Detection-Threshold Approximation for Non-Gaussian Backgrounds

Transcription

1 1 Detection-Threshold Approximation for Non-Gaussian Backgrounds D. A. Abraham Published in IEEE Journal of Oceanic Engineering Vol. 35, no. 2, pp. TBD, April 2010 Abstract The detection threshold (DT) term in the sonar equation describes the signal-to-noise ratio (SNR) required to achieve a specified probability of detection (P d ) for a given probability of false alarm (P fa ). Direct evaluation of DT requires obtaining the detector threshold (h) as a function of P fa and then using h while inverting the often complicated relationship between SNR and P d. However, easily evaluated approximations to DT exist when the background additive noise or reverberation is Gaussian (i.e., has a Rayleigh-distributed envelope). While these approximations are extremely accurate for Gaussian backgrounds, they are erroneously low when the background has a heavy-tailed probability density function. In this paper it is shown that by obtaining h appropriately from the non-gaussian background while approximating P d for a target in the non-gaussian background by that for a Gaussian background, the easily evaluated approximations to DT extend to non-gaussian backgrounds with minimal loss in accuracy. Both fluctuating and non-fluctuating targets are considered in Weibulland K-distributed backgrounds. While the P d approximation for fluctuating targets is very accurate, it is coarser for nonfluctuating targets, necessitating a correction factor to the DT approximations. Index Terms sonar equation, detection threshold, non- Gaussian, non-rayleigh, clutter, Weibull distribution, K distribution I. INTRODUCTION In sonar and radar, performance prediction is typically accomplished through a second-order analysis termed the sonar and radar equations [1, Ch. 2], [2, Ch. 4]. It is common to use simple, approximate models for the various terms in the equations rather than highly accurate, but computationally intensive alternatives (e.g., using 10 log r for transmission loss assuming cylindrical spreading where r is range [1, Sect. 5.2]). The fundamental measure of performance is the signal excess (SE), which is the difference (in decibels) between the signalto-noise power ratio (SNR) and the detection threshold (DT). The DT term is the SNR required to achieve a specific operating point on the receiver operating characteristic (ROC) curve in terms of the probability of detection (P d ) and probability of false alarm (P fa ); for example, P d = 0.9 and P fa = In order to obtain DT, it is necessary to first obtain the detector threshold (h) as a function of the desired P fa [see inset box describing the difference between the detection threshold Submitted to IEEE Journal of Oceanic Engineering March 11, 2009, revised November 10, 2009 and February 9, This work was sponsored by the Office of Naval Research under contract numbers N C-0092 and N C and a detector threshold ]. The often complicated function between SNR and P d for the given h is then inverted to find the value of SNR achieving the specified P d. Until the early 1980 s, this process was most easily accomplished by manual evaluation of curves or nomograms relating P fa, P d and DT (e.g., [3] [5]). However, in 1981 Albersheim [6] presented an accurate and easily evaluated approximation to DT. Since then there have been quantifications, extensions and refinements for the various Marcum and Swerling target models in additive Gaussian backgrounds [7] [10]. Despite the advances in desktop computational power since Albersheim first presented his empirically-derived approximation to DT, the simplicity of the approximations continues to make them popular in sonar- and radar-equation-level analysis (e.g., see the recent texts [11, Ch. 6], [12, Ch. 7]) where ease of implementation is favored over extreme accuracy. Such approximations are also imperative for performance prediction in tactical decision aids implemented using in situ measurements [13, pg. 30] [14] where computational resources are limited. In this paper, an approach for extending these approximations to account for non-gaussian backgrounds (equivalently, non-rayleighdistributed envelopes) is described and evaluated. Detection threshold or detector threshold? The detection threshold term in the sonar equation (DT) is the SNR required to achieve a specified P d and P fa. It is a metric used in the analysis of sonar or radar performance. The detector threshold (h), however, is the quantity the detection statistic (T ) must exceed in order for an alarm to be declared and depends only on P fa. It is implicitly used in performance analysis and is necessary to implement the detector. When the additive background noise, interference or reverberation is non-gaussian, the tails of the probability density function (PDF) of the envelope are often heavier than the Rayleigh distribution, leading to a larger P fa than might be expected. In active sonar, this is often termed clutter with many alternative distributions (e.g., Weibull, log-normal, Rayleigh mixture, and the K distributions [15], [16]) capable of representing the departure from the traditionally assumed Rayleigh-distributed matched filter envelope. With the focus of the present analysis on clutter in active sonar systems, consideration is limited to the detector formed by comparing

