has been used for clutter modelling and simulation [9]. Through the theory of SIRPs, a library of multivariate correlated non-Gaussian PDFs for characterizing various scenarios was developed in [17]. In [5], a canonical form for the locally optimum detector (LOD) was established. The performance of LODs in radar weak signal detection for finite sample sizes was also determined, where the radar disturbance was modelled as correlated SIRPs. Some other applications for the theory of SIRPs can be found in the areas of signal (or radar) detection and estimation problems [7,8,10,18,19,21] and speech signal processing [3,15]. Here, we consider the sampling design for weak signal non-Gaussian detection H 1 : rðtÞ ¼ ysðtÞ þ nðtÞ; aptpb; H 0 : rðtÞ ¼ nðtÞ; aptpb;
ð1Þ
where rðtÞ is the observation over the interval ½a; b; nðtÞ is a non-Gaussian random process modelled by a SIRP with zero mean and autocorrelation function Rn ðt1 ; t2 Þ; and the signal sðtÞ could be a deterministic signal or a random process with mean-value function ms ðtÞ and autocorrelation function Rs ðt1 ; t2 Þ: The signal sðtÞ is assumed to be independent of the noise nðtÞ: y is a parameter with small positive real value so that
ys(t) represents the weak signal as y approaches zero. Without loss of generality, the average power of sðtÞ and that of nðtÞ are normalized to unity so that y can be used to control the signal to noise ratio. Since the non-Gaussian noise is modelled by a SIRP, an analytical expression for the joint PDFs of N random variables obtained by sampling the noise process is available via the theory of SIRP. In Section 2, we present a brief review of the class of ASDMs and the theory of SIRPs. In Section 3, we present the sampling design procedure for the problem under consideration. In Section 4, we present two numerical examples to illustrate our sampling schemes. Concluding remarks are made in the last section.
2. Preliminaries We briefly review the class of ASDMs and some relevant mathematical preliminaries for real SIRPs as follows: 2.1. The class of Ali–Silvey distance measures The class of ASDMs [1] is defined by the general expression Dðp0 ; p1 Þ ¼ f ðE 0 ½CðLÞÞ;
(2)
where p0 ; p1 are the two probability distributions defined on the same space; Dðp0 ; p1 Þ is the Ali–Silvey distance between p0 and p1 ; f is an increasing function; E 0 denotes expectation with respect to p0 ; C is a convex function; and L is the likelihood ratio. Some examples of this class of distance measures are (1) I-divergence [14], I ¼ E 0 ½ lnðLÞ; (2) J-divergence [14], J ¼ E 0 ½ðL 1Þ lnðLÞ; pffiffiffiffi (3). Bhattacharyya distance [12], B ¼ lnðE 0 ½ LÞ; (4) Chernoff distance, [6], C h ¼ lnðE 0 ½La Þ; where 0 pap1: 2.2. Specification of the SIRP Spherically invariant random vector (SIRV) and SIRP are specified as follows [9,17]:
A SIRV is a random vector whose PDF is uniquely determined by the specification of a
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mean vector, a covariance matrix and a characteristic first order PDF. The PDF of a random vector is defined to be the joint PDF of the components of the random vector. A SIRP is a random process such that every random vector obtained by sampling this process is a SIRV.
Let Z ¼ ½Z1 ; Z 2 ; . . . ; Z N T ; where T denotes the matrix transpose, be a real zero mean Gaussian random vector with covariance matrix Kn : Let W be a non-negative random variable with PDF f W ðwÞ: W is assumed independent of Z: A SIRV can be obtained by the product X ¼ ZW : A representation theorem is used to construct the PDF of a SIRV [13,20,21], and the PDF of the product X is given by f x ðxÞ ¼
1=2
ð2pÞ jKn j Z 1 xT K1 N n x w exp f W ðwÞ dw: 2w2 0 ð3Þ
If we let q ¼ xT K1 n x; then the PDF can be rewritten as f x ðxÞ ¼
1 ð2pÞ
N=2
jKn j1=2
hN ðqÞ;
(4)
where Z hN ðqÞ ¼ 0
1
based on these discrete-time data. Let r; n and ys represent the sampled observation, noise and signal data vectors, respectively. Also, let pðrjH 1 Þ; pðrjH 0 Þ; and f N ðnÞ denote the joint PDFs of r under H 1 ; of r under H 0 and of noise, respectively. We note the ASDM between the conditional densities goes to zero as signal becomes very weak (as y approaches zero). Therefore, in this case, the sampling design criterion based on the class of ASDMs between pðrjH 1 Þ and pðrjH 0 Þ is not directly suitable. Instead of maximizing the distance measures DðpðrjH 1 Þ; pðrjH 0 ÞÞ; we maximize the first nonzero term of the series expansion of DðpðrjH 1 Þ; pðrjH 0 ÞÞ to design the sampling schemes. The objective function (the first non-zero term of the series expansion) can be expressed in the form [22, p. 2330]: DYV ¼ ci E 0 ½ðLLOD Þ2 ¼ f ðE 0 ½CðLLOD ÞÞ;
1 N=2
207
q wN exp 2 f W ðwÞ dw: 2w
(6)
where ci is a constant that depends on the signalto-noise ratio, y2 ; and the functions f and C defined in the class of ASDMs, LLOD represents the LOD statistic, and the functions are defined as f ðxÞ ¼ ci x and CðxÞ ¼ x2 : Because f is an increasing function, and C is a convex function, this objective function can be interpreted as a member of the class of ASDMs. The expressions of DYV can be evaluated completely if the LOD statistic is determined. Here, we find the LOD statistics for deterministic signal and random signal cases separately in the following.
(5)
The PDF of the random variable W is called the characteristic PDF of the SIRV. It is interesting to note that when f W ðwÞ ¼ dðw 1Þ; where dð Þ is the Dirac delta function, the resulting hN ðqÞ is equal to expðq2Þ and the PDF of X is the well-known Gaussian PDF.
3. Sampling design development In this section, we develop sampling schemes for the weak signal detection problem stated in Eq. (1). The received continuous-time signal, rðtÞ; is sampled by means of a sampling scheme. Sampled data are fed to the LOD that makes a decision
3.1. Deterministic signal in additive SIRP noise In this case, the signal is deterministic and the noise is a SIRP with zero mean and known autocorrelation function, Rn ðt1 ; t2 Þ; where the noise average power, Rn ðt; tÞ; is set to unity. Because the noise is modelled as a SIRP, the N observation samples form a SIRV. Thus, the joint PDFs of the sampled data, r ¼ ½rðt1 Þ; rðt2 Þ; . . . ; rðtN ÞT ; under hypotheses H i ; i ¼ 0; 1; are given by pðrjH i Þ ¼
1 ð2pÞ
N=2
jKn j1=2
hN ðqi Þ ¼ KhN ðqi Þ;
(7)
where the quadratic forms, q0 and q1 ; are equal to T 1 rT K1 n r and ðr ysÞ Kn ðr ysÞ; respectively; Kn is the covariance matrix of r; and it depends on the
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autocorrelation function and sampling points, t1 ; t2 ; . . . ; tN ; and hN ð Þ is as defined in Eq. (5). For this known signal case, the LOD statistic denoted by LdLOD is given by @L hNþ2 ðq0 Þ d LLOD ¼ : (8) ¼ sT K1 n r @y y¼0 hN ðq0 Þ
3.2. Random signal in additive SIRP noise In this case, sðtÞ is a random signal with meanvalue function, sm ðtÞ; and autocorrelation function, Rs ðt1 ; t2 Þ; the noise nðtÞ is modelled as a SIRP with zero mean and autocorrelation function, Rn ðt1 ; t2 Þ; and a characteristic PDF, f W ðwÞ: Again, Rs ðt; tÞ and Rn ðt; tÞ are set to 1. Then, pðrjH 0 Þ is given by Eq. (7) with i ¼ 0: Let the PDF of the N samples of sðtÞ; s ¼ ½sðt1 Þ; sðt2 Þ; . . . ; sðtN ÞT ; be denoted as f S ðsÞ: The noise is assumed to be independent of the signal. Thus, pðrjH 1 Þ can be written as Z pðrjH 1 Þ ¼ f N ðr ysÞf S ðsÞ ds s
¼ E s ½f N ðr ysÞ ¼ E s ½KhN ðq1 Þ;
ð9Þ
where f N ðrÞ ¼ f N ðrjH 0 Þ ¼ pðrjH 0 Þ; and E s denotes expectation with respect to f S ðsÞ: When the mean vector, sm ; is non-zero, the LOD statistic is essentially the same as in Eq. (8) where s is replaced with sm : In the case when sm ¼ 0 is considered, the LOD statistic, LrLOD ; can be shown to be @2 L LrLOD ¼ 2 @y y¼0 1 hNþ4 ðq0 Þ ¼ rT K1 n Ks Kn r hN ðq0 Þ h Nþ2 ðq0 Þ ; ð10Þ E s ½sT K1 n s hN ðq0 Þ where Ks is the covariance matrix of the random signal samples. Next, two examples are presented to provide some insights on the sampling methodology.