2 2 the envelope or intensity of a single sonar resolution cell to a threshold (h). The value of h required to achieve the desired P fa in the non-gaussian background is typically larger than that for the Gaussian background. As illustrated in [17], this in turn causes a reduction in P d for a given SNR or, equivalently, an increase in DT to maintain the same ROC curve operating point. To emphasize the potential for error in assuming a Gaussian background when the PDF tails are heavier, the increase in DT for fluctuating and non-fluctuating targets (respectively, FT and NFT) in Weibull- and K-distributed backgrounds is shown in Fig. 1 for P d = 0.9 and P fa = A shape parameter β = 1 for the Weibull distribution or α for the K distribution produce a Rayleigh-distributed envelope and therefore no increase in DT. However, as seen in Fig. 1, DT can be more than 10 db above the value for a Gaussian background for smaller shape parameters where the background is heavily non-gaussian. Fig. 1. Increase in DT over DT for a Gaussian background for FT and NFT in (a) Weibull background and (b) K-distributed background for an operating point of P fa = 10 6 and P d = 0.9. Computing DT for non-gaussian backgrounds requires the same steps as those for Gaussian backgrounds (described in detail in Sect. II), but with the appropriately defined relationships, first between h and P fa and then between SNR, h and P d. Although evaluating P fa for non-gaussian backgrounds is often straightforward, inverting the function for h may require a numerical search. Evaluating P d, however, can be difficult and almost certainly requires a numerical search to obtain DT. However, as shown in Sect. III, the easily evaluated approximations to DT for Gaussian backgrounds (in particular, Hmam s [10] approximation is used because of its accuracy) can be extended to account for non-gaussian backgrounds. The proposed approach obtains h appropriately according to the non-gaussian background but then approximates P d by that for the Gaussian background, which enables use of the easily evaluated approximations. In Sect. IV, the approximation is evaluated for Weibull- and K-distributed backgrounds for the FT and NFT models where the former target model results in impressively accurate approximations while the latter is seen to require a correction factor but still provides reasonable accuracy over a wide range of P fa and P d. II. BACKGROUND Active sonar or radar detection is traditionally accomplished by comparing the envelope output of a matched filter to a threshold. This may be equivalently described as comparing the envelope-squared or intensity output of the matched filter to the square of the threshold. When the received signal is comprised of a target amid a background of noise and reverberation, the matched-filter intensity output can be described as T = Ũ + 2 Aejθ (1) where A and θ represent the target signal s amplitude and phase and Ũ represents the background noise and/or reverberation. The target phase is typically assumed to be uniformly random on (0, 2π). The Marcum target (also known as the NFT) is obtained when A is deterministic while the Swerling Type II target (also known as the FT) arises when A follows a Rayleigh distribution, which is sometimes described as an exponentially distributed cross-section. For Gaussian noise and reverberation, which leads to a Rayleigh-distributed envelope under the background-only case, Ũ is complex-normally distributed with zero mean and variance σ0. 2 The detector is implemented by comparing T to a threshold h. Typically, a normalizer is used to remove any time-varying power level from T ; however, it is not necessary for the purposes of this analysis. The cumulative distribution function (CDF) of T under the background-only or null hypothesis (H 0 ), F T (t H 0 ), provides P fa as a function of the detector threshold h, P fa = 1 F T (h H 0 ). (2) When the background is Gaussian, T follows an exponential distribution with mean σ 2 0 and (2) may be inverted to obtain the detector threshold as a function of P fa, h = σ 2 0 log P fa. (3) The detector threshold is then used to obtain P d using F T (t H 1 ), the CDF of T under the target-plus-background or alternative hypothesis (H 1 ), P d = 1 F T (h H 1 ). (4) For both the FT and NFT, the CDF of T depends on σ 2 0 and the signal-to-background power ratio, S = E[A2 ] σ0 2. (5) Consider, for example, the FT where T is exponentially distributed with mean σ0(1 2 + S) under H 1. In this case, the SNR required to achieve a specified operating point (P fa, P d ) is h 0 S = 1 (6) log P d = log P fa log P d 1 (7)

3 3 where h 0 = h σ0 2 = log P fa (8) is the normalized detector threshold. Converting S to decibels results in the DT term 1 in the sonar equation for active sonar DT = 10 log 10 S (9) [ ] log Pfa = 10 log 10 1 (10) log P d No such closed-form result exists for the NFT; however, several very accurate approximations [6], [8] [10] have been developed to describe S as a function of P d and P fa. The approximations also account for the incoherent integration of n samples (e.g., summing the echoes from multiple pulses or incoherently combining frequency bins in broadband passive sonar detection), although the focus of this paper is on the n = 1 case described by (1). The result of Hmam [10] for the NFT is repeated here as it is utilized in the approach described in Sect. III that accounts for non-gaussian backgrounds. Hmam [10] approximates S for the NFT according to [ ( n ) ] 2 S h sign(0.5 P d )δ ( n ) (11) where σ = and S H ( h 0, P d, n) [ Pd ( P d ) 0.905] (12) δ = log σ. (13) The approximation described by (11) is extremely accurate, with errors less than a tenth of a db over the range P d [, 0.999], P fa [10 12, 10 3 ], and n 20. Hmam also includes an approximation to the normalized threshold h 0 for n > 1; however, in this paper (11) is only used for n = 1 so (8) suffices for computing h 0 from P fa for the Gaussian background. Table I summarizes the relationships between P fa and detector threshold h 0 and P d and P fa or h 0 with detection threshold S for the two target types in a Gaussian background. The equations and approximations developed and evaluated in the Sects. III and IV for Weibull- and K-distributed backgrounds are also shown. III. ACCOUNTING FOR NON-GAUSSIAN BACKGROUNDS Often the detector threshold in an active sonar or radar system is chosen assuming the background has a Rayleighdistributed envelope. In the presence of a heavy-tailed non- Gaussian background, the resulting P fa is larger than expected. To obtain a specified P fa in such a situation, the threshold must be increased compared with (8), which will in turn impact P d. Generalizations to (8) may be derived for non- Gaussian backgrounds through (2) by using the appropriate 1 Note that S = d where d is the detection index. background distribution. As seen in Sect. IV, this can result in simple expressions