4. Numerical examples In this section, we present two numerical examples to illustrate our sampling design for the
weak signal detection problem. The sampling points t ¼ ½t1 ; t2 ; . . . ; tN are determined so as to maximize DYV of Eq. (6); i.e., min DYV ¼ min ci E 0 ½ðLLOD Þ2 subject to t
t
apt1 ot2 o otN pb:
ð11Þ
The constant ci can be excluded and therefore the sampling design is independent of the signal-to-noise ratio, y2 : This nonlinearly constrained maximization problem can be solved by employing Fortran IMSL subroutine DNCONF. The expectation can be carried out by Monte Carlo integration [16, pp. 221–222]. For this purpose and that of evaluating the system performance by using Monte Carlo simulation later, we need to generate correlated N-multivariate student-T and K-distributed noise vectors used in the examples. The procedures of generating these distributions are not presented here and the reader is referred to [5,11]. In these examples, the sampling points are obtained by maximizing DYV and the maximum achievable sampling rate for the system, R; is set to 104 Hz. The constant ci of DYV is excluded (i.e., ci ¼ 1) in the numerical computations. A linear interpolation method is employed to obtain a continuous curve in the figures for these examples. In those figures, ‘‘Distance Sampling’’ represents the sampling scheme based on the distance measure, DYV ; ‘‘Uniform Sampling’’ is uniform sampling with sampling points, ti ¼ ði 1Þ=ðN 1Þ; i ¼ 1; . . . ; N: Example 1. Known signal in a stationary SIRP noise. In this example, we consider the observation qffiffi model given in Eq. (1) where sðtÞ ¼ 83 sin2 ðptÞ; with unit power over ½0; 1; and nðtÞ is a stationary SIRP noise with zero mean and autocorrelation function, Rn ðtÞ ¼ expðmjtjÞ: The sampled SIRP noise vector is modelled as having a multivariate student-T distribution and a multivariate Kdistribution, respectively. The parameter m represents a measure of the bandwidth of the noise process nðtÞ: The effect of the ‘‘duration-bandwidth product’’, ðb aÞm; on the detection performance will be shown. For convenience, we let the observation interval ½a; b to be ½0; 1; in which case
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the ‘‘duration-bandwidth product’’ is simply m: Now we can determine sampling schemes for this known signal detection in the two different noises separately.
The noise vector is modelled as N-multivariate student-T distribution
In this case, the conditional densities of sampled data vector can be modelled as a N-dimensional multivariate student-T distribution with parameters N and b: They are given by [5,17] pðrjH i Þ ¼ KhN ðqi Þ 1 ¼ N=2 ð2pÞ jKn j1=2
ðb 1Þb GðN=2 þ bÞ GðbÞðb 1 þ qi =2ÞðN=2þbÞ
; i ¼ 0; 1;
where N is the number of samples and b determines the tail behavior of these density functions. The larger the value of b; the smaller is the tail. Note that Eq. (12) is exactly same as Eq. (48) of [17] when n ¼ b and b2 ¼ 2b 2: From Eqs. (8) and (12), we have the LOD statistic which can be written as ðN=2 þ bÞ : ðb 1 þ q0 =2Þ
bandwidth parameter m) and the shape of the signal. In general, as the value of m increases, i.e., as the additive colored noise becomes whiter, the sampling points become more and more clustered around the point 1/2. Obviously, this result is due to the shape of the signal which has a peak in 1/2. By using the behavior of the sampling scheme, we may determine a good signal for known signal detection so that the detection performance is enhanced. This will be illustrated later. Next, sampled data are processed by the LOD that employs the Neyman–Pearson criterion to design the decision rule H1 sT K1 n r q0 _ Z1 : ðb 1 þ 2 Þ H 0
ð12Þ
LdLOD ðrÞ ¼ sT K1 n r
209
(13)
Here, let b be 1.50. By using Eqs. (11) and (13), sampling points for different values of m and N are determined but only tabulated for N ¼ 7 in Table 1. We note that the sampling points are affected by the autocorrelation function (the
The constant term, ðb þ N=2Þ; in the LOD statistic has been included into the threshold Z1 : Since the PDF of the LOD statistic cannot be determined analytically, analytical evaluation of the exact detection performance is intractable. Consequently, we evaluate the LOD detection performance by Monte Carlo simulation. Here, the probability of false alarm Pf is set to 102 and the probability of detection Pd is obtained by simulation. The number of trails for obtaining Pd and Z1 in the Monte Carlo simulation is equal to 105 : We evaluate Pd for the uniform sampling design and the sampling designs obtained earlier. Figs. 1–3 show the performance effect of varying the signal bandwidth parameter m; sample size N and signal to noise ratio y2 ; respectively. Fig. 3 shows that we can increase the sample size N to
Table 1 Sampling points for the student-T distribution noise with b ¼ 1:5 m
t1
t2
t3
t4
t5
t6
t7
DYV
1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 500.0
0.0655 0.0620 0.0492 0.3055 0.2979 0.3504 0.3951 0.4312 0.4633
0.3993 0.3946 0.3500 0.3894 0.3728 0.4054 0.4331 0.4558 0.4762
0.4633 0.4624 0.4231 0.4639 0.4382 0.4539 0.4673 0.4783 0.4883
0.5431 0.5491 0.5071 0.5374 0.5005 0.5002 0.5001 0.5001 0.5000
0.6312 0.6389 0.5814 0.6110 0.5617 0.5462 0.5328 0.5218 0.5118
0.8610 0.8652 0.6695 0.7026 0.6294 0.5954 0.5674 0.5445 0.5239
0.9566 0.9574 0.9505 0.9793 0.7090 0.6520 0.6061 0.5694 0.5369
14.1280 8.5531 7.4990 10.7105 17.2540 27.3292 32.8735 35.8882 37.4591
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obtain a higher probability of detection when the other parameters are fixed. But, the difficulty of solving the nonlinearly constrained maximization problem also increases. Thus, a more powerful computer or a more efficient algorithm would be needed.
0.5 Distance Sampling: Uniform Sampling:
Probability of Detection
0.45 0.4 0.35 0.3
Student-T
0.25
0.2 K-distribution
0.15 0.1 0.05 100
101
102
µ
Fig. 1. Pd as a function of m with y ¼ 0:45 (SNR=6:94 dB), N ¼ 20 and Pf ¼ 102 for Example 1.
0.6
Probability of Detection
The conditional densities of sampled data vector are modelled as a N-dimensional multivariate Kdistribution with parameters N and g: They are given by pðrjH i Þ ¼ KhN ðqi Þ 1 ¼ N=2 ð2pÞ jKn j1=2
Distance Sampling:
2ðN=2gþ1Þ=2þ1 gðN=2þgÞ=2 K N=2g ð GðgÞðN=2gÞ=2 qi
Uniform Sampling:
0.5
i ¼ 0; 1;
Student-T
0.4
K-distribution
0.3 0.2 0.1 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
θ
Fig. 2. Pd as a function of y with m ¼ 200; N ¼ 40 and Pf ¼ 102 for Example 1.
0.5 Distance Sampling: Uniform Sampling:
0.45
Probability of Detection
The noise vector is modelled as N-multivariate K-distribution.
0.4 Student-T
0.35 0.3
K-distribution
0.25 0.2 0.15 0.1 0.05 8
10
12
14
16
18
20
22
24
26
28
30
N
Fig. 3. Pd as a function of sample size N with m ¼ 100:0; y ¼ 0:45 (SNR=6:94 dB) and Pf ¼ 102 for Example 1.
pffiffiffiffiffiffiffiffiffi 2gqi Þ
;
ð14Þ
where N is the number of samples; K N=2g ð Þ represents the modified Bessel function of the second kind of order N=2 g; g determines the tail behavior of these density functions. As g ! 1; the K-distribution tends to the Gaussian distribution. When g close to 0.5, the K-distribution deviates from that of the Gaussian distribution in the sense of having very large tails. In this example, g is set to be 1.50. From Eqs. (8) and (14), the LOD statistic can be written as pffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2gK N=2gþ1 ð 2gq0 Þ d T 1 pffiffiffiffiffiffiffiffiffi : LLOD ðrÞ ¼ s Kn r pffiffiffiffiffi (15) q0 K N=2g ð 2gq0 Þ In a similar way, we determined the sampling points for g ¼ 1:50; and different values of m and N; respectively. A sampling design for N ¼ 7 only is shown in Table 2. Again, sampled data are processed by the LOD that employs the Neyman–Pearson criterion to design the decision rule pffiffiffiffiffiffiffiffiffi H1 T 1 K N=2gþ1 ð 2gq0 Þ pffiffiffiffiffiffiffiffiffi _ Z2 : s Kn r pffiffiffiffiffi q0 K N=2g ð 2gq0 Þ H 0 pffiffiffiffiffi The constant term, 2g; in the LOD statistic has been included into the threshold Z2 : By using Monte Carlo simulation, we evaluate Pd for the
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Table 2 Sampling points for the K-distribution noise with g ¼ 1:5 m
t1
t2
t3
t4
t5
t6
t7
DYV
1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 500.0
0.0378 0.0698 0.0563 0.2780 0.2906 0.3484 0.3943 0.4310 0.4633
0.1012 0.3344 0.3248 0.3569 0.3703 0.4052 0.4332 0.4560 0.4764
0.4296 0.3877 0.4053 0.4362 0.4405 0.4555 0.4683 0.4789 0.4886
0.5693 0.4317 0.4694 0.4965 0.5014 0.5015 0.5010 0.5007 0.5004
0.9059 0.5160 0.5612 0.5719 0.5687 0.5496 0.5348 0.5229 0.5123
0.9385 0.6062 0.6558 0.6442 0.6360 0.5988 0.5693 0.5456 0.5244
0.9797 0.9359 0.9524 0.7235 0.7143 0.6550 0.6079 0.5705 0.5375
15.1248 8.9837 8.1510 11.7936 18.7030 29.4589 35.3222 38.4924 40.1373
0.4
Probability of Detection
0.35
Student-T: K-distribution: Gaussian assumption: ,
,
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
θ
Fig. 4. Pd comparison of sampling schemes for the Kdistributed, student-T and Gaussian noises in Example 1 with m ¼ 200; N ¼ 20 and Pf ¼ 102 : 0.5 0.45 0.4 Probability of Detection
uniform sampling design and the sampling designs obtained earlier. Detection performances for the above sampling schemes are compared in Figs. 1–3. For the purpose of comparison, sampling points for the N-multivariate Gaussian noise are also obtained by maximizing sT K1 n s [22]. Interestingly enough, we note that the sampling points for the Gaussian, K-distributed and student-T noises are fairly close to each other. Our numerical experience indicates that small sampling variation does not affect detection performance significantly. This can be seen from Fig. 1 where the distance sampling scheme has some differences from the uniform sampling scheme, but they have very close detection performance when m is small. A detection performance comparison between the sampling schemes for K-distributed and student-T noises, and those with a Gaussian noise assumption is also shown in Figs. 4 and 5. Thus, these numerical results show that the sampling design is insensitive to the characteristic PDF of the SIRP noise samples, and imply that we can design a sampling scheme for the SIRP noise case by employing the simpler design criterion for Gaussian noise case. Therefore, the complexity of solving the nonlinearly constrained maximization problem decreases significantly. To show that the sampling scheme may apply to determine a good signal for known signal detection, we also carried out the sampling design and detection performance evaluation for constant level signal (i.e., sðtÞ ¼ 1; 0ptp1) detection. The sampling design for the constant level signal case comes out close to an uniform sampling. The
Student-T : K-distribution : Gaussian assumption:
0.35 0.3
Student-T
0.25 0.2 K-distribution
0.15 0.1 0.05 100
101
102 µ
Fig. 5. Pd comparison of sampling schemes for the Kdistributed, student-T and Gaussian noises in Example 1 with y ¼ 0:4 (SNR=6:94 dB), N ¼ 20 and Pf ¼ 102 :
detection performance for the above sampling designs are shown in Fig. 6. In the figure, s1ðtÞ and qffiffi s2ðtÞ represent 83 sin2 ðptÞ and the constant level
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Probability of Detection
0.4 0.35
Distance Sampling for S1(t): Uniform Sampling: Distance Sampling for S2(t):
interval, jti tj j: Therefore, we may set t1 ¼ 0: We obtain the sampling designs by employing the similar procedure used earlier. The sampling points are affected by the autocorrelation functions (the bandwidth parameters, a and m). In general, as the values of m or b increase, the sampling points become more and more clustered. As earlier, the Neyman–Pearson decision rule in LOD is
,
,
0.3 Student-T
0.25 0.2 0.15
K-distribution
0.1 0.05 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
θ
ðN=2 þ b þ 1Þ T 1 r Kn Ks K1 n r ðb 1 þ q0 =2Þ2
Fig. 6. Pd comparison of sampling schemes for the signals s1ðtÞ and s2ðtÞ in Example 1 with m ¼ 200; N ¼ 40 and Pf ¼ 102 :
Example 2. Weak stationary Gaussian signal in stationary SIRP noise. We use the same observation model as in Example 1 except that the signal, sðtÞ; is a Gaussian process with zero mean and autocorrelation function, Rs ðtÞ ¼ sinc2 ðatÞ: The parameters, m and a; represent a measure of the bandwidth of the noise process nðtÞ and the signal process sðtÞ; respectively. Because the sampled SIRP noise vector is modelled as having a multivariate student-T distribution, we can employ Eqs. (7) with i = 0, (9), (10) and (12) to obtain the LOD statistic: LrLOD
ðN=2 þ b þ 1ÞðN=2 þ bÞ T 1 ¼ r Kn Ks K1 n r ðb 1 þ q0 =2Þ2 ðN=2 þ bÞ : ð16Þ tr½Ks K1 n ðb 1 þ q0 =2Þ
Consider different combinations of parameter values, a ¼ 0:32; 0:64; 1:28; 3:2; 6:4 and 12.8; m ¼ 1; 2; 5; 10; 20; 50; 100; 200: When both the signal and the noise are zero-mean stationary processes, the sampling design depends only on the sampling
The constant term, ðN=2 þ bÞ; of LOD statistic has been included into the threshold Z3 : As in Example 1, we also note that sampling points for the Gaussian noise are fairly close to those for student-T noise. Similarly, we evaluated Pd for the uniform sampling design and the sampling designs obtained in this example. Figs. 7 and 8 show the performance effect of varying the bandwidth parameters m and a; respectively. The curve noted ‘‘Distance Sampling’’ in Fig. 8 is based on the sampling design for the Gaussian noise, i.e., sampling points are obtained by 2 maximizing tr½ðKs K1 n Þ [22]. From the figures for the above examples, we can observe that the sampling design based on the distance DYV results in an improved detection performance, especially when m and a are large the
0.2 0.18 0.16
Probability of Detection
signal, respectively. The detection performance comparison shows that we may apply the sampling scheme to select a good signal shape with a constrained power for known signal detection so that its detection performance is enhanced. This strategy is useful particularly in the wide band noise situation.