or require a numerical search to obtain the threshold. Generalizations to (7), (11) and their brethren for DT may be similarly derived through extensions to (4) that account for the non-gaussian background. However, this is a daunting task owing to the difficulty in evaluating (4) for targets in non-gaussian backgrounds. Fortunately, a simpler approach exists in assuming that P d is similar to that obtained when the background is Gaussian. This approach was used in [18] to approximate the performance of an m-of-n fusion processor operating in a background of dependent K-distributed data and seen to be quite accurate, especially for the FT. To understand why this is a reasonable approximation, first consider the real part of the complex matched-filter envelope found in (1), X = Re {Ũ + Ae jθ } = U r + A cos θ (14) where U r is a Gaussian random variable (RV) with zero mean and variance 0.5 (so σ0 2 = 1) and the latter component is constant for the NFT or Gaussian with zero mean and variance equal to S/2 for the FT. Heavier-tailed non-gaussian backgrounds may then be represented as X = V U r + A cos θ (15) where V is a positive random variable with unit mean. When V is gamma distributed with shape α and scale 1/α, the background is K distributed, following from the product form described in [19]. Appropriately choosing the PDF of V will lead to other standard distributions (e.g., a Rayleigh mixture is obtained when V is a discrete RV). The PDF of X under both a Rayleigh background and K- distributed clutter (respectively, (14) and (15)) is shown in Fig. 2 for the FT for 0, 5, and 10 db SNR and a K-distribution shape parameter α = 1, which represents very heavy-tailed clutter. The PDFs for the two backgrounds are very similar, with differences only noticeable in the tails at the lower values of SNR. Based on this, P d for the FT in a K-distributed background should be very similar to that for the Gaussian background, even when the background PDF is significantly different. Accuracy is expected to degrade, however, if P d is so large or small that the PDF is evaluated in the tails. A similar effect is expected for other non-gaussian backgrounds, implying that the SNR required to achieve a specified P d and P fa for the FT may be approximated by simply using (6), S h NG log P d 1, (16) where the normalized intensity threshold ( h NG ) is obtained through a functional inversion of (2) using the CDF of the intensity under H 0, h NG = F 1 T (1 P fa H 0 )/σ 2 0. (17) Examples in the following section describe the process in more detail for both Weibull- and K-distributed clutter in addition to evaluating the accuracy of the approximation.

4 4 TABLE I SUMMARY OF EQUATIONS AND APPROXIMATIONS FOR DETECTOR INTENSITY-THRESHOLD AND REQUIRED SNR (DT) FOR FT AND NFT IN GAUSSIAN- (RAYLEIGH ENVELOPE), WEIBULL-, AND K-DISTRIBUTED BACKGROUNDS. EQUATION NUMBERS ARE INCLUDED FOR EASY REFERENCE TO THE TEXT. Quantity Gaussian/Rayleigh Weibull K Detector threshold h0 = log P fa (8) hw = ( log P fa) β 1 S for FT = log P fa Γ(1+β 1 ) (24) hk h 0 + a 1 h a (7) ( log P fa) β 1 1 (25) h K 1 (34) log P d Γ(1+β 1 ) log P d log P d S for NFT S H ( h0, P d, 1 ) (11) S H ( hw, P d (P d, β), 1 ) (29) S H ( hk, P d (P d, β K ), 1 ) (37) α a 3 h (32) PDF tails increase, SNR decreases, or for very small or large values of P d. As shown in the following section, the structured manner in which the PDFs depart from those for the Gaussian background may be exploited to form a simple correction factor adjusting the value of P d used in (18) and resulting in improved accuracy for heavy-tailed clutter. Fig. 2. PDF of the real part of the complex matched-filter envelope for FT in Gaussian- and K-distributed backgrounds. As noted in [18], the variability of the Rayleigh-distributed A in (15) for the FT dwarfs the additional randomness contributed by V being random, but constrained to have unit mean. Unfortunately, A is constant for the NFT so the variability in V plays a greater role. The PDF of the real part of the complex envelope again illustrates this impact, as shown in Fig. 3 for α = 1, 2, and 10 with an SNR of 20 db and zero phase error (i.e., θ = 0). Noting the form of (15), changing the SNR for the NFT simply shifts the PDFs seen in Fig. 3 to the left or right, leaving the shape unchanged. As expected, when α increases, the K-distribution cases tend toward the Gaussian PDF. However, for small α, the PDF tails are clearly heavier than those of the Gaussian PDF, indicating that P d for the NFT in a non-gaussian background will be somewhat different than in a Gaussian background. Despite this difference, the approximations to S found in [6], [8] [10] may still be utilized to obtain DT when the background is non-gaussian. A first order approximation, using Hmam s result from (11), takes the form ) S S H ( hng, P d, 1 (18) Based on the relationships illustrated in Fig. 3, this approximation should be good for near-rayleigh backgrounds or when the SNR is high, but become increasingly worse as the clutter Fig. 3. PDF of the real part of the complex matched-filter envelope for NFT in Gaussian- and K-distributed backgrounds. A. Evaluating the accuracy of the DT approximations In order to evaluate the approximations to S, the detector threshold must be obtained as a function of P fa by inverting (2) as described in (17). Given the detector threshold, the CDF of T under H 1 is then utilized to find the value of S achieving the desired value of P d. As a result of the complicated form of the CDF for targets in most non-gaussian backgrounds, an iterative search is required. Unfortunately, these CDFs do not usually have simple analytical forms; it is common to see them described as integrals, infinite summations, and even Hankel transforms of characteristic functions [17], [20] [22], [23, Sect. 8.5]. Evaluating P d, can therefore be numerically intensive and at times provide inaccurate results, particularly for the NFT in very heavy-tailed clutter. These issues simply in evaluating P d for common target models in non-gaussian backgrounds emphasize the utility of the approximations to DT developed in this paper.