H1 tr½Ks K1 n _ Z3 : ðb 1 þ q0 =2Þ H 0
0.14 0.12
Distance Sampling
0.1 Uniform Sampling
0.08 0.06 0.04 0.02 100
101
µ
102
Fig. 7. Pd as a function of m with y ¼ 0:45 (SNR = 6:94 dB), a ¼ 3:2; b ¼ 1:5; N ¼ 20 and Pf ¼ 102 for Example 2.
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samples. This implies that sampling design for the weak signal detection in SIRP noise could be simplified to that for the weak Gaussian detection problem. Therefore, the complexity of maximization problem in the design procedure reduces significantly.
0.25 Distance Sampling
Probability of Detection
213
0.2
0.15 Uniform Sampling
0.1
0.05
Acknowledgements 0 2
3
4
5
6
7
8
9
10
11
α
Fig. 8. Pd as a function of a with y ¼ 0:45 (SNR = 6:94 dB), m ¼ 200; b ¼ 1:5; N ¼ 30 and Pf ¼ 102 for Example 2.
improvement is more significant. This is due to the fact that uniform sampling does not use the knowledge about signal and noise correlations (the information provided by m and a). The numerical results also show that sampling designs based on the distance are affected by the bandwidth parameters m and a much more significantly than by the statistics (the characteristic PDF f W ðwÞ; the distribution shape parameters b and g) of the SIRP noise samples.
5. Conclusions In this paper, we have proposed a technique to design sampling schemes with finite sample size N for the weak signal detection in SIRP noise. A member of the class of ASDMs suitable for this problem was developed. This distance measure was obtained by expanding the ASDM between the class conditional densities in terms of a power series and considering the first nonzero term. This objective function is maximized to obtain the sampling scheme for the detection problem under consideration. In the examples, it was demonstrated that this distance measure can be used successfully to design the sampling scheme for the problem under consideration and the sampling designs outperformed the uniform sampling scheme. It was also numerically shown that the sampling design is insensitive to the statistics (the characteristic PDF) of the SIRP noise
The author would like to thank the anonymous reviewers for their constructive comments. The author also expresses his gratitude to Dr. P.K. Varshney for his extremely helpful comments and suggestions. All these comments have significantly strengthened this paper.
References [1] S.M. Ali, S.D. Silvey, A general class of coefficients of divergence of one distribution from another, J. Roy. Statist. Soc. Ser. B 28 (1966) 131–143. [2] R. Bahr, J. Bucklew, Optimal sampling schemes for Gaussian hypothesis testing problem, IEEE Trans. Acoust. Speech Signal Process. 38 (10) (October 1990) 1677–1686. [3] H. Brehm, Description of Spherically Invariant Random Processes by Means of G-functions, Springer Lecture Notes, vol. 969, 1982, pp. 39–73. [4] S. Cambanis, E. Masry, Sampling designs for the detection of signal in noise, IEEE Trans. Inform. Theory IT-29 (1) (January 1983) 83–104. [5] P.R. Chakravarthi, Subclutter visibility: the problem of weak signal detection, Ph.D. Dissertation, EECS Department, Syracuse University, New York, March 1993. [6] H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statist. 23 (1952) 493–507. [7] E. Conte, A. De Maio, C. Galdi, Radar detection of fluctuating targets in compound-Gaussian clutter, The Record of the IEEE 2000 International Radar Conference, 2000, pp. 392–397. [8] E. Conte, A. De Maio, C. Galdi, Signal detection in compound-Gaussian noise: Neyman–Pearson and CFAR detectors, IEEE Trans. Signal Process. 48 (February 2000) pp. 419–428. [9] E. Conte, M. Longo, Characterization of radar clutter as a spherically invariant random process, IEE Proc. F Comm. Radar Signal Process. 134 (1987) 191–197. [10] J. Goldman, Detection in the presence of spherically symmetric random processes, IEEE Trans. Inform. Theory IT-22 (September 1976) 52–58.
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[11] N. Johnson, S. Kotz, N. Balkrishnan, Continuous Univariate Distributions, vol. 1, second ed., Wiley, New York, 1994. [12] T. Kailath, The divergence and Bhattacharyya distance measure in signal selection, IEEE Trans. Comm. COM-15 (1) (February 1967) 52–60. [13] K. Kingman, On random sequences with spherical symmetry, Biometrika 59 (1972) 492–494. [14] S. Kullback, Information Theory and Statistics, Wiley, New York, 1960. [15] J.P. LeBlanc, P.L. De Leon, Speech separation by kurtosis maximization, Proceedings of the 1998 IEEE Acoustics Speech and Signal Processing, vol. 2, February 1998, pp. 1029–1032. [16] A. Papoulis, Probability, Random Variables and Stochastic Process, third ed., McGraw-Hill, New York, 1991. [17] M. Rangaswamy, D.D. Weiner, A. Ozturk, Spherically invariant random processes for modeling and distribution identification of non-Gaussian random vectors, IEEE
[18]
[19]
[20]
[21]
[22]
Trans. Aerospace Electronic Systems AES-29 (January 1993) 111–124. K.J. Sangston, K. Gerlach, Results on the detection of signals in spherically invariant random noise, NRL Report, vol. 9202, 1989, pp. 1–12. J. Tang, Z. Zhu, The locally optimum detectors in clutter characterized as SIRP, Proceedings of the IEEE 1995 National Aerospace and Electronics Conference, vol. 1, 1995, pp. 244–251. G. Wise, N. Gallagher, On spherically invariant random processes, IEEE Trans. Inform. Theory IT-24 (1978) 118–120. K. Yao, A representation theorem and its application to spherically invariant random processes, IEEE Trans. Inform. Theory IT-19 (1973) 600–608. C.T. Yu, P.K. Varshney, Sampling design for Gaussian detection problems, IEEE Trans. Signal Process. 45 (9) (1997) 2326–2337.
Signal Processing 85 (2005) 205–214 www.elsevier.com/locate/sigpro
Sampling design for weak signal detection in SIRP noise$ Chao-Tang Yu Department of Electronic Engineering, Southern Taiwan University of Technology, No. 1, Nan-Tai Street, Yung-Kang City, Tainan, Taiwan Received 28 February 2002; received in revised form 27 September 2004
Abstract In this paper, we present a sampling design for weak signal detection in additive spherically invariant random process (SIRP) noise. Our approach for the design of sampling schemes is based on the class of Ali–Silvey distance measures. A design objective function for this problem is developed. This objective function is maximized to obtain the sampling scheme for the detection problem under consideration. Two examples are presented to illustrate our sampling design. Numerical results show that our sampling design outperforms the uniform sampling. It is also numerically shown that the sampling design is insensitive to the statistics (the characteristic probability density function) of the SIRP noise samples. r 2004 Elsevier B.V. All rights reserved. Keywords: Sampling design; SIRP; Detection
1. Introduction Sampling design for the Gaussian detection problem has been studied in [2,4,22], where the received signals under hypotheses H 1 and H 0 were Gaussian random processes. Both random and deterministic sampling schemes for the detection of known signals in Gaussian noise were considered in [4]. Using the decision statistic based on the $ Research sponsored by National Science Council, Taiwan, R.O.C., under Grant No. NSC 88-2213-E-218-010. Tel.: +886 6 2533131x3137; fax: +886 6 2426911, +886 6 2537461. E-mail addresses: [email protected], [email protected] (C.-T. Yu).