5 5 To avoid the issues of varying (and potentially unknown) accuracy associated with direct evaluation of P d, S is instead obtained through a Monte-Carlo-integration (i.e., simulation) evaluation of P d [24, Sect. 5.1]. By generating samples T i = V i Ũ i + 2 SÃie jθi (19) for i = 1,..., n s, P d can be approximated by counting the number of threshold crossings and dividing by n s, ˆP d = 1 n s n s i=1 I (T i h ) NG (20) where I(T i > h NG ) is an indicator function returning one when the argument is true and zero otherwise. In (19), V i and Ũ i are generated so that the background is either Weibullor K-distributed but with unit power (i.e., E[V i Ũi 2 ] = 1). The normalized target amplitude is a constant for the NFT, Ã i = 1, and Rayleigh distributed for the FT with unit power, E[ Ãi 2 ] = 1. θ i is uniformly random on (0, 2π). An estimate of DT may be formed by varying S in (19) until the righthand-side of (20) equals the desired P d. Although the resulting quantity is random, it is an unbiased estimate with predictable error variance. The number of simulation trials (n s ) is chosen to limit the error variance in the estimate of DT (call this estimate ŜdB). The variability of ŜdB may be inferred by describing it as a function of ˆP d, which is a binomial random variable divided by n s. Approximating the function by a first-order Taylor series, the standard deviation of ŜdB is approximately Std[ŜdB] g ( ˆP d ) Std[ ˆP d ] g P d (1 P d ) (P d ) (21) n s where g(p d ) relates P d to DT (in db), essentially 10 log 10 of the functional inverse of (4) for S. Setting n s = 10 6 and approximating g(p d ) by 10 log 10 of (16) or (18), the maximum standard deviation of ŜdB over the shape parameter range shown in Table II was less than 0.01 db for the NFT for all cases, except for the combination of P fa = 10 2 and P d = 0.1 where the standard deviation was db. For the FT, the standard deviation was similar in the NFT exception at P d = 0.1. However, owing to the sharpness of g(p d ) for the FT at high P d, the standard deviation rose above 0.01 db, starting at approximately P d = 0.8, to an unacceptably high level. Thus, for values of P d 0.8, n s = 10 7 samples were used for the FT, which kept the error standard deviation below 0.01 db except for P d = 0.99 for which the standard deviation of the error rose to db. The results were nearly identical for both the Weibull- and K-distributed backgrounds. With n s 10 6, the P d estimate is essentially Gaussian, so it is very unlikely that the errors would rise above three standard deviations. Thus, the simulation approach is expected to provide results with accuracy better than one twentieth of a db in all of the cases articulated in Table II. IV. EXAMPLES In this section, the accuracy of the approximations to S described by (16) and (18) is evaluated for the FT and NFT in Weibull- and K-distributed backgrounds. The range of P fa and P d evaluated are found in Table II, along with the range of shape parameters for the Weibull and K distributions. Note that extremely low values of shape parameter (i.e., very heavytailed background PDFs) are evaluated. Although there may be some doubt as to the physical realism of such heavy-tailed data, the intent in this paper is to determine the extent to which the proposed approximations to DT are accurate. TABLE II PARAMETER RANGES FOR EVALUATING S. Parameter Range P fa to 10 2 P d 0.1 to 0.99 Weibull shape β [0.25, 1] K shape α [0.1, 100] Simulation trials: FT for P d 0.8 n s = 10 7 all other cases n s = 10 6 A. Weibull-distributed background The matched-filter-intensity PDF and CDF under H 0 for a Weibull-distributed background are and f T (t H 0 ) = βtβ 1 λ β e (t/λ) β (22) F T (t H 0 ) = 1 e (t/λ)β (23) where β is the shape parameter and λ is the scale parameter. Since the Weibull distribution is closed under a power-law transformation (i.e., if W is Weibull distributed, then so will be W p ), the matched-filter envelope is also Weibull distributed [21] but with a shape parameter equal to 2β. Noting that the mean of the Weibull distribution is λ/γ(1+ 1/β), the normalized detector threshold is easily obtained from (23) as a function of P fa and β, h W = ( log P fa) β 1 Γ (1 + β 1 ). (24) 1) Fluctuating target: Using (24) in (16) results in the following approximation for a FT in a Weibull-distributed background, S ( log P fa) β 1 Γ (1 + β 1 1 = ) log P S. (25) d To evaluate the accuracy of the approximation, consider the maximum absolute error in decibels between the approximation of (25) and that obtained from the simulation analysis, ɛ db = max P d,β ŜdB S db (26) where S db = 10 log 10 S and the maximization occurs over all the combinations of P d and β evaluated or as otherwise