sampled data, instead of the continuous-time detector (matched filter or correlation receiver), naturally results in a degradation of performance. Deterministic and random sampling schemes were designed so as to minimize the degradation in the performance of the detector. In [2], an asymptotically optimal periodic sampling design for the Gaussian hypothesis testing problem was presented. Under very general conditions, the probabilities of error decrease exponentially with increasing sample size. For this problem, an asymptotically optimal periodic sampling scheme was designed by maximizing the rate of exponential decrease. In [22], a universal approach based on the Ali–Silvey distance measure (ASDM) for
0165-1684/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2004.09.012
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the design of sampling schemes for Gaussian hypothesis testing problems was presented. In the strong signal case, sampling points were determined that maximize the distance measures between the conditional densities. In the weak signal case, a new member of the class of ASDMs suitable for the weak Gaussian signal detection problem was developed. The sampling scheme was designed by maximizing the distance measure. In the weak signal detection problem, the assumption of Gaussian noise is not suitable for many situations. Here, we are interested in sampling design for weak signal detection in nonGaussian noise characterized as a spherically invariant random process (SIRP). When we sample a SIRP, the probability density function (PDF) of the sampled data vector is uniquely determined by the specification of a mean vector, a covariance matrix and a characteristic first order PDF. SIRP has been used for clutter modelling and simulation [9]. Through the theory of SIRPs, a library of multivariate correlated non-Gaussian PDFs for characterizing various scenarios was developed in [17]. In [5], a canonical form for the locally optimum detector (LOD) was established. The performance of LODs in radar weak signal detection for finite sample sizes was also determined, where the radar disturbance was modelled as correlated SIRPs. Some other applications for the theory of SIRPs can be found in the areas of signal (or radar) detection and estimation problems [7,8,10,18,19,21] and speech signal processing [3,15]. Here, we consider the sampling design for weak signal non-Gaussian detection H 1 : rðtÞ ¼ ysðtÞ þ nðtÞ; aptpb; H 0 : rðtÞ ¼ nðtÞ; aptpb;
ð1Þ
where rðtÞ is the observation over the interval ½a; b; nðtÞ is a non-Gaussian random process modelled by a SIRP with zero mean and autocorrelation function Rn ðt1 ; t2 Þ; and the signal sðtÞ could be a deterministic signal or a random process with mean-value function ms ðtÞ and autocorrelation function Rs ðt1 ; t2 Þ: The signal sðtÞ is assumed to be independent of the noise nðtÞ: y is a parameter with small positive real value so that
ys(t) represents the weak signal as y approaches zero. Without loss of generality, the average power of sðtÞ and that of nðtÞ are normalized to unity so that y can be used to control the signal to noise ratio. Since the non-Gaussian noise is modelled by a SIRP, an analytical expression for the joint PDFs of N random variables obtained by sampling the noise process is available via the theory of SIRP. In Section 2, we present a brief review of the class of ASDMs and the theory of SIRPs. In Section 3, we present the sampling design procedure for the problem under consideration. In Section 4, we present two numerical examples to illustrate our sampling schemes. Concluding remarks are made in the last section.
2. Preliminaries We briefly review the class of ASDMs and some relevant mathematical preliminaries for real SIRPs as follows: 2.1. The class of Ali–Silvey distance measures The class of ASDMs [1] is defined by the general expression Dðp0 ; p1 Þ ¼ f ðE 0 ½CðLÞÞ;
(2)
where p0 ; p1 are the two probability distributions defined on the same space; Dðp0 ; p1 Þ is the Ali–Silvey distance between p0 and p1 ; f is an increasing function; E 0 denotes expectation with respect to p0 ; C is a convex function; and L is the likelihood ratio. Some examples of this class of distance measures are (1) I-divergence [14], I ¼ E 0 ½ lnðLÞ; (2) J-divergence [14], J ¼ E 0 ½ðL 1Þ lnðLÞ; pffiffiffiffi (3). Bhattacharyya distance [12], B ¼ lnðE 0 ½ LÞ; (4) Chernoff distance, [6], C h ¼ lnðE 0 ½La Þ; where 0 pap1: 2.2. Specification of the SIRP Spherically invariant random vector (SIRV) and SIRP are specified as follows [9,17]:
A SIRV is a random vector whose PDF is uniquely determined by the specification of a
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mean vector, a covariance matrix and a characteristic first order PDF. The PDF of a random vector is defined to be the joint PDF of the components of the random vector. A SIRP is a random process such that every random vector obtained by sampling this process is a SIRV.
Let Z ¼ ½Z1 ; Z 2 ; . . . ; Z N T ; where T denotes the matrix transpose, be a real zero mean Gaussian random vector with covariance matrix Kn : Let W be a non-negative random variable with PDF f W ðwÞ: W is assumed independent of Z: A SIRV can be obtained by the product X ¼ ZW : A representation theorem is used to construct the PDF of a SIRV [13,20,21], and the PDF of the product X is given by f x ðxÞ ¼
1=2
ð2pÞ jKn j Z 1 xT K1 N n x w exp f W ðwÞ dw: 2w2 0 ð3Þ
If we let q ¼ xT K1 n x; then the PDF can be rewritten as f x ðxÞ ¼
1 ð2pÞ
N=2
jKn j1=2
hN ðqÞ;
(4)
where Z hN ðqÞ ¼ 0
1
based on these discrete-time data. Let r; n and ys represent the sampled observation, noise and signal data vectors, respectively. Also, let pðrjH 1 Þ; pðrjH 0 Þ; and f N ðnÞ denote the joint PDFs of r under H 1 ; of r under H 0 and of noise, respectively. We note the ASDM between the conditional densities goes to zero as signal becomes very weak (as y approaches zero). Therefore, in this case, the sampling design criterion based on the class of ASDMs between pðrjH 1 Þ and pðrjH 0 Þ is not directly suitable. Instead of maximizing the distance measures DðpðrjH 1 Þ; pðrjH 0 ÞÞ; we maximize the first nonzero term of the series expansion of DðpðrjH 1 Þ; pðrjH 0 ÞÞ to design the sampling schemes. The objective function (the first non-zero term of the series expansion) can be expressed in the form [22, p. 2330]: DYV ¼ ci E 0 ½ðLLOD Þ2 ¼ f ðE 0 ½CðLLOD ÞÞ;
1 N=2
207
q wN exp 2 f W ðwÞ dw: 2w
(6)
where ci is a constant that depends on the signalto-noise ratio, y2 ; and the functions f and C defined in the class of ASDMs, LLOD represents the LOD statistic, and the functions are defined as f ðxÞ ¼ ci x and CðxÞ ¼ x2 : Because f is an increasing function, and C is a convex function, this objective function can be interpreted as a member of the class of ASDMs. The expressions of DYV can be evaluated completely if the LOD statistic is determined. Here, we find the LOD statistics for deterministic signal and random signal cases separately in the following.
(5)
The PDF of the random variable W is called the characteristic PDF of the SIRV. It is interesting to note that when f W ðwÞ ¼ dðw 1Þ; where dð Þ is the Dirac delta function, the resulting hN ðqÞ is equal to expðq2Þ and the PDF of X is the well-known Gaussian PDF.