6 6 specified. Table III displays ɛ db for each of the P fa levels evaluated. The approximation of (25) is surprisingly accurate with errors less than 0.1 db for all values of P fa except As might be expected, the maximum error for P fa = 10 2 occurred for P d = 0.1 and β = For the smaller values of P fa, the maximum error occurred for P d = An example set of DT curves is shown in Fig. 4 for P fa = 10 6 and various values of P d. On this scale, it is not possible to see the difference between the approximation and the simulated result. TABLE III MAXIMUM ABSOLUTE ERROR IN DT APPROXIMATION (ɛ db ) FOR FT AND NFT MODELS IN WEIBULL-DISTRIBUTED BACKGROUND OVER β [0.25, 1] AND P d [0.1, 0.99]. P fa FT NFT no corr. w/corr. 1e e e e e e e e e Fig. 4. DT for FT in Weibull-distributed background for P fa = 10 6 and various values of P d. The DT approximation is nearly identical to the simulation results. 2) Non-fluctuating target: Direct application of Hmam s DT approximation for the NFT using the Weibull-background detector threshold (i.e., (18) using (24)) achieves a reasonable accuracy for smaller values of P fa, as seen in Table III (column labeled no corr. under NFT). The smallest errors are around one quarter db and occur for the smallest P fa while the errors range above one decibel at the largest P fa. The maximum error typically occurred at either P d = 0.1 (larger values of P fa ) or at P d = 0.99 (smaller values of P fa ) with the errors generally smaller away from these boundaries. For example, for P d = 0.5, the errors were nearly identical to those reported for the FT. The magnitude of these errors compared with that achieved for the FT and the structured manner in which P d for the NFT varies when progressing from a Gaussian background to a heavier-tailed background motivates the development of a correction factor to improve the accuracy of the approximation. An examination of Fig. 3, illustrates that P d for a non- Gaussian background could be above or below that for a Gaussian background with the same SNR, depending on the threshold. This has an inverse effect on DT; that is, when P d is slightly higher for the non-gaussian background, the DT will be slightly smaller than (18) using (24). However, the disparity should disappear as β 1 and may be small near thresholds producing P d = 0.5. Empirically, it was determined that by modifying the value of P d used in (18) according to P d(p d, β) = P βc(p d ) d (27) where the exponent on β is { } 1 c(p d ) = [0.5 P d ] min 6P d + 0.2, 2.6, 18(1 P d ) (28) the approximations, now described by ) S S H ( hw, P d (P d, β), 1 (29) could be significantly improved. From the form of (27) and (28), it is clear that no correction is applied (i.e., P d (P d, β) = P d ) when β = 1 or P d = 0.5. As seen in Table III, the errors with the correction of (27) (29) are less than half of those without the correction, ranging down to one tenth of a decibel at the lowest P fa. As seen in the example set of DT curves found in Fig. 5 for P fa = 10 6 and various values of P d, the errors are greatest for the smallest and largest values of P d. However, in a wide region between these extremes, the errors are even smaller; for example, for P fa 10 3 and P d [0.2, 0.9], the errors are all less than one tenth of a decibel. The maximum error over P d is shown in Fig. 6 as a function of β for various values of P fa. As expected, the errors approach the simulation resolution as β 1, but can also be seen to decrease somewhat as β decreases below about 0.5. Without the correction factor, these errors would rise as β decreases. With or without the correction, the approximation accuracy improves as P fa decreases. This arises from the resulting higher values of detector threshold and therefore S. In this regime, the improved fit of the P d approximation (i.e., assuming a Gaussian background) indicates that the approximations may retain accuracy for P fa < However, the same is clearly not true for larger values of P fa. B. K-distributed background The matched-filter-intensity PDF and CDF under H 0 for a K-distributed background are f T (t H 0 ) = 2 ( ) α 1 ) t 2 t K α 1 (2 (30) λγ(α) λ λ

7 7 Fig. 5. DT for NFT in Weibull-distributed background for Pf a = 10 6 and various values of Pd. Fig. 6. Maximum absolute error in DT approximation ( db (β)) for the NFT in Weibull-distributed background over Pd as a function of β for various Pf a. and 2 FT (t H0 ) = 1 Γ(α) α2 t Kα λ r! t 2 λ a1 h a αa3 h 0 Fig. 7. Detector threshold for K-distributed background compared with the approximation using first two rows of Table IV as a function of α for various Pf a. (31) where α is the shape parameter and λ is the scale parameter. The normalized detector threshold is obtained by setting λ = σ02 /α and using (31) in (17). In contrast to the Weibull background, there is no closed-form solution for the detector threshold in a K-distributed background, which forces the use of numerical routines. However, in the spirit of Hmam s [10] development of approximations to the detector threshold for the Gaussian background with incoherent integration of multiple samples, an approximation of the form h K h 0 + is proposed for a K-distributed background with the parameters a1, a2 and a3 chosen according to Pf a as found in the first two rows of Table IV. Equation (32) was derived empirically and is seen to (i) be greater than the intensity threshold for a Rayleigh-distributed envelope (h 0 ) when a1 > 0, illustrating the increase necessary to account for the heavier tails in the K distribution, and (ii) tend to h 0 when a3 > 0 and α, where the K distribution simplifies to the Rayleigh. The accuracy of the approximation over a wide range of α and Pf a is shown graphically in Fig. 7 where the coefficients found in the first two rows of Table IV are used. The values of the parameters shown in Table IV were obtained by minimizing the maximum error in the specified region of Pf a (evaluating each order of magnitude) and for α [0.2, 100]. The maximum error over each region as listed in the table is less than 0.2 db. As seen in Fig. 8, the error is also less than 0.2 db when α [0.1, 0.2] except for Pf a = 10 2 and 10 5 where the approximation fails below α = 0.2. Clearly, using the approximation of (32) outside of the regions in which it has been evaluated could result in larger errors than those reported herein. (32) TABLE IV PARAMETERS FOR K- DISTRIBUTION DETECTOR THRESHOLD APPROXIMATION. Pf a range a1 a2 a3 Maximum error (db) when α 0.2 1e-2 to 1e-5 1e-6 to 1e-10 1e When greater accuracy is necessary, parameters can be found for (32) that minimize the error for a specific Pf a. The parameters for Pf a = 10 6 are shown in Table IV as an example. The parameters optimizing each order of magnitude

8 8 over the full range of Pf a considered are listed in the Appendix. For example, as seen in Fig. 8, restricting application to one Pf a can reduce the error to less than one twentieth of a decibel over α [0.2, 100]. Such an approximation is useful when implementing an adaptive detector threshold (similar to a statistical normalizer [25], [26]) to approximate a constant false-alarm rate; that is, obtaining a local estimate of α and increasing the detector threshold above h 0 according to α using (32). The approximation of (32) is also useful to start a Newton-Raphson iteration to increase the accuracy for a given Pf a and α, FT σ02 h K H0 (1 Pf a ). (33) h K := h K ft σ02 h K H0 For all of the cases considered, starting (33) with (32) resulted in six decimal places of accuracy (for the threshold in db) within five iterations. An example set of DT curves is shown in Fig. 9 for Pf a = 10 6 and various values of Pd where the approximation to DT uses the exact threshold. Similar to the Weibull-background results, it is not possible to see the difference between the approximation and the simulated result on this scale. TABLE V M AXIMUM ABSOLUTE ERROR IN DT APPROXIMATION ( db ) FOR FT MODEL IN K- DISTRIBUTED BACKGROUND OVER α [0.1, 100] AND Pd [0.1, 0.99]. Pf a 1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 1e-10 Exact h K 0.06 Approximate h K Fig. 8. Error in the detector-threshold approximation for K-distributed background as a function of α for various Pf a. 1) Fluctuating target: For the FT, (16) approximates DT where the normalized threshold is obtained from (32) as an approximation or starting with (32) and iterating (33) for the precise result, h K S 1. (34) log Pd The maximum error of the DT approximation over Pd and α (i.e., (26) where the maximization is over α instead of β) is found in Table V as a function of Pf a where it is seen to be approximately one twentieth of a decibel for the exact threshold and, excepting Pf a = 10 2, between one tenth and one quarter of a decibel for the approximate h K. When Pd 0.97 and Pf a 10 3, the maximum error for the exact threshold drops to db. Note that restricting the error evaluation to α 0.2 for Pf a = 10 2 when using the approximate h K, as might be suggested by the above analysis, more than halves the error from 0.72 to Fig. 9. DT for FT in K-distributed background for Pf a = 10 6 and various values of Pd. The DT approximation is nearly identical to the simulation results. 2) Non-fluctuating target: Direct application of Hmam s DT approximation for the K-distributed background (i.e., (18) using (32) or (33)) leads to similar errors to the Weibullbackground. As seen in Table VI, without a correction the errors are over one decibel for the largest Pf a and range down to one quarter of a decibel for the smallest Pf a. Ideally, the correction factor would be similar to that used for the Weibull distribution. A reasonable approach is to obtain a value of the Weibull shape parameter (call it βk ) that produces a PDF similar to the K PDF for a given α. By equating the first two intensity moments of the distributions, it is possible to relate βk to α according to α= 2Γ2 (1 + 1/βK ). Γ(1 + 2/βK ) 2Γ2 (1 + 1/βK ) (35)