3. Sampling design development In this section, we develop sampling schemes for the weak signal detection problem stated in Eq. (1). The received continuous-time signal, rðtÞ; is sampled by means of a sampling scheme. Sampled data are fed to the LOD that makes a decision
3.1. Deterministic signal in additive SIRP noise In this case, the signal is deterministic and the noise is a SIRP with zero mean and known autocorrelation function, Rn ðt1 ; t2 Þ; where the noise average power, Rn ðt; tÞ; is set to unity. Because the noise is modelled as a SIRP, the N observation samples form a SIRV. Thus, the joint PDFs of the sampled data, r ¼ ½rðt1 Þ; rðt2 Þ; . . . ; rðtN ÞT ; under hypotheses H i ; i ¼ 0; 1; are given by pðrjH i Þ ¼
1 ð2pÞ
N=2
jKn j1=2
hN ðqi Þ ¼ KhN ðqi Þ;
(7)
where the quadratic forms, q0 and q1 ; are equal to T 1 rT K1 n r and ðr ysÞ Kn ðr ysÞ; respectively; Kn is the covariance matrix of r; and it depends on the
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autocorrelation function and sampling points, t1 ; t2 ; . . . ; tN ; and hN ð Þ is as defined in Eq. (5). For this known signal case, the LOD statistic denoted by LdLOD is given by @L hNþ2 ðq0 Þ d LLOD ¼ : (8) ¼ sT K1 n r @y y¼0 hN ðq0 Þ
3.2. Random signal in additive SIRP noise In this case, sðtÞ is a random signal with meanvalue function, sm ðtÞ; and autocorrelation function, Rs ðt1 ; t2 Þ; the noise nðtÞ is modelled as a SIRP with zero mean and autocorrelation function, Rn ðt1 ; t2 Þ; and a characteristic PDF, f W ðwÞ: Again, Rs ðt; tÞ and Rn ðt; tÞ are set to 1. Then, pðrjH 0 Þ is given by Eq. (7) with i ¼ 0: Let the PDF of the N samples of sðtÞ; s ¼ ½sðt1 Þ; sðt2 Þ; . . . ; sðtN ÞT ; be denoted as f S ðsÞ: The noise is assumed to be independent of the signal. Thus, pðrjH 1 Þ can be written as Z pðrjH 1 Þ ¼ f N ðr ysÞf S ðsÞ ds s
¼ E s ½f N ðr ysÞ ¼ E s ½KhN ðq1 Þ;
ð9Þ
where f N ðrÞ ¼ f N ðrjH 0 Þ ¼ pðrjH 0 Þ; and E s denotes expectation with respect to f S ðsÞ: When the mean vector, sm ; is non-zero, the LOD statistic is essentially the same as in Eq. (8) where s is replaced with sm : In the case when sm ¼ 0 is considered, the LOD statistic, LrLOD ; can be shown to be @2 L LrLOD ¼ 2 @y y¼0 1 hNþ4 ðq0 Þ ¼ rT K1 n Ks Kn r hN ðq0 Þ h Nþ2 ðq0 Þ ; ð10Þ E s ½sT K1 n s hN ðq0 Þ where Ks is the covariance matrix of the random signal samples. Next, two examples are presented to provide some insights on the sampling methodology.
4. Numerical examples In this section, we present two numerical examples to illustrate our sampling design for the
weak signal detection problem. The sampling points t ¼ ½t1 ; t2 ; . . . ; tN are determined so as to maximize DYV of Eq. (6); i.e., min DYV ¼ min ci E 0 ½ðLLOD Þ2 subject to t
t
apt1 ot2 o otN pb:
ð11Þ
The constant ci can be excluded and therefore the sampling design is independent of the signal-to-noise ratio, y2 : This nonlinearly constrained maximization problem can be solved by employing Fortran IMSL subroutine DNCONF. The expectation can be carried out by Monte Carlo integration [16, pp. 221–222]. For this purpose and that of evaluating the system performance by using Monte Carlo simulation later, we need to generate correlated N-multivariate student-T and K-distributed noise vectors used in the examples. The procedures of generating these distributions are not presented here and the reader is referred to [5,11]. In these examples, the sampling points are obtained by maximizing DYV and the maximum achievable sampling rate for the system, R; is set to 104 Hz. The constant ci of DYV is excluded (i.e., ci ¼ 1) in the numerical computations. A linear interpolation method is employed to obtain a continuous curve in the figures for these examples. In those figures, ‘‘Distance Sampling’’ represents the sampling scheme based on the distance measure, DYV ; ‘‘Uniform Sampling’’ is uniform sampling with sampling points, ti ¼ ði 1Þ=ðN 1Þ; i ¼ 1; . . . ; N: Example 1. Known signal in a stationary SIRP noise. In this example, we consider the observation qffiffi model given in Eq. (1) where sðtÞ ¼ 83 sin2 ðptÞ; with unit power over ½0; 1; and nðtÞ is a stationary SIRP noise with zero mean and autocorrelation function, Rn ðtÞ ¼ expðmjtjÞ: The sampled SIRP noise vector is modelled as having a multivariate student-T distribution and a multivariate Kdistribution, respectively. The parameter m represents a measure of the bandwidth of the noise process nðtÞ: The effect of the ‘‘duration-bandwidth product’’, ðb aÞm; on the detection performance will be shown. For convenience, we let the observation interval ½a; b to be ½0; 1; in which case
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the ‘‘duration-bandwidth product’’ is simply m: Now we can determine sampling schemes for this known signal detection in the two different noises separately.
The noise vector is modelled as N-multivariate student-T distribution
In this case, the conditional densities of sampled data vector can be modelled as a N-dimensional multivariate student-T distribution with parameters N and b: They are given by [5,17] pðrjH i Þ ¼ KhN ðqi Þ 1 ¼ N=2 ð2pÞ jKn j1=2
ðb 1Þb GðN=2 þ bÞ GðbÞðb 1 þ qi =2ÞðN=2þbÞ
; i ¼ 0; 1;
where N is the number of samples and b determines the tail behavior of these density functions. The larger the value of b; the smaller is the tail. Note that Eq. (12) is exactly same as Eq. (48) of [17] when n ¼ b and b2 ¼ 2b 2: From Eqs. (8) and (12), we have the LOD statistic which can be written as ðN=2 þ bÞ : ðb 1 þ q0 =2Þ
bandwidth parameter m) and the shape of the signal. In general, as the value of m increases, i.e., as the additive colored noise becomes whiter, the sampling points become more and more clustered around the point 1/2. Obviously, this result is due to the shape of the signal which has a peak in 1/2. By using the behavior of the sampling scheme, we may determine a good signal for known signal detection so that the detection performance is enhanced. This will be illustrated later. Next, sampled data are processed by the LOD that employs the Neyman–Pearson criterion to design the decision rule H1 sT K1 n r q0 _ Z1 : ðb 1 þ 2 Þ H 0
ð12Þ
LdLOD ðrÞ ¼ sT K1 n r
209
(13)
Here, let b be 1.50. By using Eqs. (11) and (13), sampling points for different values of m and N are determined but only tabulated for N ¼ 7 in Table 1. We note that the sampling points are affected by the autocorrelation function (the
The constant term, ðb þ N=2Þ; in the LOD statistic has been included into the threshold Z1 : Since the PDF of the LOD statistic cannot be determined analytically, analytical evaluation of the exact detection performance is intractable. Consequently, we evaluate the LOD detection performance by Monte Carlo simulation. Here, the probability of false alarm Pf is set to 102 and the probability of detection Pd is obtained by simulation. The number of trails for obtaining Pd and Z1 in the Monte Carlo simulation is equal to 105 : We evaluate Pd for the uniform sampling design and the sampling designs obtained earlier. Figs. 1–3 show the performance effect of varying the signal bandwidth parameter m; sample size N and signal to noise ratio y2 ; respectively. Fig. 3 shows that we can increase the sample size N to
Table 1 Sampling points for the student-T distribution noise with b ¼ 1:5 m
t1
t2
t3
t4
t5
t6
t7
DYV
1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 500.0
0.0655 0.0620 0.0492 0.3055 0.2979 0.3504 0.3951 0.4312 0.4633
0.3993 0.3946 0.3500 0.3894 0.3728 0.4054 0.4331 0.4558 0.4762
0.4633 0.4624 0.4231 0.4639 0.4382 0.4539 0.4673 0.4783 0.4883
0.5431 0.5491 0.5071 0.5374 0.5005 0.5002 0.5001 0.5001 0.5000
0.6312 0.6389 0.5814 0.6110 0.5617 0.5462 0.5328 0.5218 0.5118
0.8610 0.8652 0.6695 0.7026 0.6294 0.5954 0.5674 0.5445 0.5239
0.9566 0.9574 0.9505 0.9793 0.7090 0.6520 0.6061 0.5694 0.5369
14.1280 8.5531 7.4990 10.7105 17.2540 27.3292 32.8735 35.8882 37.4591
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obtain a higher probability of detection when the other parameters are fixed. But, the difficulty of solving the nonlinearly constrained maximization problem also increases. Thus, a more powerful computer or a more efficient algorithm would be needed.