9 9 Inverting (35) to obtain βk as a function of α clearly requires a numerical solution or table look-up. However, owing to the smooth nature of the Pd -correction found in (27) with respect to β and the secondary nature of the correction, the following approximation was found to be adequate over the range of shape parameters being considered, βk = e [log(500/α)] 4 /1850. (36) The approximation has less than a 5% error on α [0.1, 100] and less than 1% when α DT for the NFT in a K-distributed background is then approximated by using (36) in (29), S SH h K, Pd0 (Pd, βk ), 1. (37) Using the correction factor of (27) with the exact detector threshold, as seen in Table VI, results in a similar reduction in error to the Weibull case: the errors are halved for all values of Pf a considered and range from 0.5 db at the highest Pf a down to approximately one tenth of a decibel at the smallest Pf a. Although the errors decrease with Pf a when the exact threshold is used, the approximate threshold of (32) results in a slight oscillation in the errors. The errors for Pd = 0.99 generally exceed those for smaller values; as previously noted, the approximations are expected to lose accuracy at such high values of Pd. The example DT curves found in Fig. 10, which utilize the exact detector threshold and the correction noted in (37), illustrate that the approximations degrade as Pd tends toward high or low values and as α decreases, especially below one. The maximum errors over Pd seen in Fig. 11 as a function of α and Pf a are similar to those for the Weibull distribution; however, the reduction in error observed for very heavy tailed Weibull backgrounds seems to be just starting at α = 0.1. Fig. 10. DT for NFT in K-distributed background for Pf a = 10 6 and various values of Pd. TABLE VI M AXIMUM ABSOLUTE ERROR IN DT APPROXIMATION ( db ) FOR NFT MODEL IN K- DISTRIBUTED BACKGROUND OVER α [0.1, 100] AND Pd [0.1, 0.99]. Pf a 1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 1e-10 Exact h K no corr. w/corr Approximate h K no corr. w/corr Fig. 11. Maximum absolute error in DT approximation ( db (α)) for the NFT in K-distributed background over all values of Pd as a function of α for various Pf a. V. C ONCLUSIONS An approach was described to approximate the DT term in the sonar or radar equation (i.e., the SNR required to achieve a specified Pd for a given Pf a ) for non-gaussian backgrounds. By approximating Pd for targets in non-gaussian backgrounds by that for Gaussian backgrounds, the DT approximations developed for Gaussian backgrounds may be used along with an appropriately derived detector threshold to approximate DT in a non-gaussian background. The approximations, as summarized in Table I, were seen to be easily evaluated and quite accurate over a wide span of Pf a, Pd, and for Weibull- and K-distributed backgrounds ranging from nearrayleigh envelope distributions to very heavy-tailed PDFs. The maximum errors (over the evaluated range of Pd and

10 10 background shape parameter) for fluctuating targets were approximately one-twentieth of a decibel, while non-fluctuating targets required a correction factor to produce coarser, but still adequate, approximations with errors ranging from 0.5 db at the largest value of P fa = 10 2 down to one tenth of a decibel at the smallest P fa = Although the approximation accuracy for the NFT is somewhat worse than that for the FT, it is certainly adequate for the intended use as a theoretical or hypothetical target model representing benchmark performance for consistent (i.e., deterministic) target echoes in sonar-equation analysis, particularly in light of the errors common in other terms in the sonar equation (e.g., see [27] where reverberation modeling errors dwarf those illustrated here). An approximation to the detector threshold for the K distribution was proposed, providing errors less than one fifth of a decibel over a wide range of P fa and α. These results will be useful in predicting the performance of sonar or radar systems when the background is non- Gaussian. The idea of approximating the PDF of a target in a non-gaussian background by that for a Gaussian background may be particularly useful in areas such as target tracking, classification or image segmentation. While the present results for the K and Weibull distributions cover a wide range of non-gaussian background PDFs, the basic DT approximations should extend to other non-gaussian background distributions. The NFT correction factor, which halved the maximum errors (in db), would then most easily be developed by using that presented for the Weibull background where an equivalent Weibull shape parameter is obtained via moment matching, as illustrated for the K distribution. As the present analysis is limited to the detector formed by comparing a single sonar resolution cell to a threshold, it is most appropriate for use in active sonar systems. However, noting the efficacy of the approach as it was applied to the analysis of an m-of-n fusion processor operating on dependent K-distributed data [18], these results are expected to extend to detectors incorporating incoherent integration (e.g., postmatched-filter integration in active sonar or broadband energy detection in passive