0.5 Distance Sampling: Uniform Sampling:
Probability of Detection
0.45 0.4 0.35 0.3
Student-T
0.25
0.2 K-distribution
0.15 0.1 0.05 100
101
102
µ
Fig. 1. Pd as a function of m with y ¼ 0:45 (SNR=6:94 dB), N ¼ 20 and Pf ¼ 102 for Example 1.
0.6
Probability of Detection
The conditional densities of sampled data vector are modelled as a N-dimensional multivariate Kdistribution with parameters N and g: They are given by pðrjH i Þ ¼ KhN ðqi Þ 1 ¼ N=2 ð2pÞ jKn j1=2
Distance Sampling:
2ðN=2gþ1Þ=2þ1 gðN=2þgÞ=2 K N=2g ð GðgÞðN=2gÞ=2 qi
Uniform Sampling:
0.5
i ¼ 0; 1;
Student-T
0.4
K-distribution
0.3 0.2 0.1 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
θ
Fig. 2. Pd as a function of y with m ¼ 200; N ¼ 40 and Pf ¼ 102 for Example 1.
0.5 Distance Sampling: Uniform Sampling:
0.45
Probability of Detection
The noise vector is modelled as N-multivariate K-distribution.
0.4 Student-T
0.35 0.3
K-distribution
0.25 0.2 0.15 0.1 0.05 8
10
12
14
16
18
20
22
24
26
28
30
N
Fig. 3. Pd as a function of sample size N with m ¼ 100:0; y ¼ 0:45 (SNR=6:94 dB) and Pf ¼ 102 for Example 1.
pffiffiffiffiffiffiffiffiffi 2gqi Þ
;
ð14Þ
where N is the number of samples; K N=2g ð Þ represents the modified Bessel function of the second kind of order N=2 g; g determines the tail behavior of these density functions. As g ! 1; the K-distribution tends to the Gaussian distribution. When g close to 0.5, the K-distribution deviates from that of the Gaussian distribution in the sense of having very large tails. In this example, g is set to be 1.50. From Eqs. (8) and (14), the LOD statistic can be written as pffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 2gK N=2gþ1 ð 2gq0 Þ d T 1 pffiffiffiffiffiffiffiffiffi : LLOD ðrÞ ¼ s Kn r pffiffiffiffiffi (15) q0 K N=2g ð 2gq0 Þ In a similar way, we determined the sampling points for g ¼ 1:50; and different values of m and N; respectively. A sampling design for N ¼ 7 only is shown in Table 2. Again, sampled data are processed by the LOD that employs the Neyman–Pearson criterion to design the decision rule pffiffiffiffiffiffiffiffiffi H1 T 1 K N=2gþ1 ð 2gq0 Þ pffiffiffiffiffiffiffiffiffi _ Z2 : s Kn r pffiffiffiffiffi q0 K N=2g ð 2gq0 Þ H 0 pffiffiffiffiffi The constant term, 2g; in the LOD statistic has been included into the threshold Z2 : By using Monte Carlo simulation, we evaluate Pd for the
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Table 2 Sampling points for the K-distribution noise with g ¼ 1:5 m
t1
t2
t3
t4
t5
t6
t7
DYV
1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 500.0
0.0378 0.0698 0.0563 0.2780 0.2906 0.3484 0.3943 0.4310 0.4633
0.1012 0.3344 0.3248 0.3569 0.3703 0.4052 0.4332 0.4560 0.4764
0.4296 0.3877 0.4053 0.4362 0.4405 0.4555 0.4683 0.4789 0.4886
0.5693 0.4317 0.4694 0.4965 0.5014 0.5015 0.5010 0.5007 0.5004
0.9059 0.5160 0.5612 0.5719 0.5687 0.5496 0.5348 0.5229 0.5123
0.9385 0.6062 0.6558 0.6442 0.6360 0.5988 0.5693 0.5456 0.5244
0.9797 0.9359 0.9524 0.7235 0.7143 0.6550 0.6079 0.5705 0.5375
15.1248 8.9837 8.1510 11.7936 18.7030 29.4589 35.3222 38.4924 40.1373
0.4
Probability of Detection
0.35
Student-T: K-distribution: Gaussian assumption: ,
,
0.3 0.25 0.2 0.15 0.1 0.05 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
θ
Fig. 4. Pd comparison of sampling schemes for the Kdistributed, student-T and Gaussian noises in Example 1 with m ¼ 200; N ¼ 20 and Pf ¼ 102 : 0.5 0.45 0.4 Probability of Detection
uniform sampling design and the sampling designs obtained earlier. Detection performances for the above sampling schemes are compared in Figs. 1–3. For the purpose of comparison, sampling points for the N-multivariate Gaussian noise are also obtained by maximizing sT K1 n s [22]. Interestingly enough, we note that the sampling points for the Gaussian, K-distributed and student-T noises are fairly close to each other. Our numerical experience indicates that small sampling variation does not affect detection performance significantly. This can be seen from Fig. 1 where the distance sampling scheme has some differences from the uniform sampling scheme, but they have very close detection performance when m is small. A detection performance comparison between the sampling schemes for K-distributed and student-T noises, and those with a Gaussian noise assumption is also shown in Figs. 4 and 5. Thus, these numerical results show that the sampling design is insensitive to the characteristic PDF of the SIRP noise samples, and imply that we can design a sampling scheme for the SIRP noise case by employing the simpler design criterion for Gaussian noise case. Therefore, the complexity of solving the nonlinearly constrained maximization problem decreases significantly. To show that the sampling scheme may apply to determine a good signal for known signal detection, we also carried out the sampling design and detection performance evaluation for constant level signal (i.e., sðtÞ ¼ 1; 0ptp1) detection. The sampling design for the constant level signal case comes out close to an uniform sampling. The
Student-T : K-distribution : Gaussian assumption:
0.35 0.3
Student-T
0.25 0.2 K-distribution
0.15 0.1 0.05 100
101
102 µ
Fig. 5. Pd comparison of sampling schemes for the Kdistributed, student-T and Gaussian noises in Example 1 with y ¼ 0:4 (SNR=6:94 dB), N ¼ 20 and Pf ¼ 102 :
detection performance for the above sampling designs are shown in Fig. 6. In the figure, s1ðtÞ and qffiffi s2ðtÞ represent 83 sin2 ðptÞ and the constant level
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212
Probability of Detection
0.4 0.35
Distance Sampling for S1(t): Uniform Sampling: Distance Sampling for S2(t):
interval, jti tj j: Therefore, we may set t1 ¼ 0: We obtain the sampling designs by employing the similar procedure used earlier. The sampling points are affected by the autocorrelation functions (the bandwidth parameters, a and m). In general, as the values of m or b increase, the sampling points become more and more clustered. As earlier, the Neyman–Pearson decision rule in LOD is
,
,
0.3 Student-T
0.25 0.2 0.15
K-distribution
0.1 0.05 0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
θ
ðN=2 þ b þ 1Þ T 1 r Kn Ks K1 n r ðb 1 þ q0 =2Þ2
Fig. 6. Pd comparison of sampling schemes for the signals s1ðtÞ and s2ðtÞ in Example 1 with m ¼ 200; N ¼ 40 and Pf ¼ 102 :
Example 2. Weak stationary Gaussian signal in stationary SIRP noise. We use the same observation model as in Example 1 except that the signal, sðtÞ; is a Gaussian process with zero mean and autocorrelation function, Rs ðtÞ ¼ sinc2 ðatÞ: The parameters, m and a; represent a measure of the bandwidth of the noise process nðtÞ and the signal process sðtÞ; respectively. Because the sampled SIRP noise vector is modelled as having a multivariate student-T distribution, we can employ Eqs. (7) with i = 0, (9), (10) and (12) to obtain the LOD statistic: LrLOD
ðN=2 þ b þ 1ÞðN=2 þ bÞ T 1 ¼ r Kn Ks K1 n r ðb 1 þ q0 =2Þ2 ðN=2 þ bÞ : ð16Þ tr½Ks K1 n ðb 1 þ q0 =2Þ
Consider different combinations of parameter values, a ¼ 0:32; 0:64; 1:28; 3:2; 6:4 and 12.8; m ¼ 1; 2; 5; 10; 20; 50; 100; 200: When both the signal and the noise are zero-mean stationary processes, the sampling design depends only on the sampling
The constant term, ðN=2 þ bÞ; of LOD statistic has been included into the threshold Z3 : As in Example 1, we also note that sampling points for the Gaussian noise are fairly close to those for student-T noise. Similarly, we evaluated Pd for the uniform sampling design and the sampling designs obtained in this example. Figs. 7 and 8 show the performance effect of varying the bandwidth parameters m and a; respectively. The curve noted ‘‘Distance Sampling’’ in Fig. 8 is based on the sampling design for the Gaussian noise, i.e., sampling points are obtained by 2 maximizing tr½ðKs K1 n Þ [22]. From the figures for the above examples, we can observe that the sampling design based on the distance DYV results in an improved detection performance, especially when m and a are large the
0.2 0.18 0.16
Probability of Detection
signal, respectively. The detection performance comparison shows that we may apply the sampling scheme to select a good signal shape with a constrained power for known signal detection so that its detection performance is enhanced. This strategy is useful particularly in the wide band noise situation.
H1 tr½Ks K1 n _ Z3 : ðb 1 þ q0 =2Þ H 0
0.14 0.12
Distance Sampling
0.1 Uniform Sampling
0.08 0.06 0.04 0.02 100
101
µ
102
Fig. 7. Pd as a function of m with y ¼ 0:45 (SNR = 6:94 dB), a ¼ 3:2; b ¼ 1:5; N ¼ 20 and Pf ¼ 102 for Example 2.
ARTICLE IN PRESS C.-T. Yu / Signal Processing 85 (2005) 205–214
samples. This implies that sampling design for the weak signal detection in SIRP noise could be simplified to that for the weak Gaussian detection problem. Therefore, the complexity of maximization problem in the design procedure reduces significantly.
0.25 Distance Sampling
Probability of Detection
213
0.2
0.15 Uniform Sampling
0.1
0.05
Acknowledgements 0 2
3
4
5
6
7
8
9
10
11
α
Fig. 8. Pd as a function of a with y ¼ 0:45 (SNR = 6:94 dB), m ¼ 200; b ¼ 1:5; N ¼ 30 and Pf ¼ 102 for Example 2.
improvement is more significant. This is due to the fact that uniform sampling does not use the knowledge about signal and noise correlations (the information provided by m and a). The numerical results also show that sampling designs based on the distance are affected by the bandwidth parameters m and a much more significantly than by the statistics (the characteristic PDF f W ðwÞ; the distribution shape parameters b and g) of the SIRP noise samples.
5. Conclusions In this paper, we have proposed a technique to design sampling schemes with finite sample size N for the weak signal detection in SIRP noise. A member of the class of ASDMs suitable for this problem was developed. This distance measure was obtained by expanding the ASDM between the class conditional densities in terms of a power series and considering the first nonzero term. This objective function is maximized to obtain the sampling scheme for the detection problem under consideration. In the examples, it was demonstrated that this distance measure can be used successfully to design the sampling scheme for the problem under consideration and the sampling designs outperformed the uniform sampling scheme. It was also numerically shown that the sampling design is insensitive to the statistics (the characteristic PDF) of the SIRP noise
The author would like to thank the anonymous reviewers for their constructive comments. The author also expresses his gratitude to Dr. P.K. Varshney for his extremely helpful comments and suggestions. All these comments have significantly strengthened this paper.
References [1] S.M. Ali, S.D. Silvey, A general class of coefficients of divergence of one distribution from another, J. Roy. Statist. Soc. Ser. B 28 (1966) 131–143. [2] R. Bahr, J. Bucklew, Optimal sampling schemes for Gaussian hypothesis testing problem, IEEE Trans. Acoust. Speech Signal Process. 38 (10) (October 1990) 1677–1686. [3] H. Brehm, Description of Spherically Invariant Random Processes by Means of G-functions, Springer Lecture Notes, vol. 969, 1982, pp. 39–73. [4] S. Cambanis, E. Masry, Sampling designs for the detection of signal in noise, IEEE Trans. Inform. Theory IT-29 (1) (January 1983) 83–104. [5] P.R. Chakravarthi, Subclutter visibility: the problem of weak signal detection, Ph.D. Dissertation, EECS Department, Syracuse University, New York, March 1993. [6] H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statist. 23 (1952) 493–507. [7] E. Conte, A. De Maio, C. Galdi, Radar detection of fluctuating targets in compound-Gaussian clutter, The Record of the IEEE 2000 International Radar Conference, 2000, pp. 392–397. [8] E. Conte, A. De Maio, C. Galdi, Signal detection in compound-Gaussian noise: Neyman–Pearson and CFAR detectors, IEEE Trans. Signal Process. 48 (February 2000) pp. 419–428. [9] E. Conte, M. Longo, Characterization of radar clutter as a spherically invariant random process, IEE Proc. F Comm. Radar Signal Process. 134 (1987) 191–197. [10] J. Goldman, Detection in the presence of spherically symmetric random processes, IEEE Trans. Inform. Theory IT-22 (September 1976) 52–58.
ARTICLE IN PRESS 214
C.-T. Yu / Signal Processing 85 (2005) 205–214
[11] N. Johnson, S. Kotz, N. Balkrishnan, Continuous Univariate Distributions, vol. 1, second ed., Wiley, New York, 1994. [12] T. Kailath, The divergence and Bhattacharyya distance measure in signal selection, IEEE Trans. Comm. COM-15 (1) (February 1967) 52–60. [13] K. Kingman, On random sequences with spherical symmetry, Biometrika 59 (1972) 492–494. [14] S. Kullback, Information Theory and Statistics, Wiley, New York, 1960. [15] J.P. LeBlanc, P.L. De Leon, Speech separation by kurtosis maximization, Proceedings of the 1998 IEEE Acoustics Speech and Signal Processing, vol. 2, February 1998, pp. 1029–1032. [16] A. Papoulis, Probability, Random Variables and Stochastic Process, third ed., McGraw-Hill, New York, 1991. [17] M. Rangaswamy, D.D. Weiner, A. Ozturk, Spherically invariant random processes for modeling and distribution identification of non-Gaussian random vectors, IEEE
[18]
[19]
[20]
[21]
[22]
Trans. Aerospace Electronic Systems AES-29 (January 1993) 111–124. K.J. Sangston, K. Gerlach, Results on the detection of signals in spherically invariant random noise, NRL Report, vol. 9202, 1989, pp. 1–12. J. Tang, Z. Zhu, The locally optimum detectors in clutter characterized as SIRP, Proceedings of the IEEE 1995 National Aerospace and Electronics Conference, vol. 1, 1995, pp. 244–251. G. Wise, N. Gallagher, On spherically invariant random processes, IEEE Trans. Inform. Theory IT-24 (1978) 118–120. K. Yao, A representation theorem and its application to spherically invariant random processes, IEEE Trans. Inform. Theory IT-19 (1973) 600–608. C.T. Yu, P.K. Varshney, Sampling design for Gaussian detection problems, IEEE Trans. Signal Process. 45 (9) (1997) 2326–2337.