sonar). The primary difficulty, however, lies in accurate evaluation of the true values of both detector threshold and DT. Of course any extrapolation beyond the conditions evaluated in this paper requires an appropriate error analysis. ACKNOWLEDGMENT The author thanks Dr. J. R. Preston (ARL/PSU) for pointing out the work of Pryor [5] and the reviewers for their constructive comments. REFERENCES [1] R. J. Urick, Principles of Underwater Sound. New York: McGraw-Hill, Inc., [2] P. Z. Peebles, Jr., Radar Principles. New York: John Wiley & Sons, Inc., [3] D. E. Bailey and N. C. Randall, Nomogram determines probability of detecting signals in noise, Electronics, p. 66, March [4] G. H. Robertson, Operating characteristic for a linear detector, Bell System Technical Journal, pp , April [5] C. N. Pryor, Calculation of the minimum detectable signal for practical spectrum analyzers, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland, Tech. Rpt , August [6] W. J. Albersheim, A closed-form approximation to Robertson s detection characteristics, Proceedings of the IEEE, vol. 69, no. 7, p. 839, [7] D. W. Tufts and A. J. Cann, On Albersheim s detection equation, IEEE Transactions Aerospace and Electronic Systems, vol. AES-19, no. 4, pp , [8] D. A. Shnidman, Determination of required SNR values, IEEE Transactions on Aerospace and Electronic System, vol. 38, no. 3, pp , [9] H. Hmam, Approximating the SNR value in detection problems, IEEE Transactions Aerospace and Electronic Systems, vol. 39, no. 4, pp , [10], SNR calculation procedure for target types 0, 1, 2, 3, IEEE Transactions on Aerospace and Electronic System, vol. 41, no. 3, pp , [11] M. A. Richards, Fundamentals of Radar Signal Processing. New York: McGraw-Hill, [12] M. A. Ainslie, Principles of Sonar Performance Modelling. New York: Springer, [13] Ocean Studies Board Commission on Geosciences, Environment, and Resources, National Research Council, Oceanography and Mine Warfare. Washington, D.C.: National Academy Press, [Online]. Available: [14] W. E. Brown and M. L. Barlett, Midfrequency through-the-sensor scattering measurements: A new approach, IEEE Journal of Oceanic Engineering, vol. 30, no. 4, pp , [15] M. Gensane, A statistical study of acoustic signals backscattered from the sea bottom, IEEE Journal of Oceanic Engineering, vol. 14, no. 1, pp , January [16] A. P. Lyons and D. A. Abraham, Statistical characterization of highfrequency shallow-water seafloor backscatter, Journal of the Acoustical Society of America, vol. 106, no. 3, pp , September [17] D. A. Abraham, Signal excess in K-distributed reverberation, IEEE Journal of Oceanic Engineering, vol. 28, no. 3, pp , July [18], Distributed active sonar detection in dependent K-distributed clutter, IEEE Journal of Oceanic Engineering, vol. 34, no. 3, pp , July [19] K. D. Ward, Compound representation of high resolution sea clutter, Electronics Letters, vol. 17, no. 16, pp , August [20] G. V. Trunk, Non-Rayleigh sea clutter: Properties and detection of targets, Naval Research Laboratory, Report 7986, 1976, reprinted in Automatic Detection and Radar Data Processing, D. C. Schleher, Ed., Artech House, Dedham, [21] D. C. Schleher, Radar detection in Weibull clutter, IEEE Transactions on Aerospace and Electronic Systems, vol. AES-12, no. 6, pp , [22] D. M. Drumheller, Padé approximations to matched filter amplitude probability functions, IEEE Transactions on Aerospace and Electronic Systems, vol. 35, no. 3, pp , July [23] K. D. Ward, R. J. A. Tough, and S. Watts, Sea Clutter: Scattering, the K Distribution and Radar Performance. London: The Institution of Engineering and Technology, [24] B. D. Ripley, Stochastic Simulation. New York: John Wiley & Sons, [25] D. A. Abraham, Statistical normalization of non-rayleigh reverberation, in Proceedings of Oceans 97 Conference, Halifax, Nova Scotia, October 1997, pp [26] T. J. Barnard and F. Khan, Statistical normalization of spherically invariant non-gaussian clutter, IEEE Journal of Oceanic Engineering, vol. 29, no. 2, pp , April [27] C. W. Holland, Fitting data, but poor predictions: Reverberation prediction uncertainty when seabed parameters are derived from reverberation measurements, The Journal of the Acoustical Society of America, vol. 123, no. 5, pp , May APPENDIX Parameters for approximating the detector threshold for a K- distributed background The approximation to the intensity threshold for a K- distributed background described in (32) as a function of P fa and α provides an easily evaluated alternative to numerically

11 11 inverting the K-distribution intensity CDF. Parameters for (32) providing minimal error for specific values of P fa are listed in Table VII. By articulating the parameters for a specific P fa, the error can be limited to less than one twentieth of a decibel for P fa 10 6 and less than one tenth of decibel for P fa TABLE VII PARAMETERS FOR K-DISTRIBUTION DETECTOR THRESHOLD APPROXIMATION. P fa a 1 a 2 a 3 Maximum error (db) when α 0.2 1e e e e e e e e e