1

Detection-Threshold Approximation forNon-Gaussian Backgrounds

D. A. Abraham

Published in IEEE Journal of Oceanic EngineeringVol. 35, no. 2, pp. TBD, April 2010

Abstract— The detection threshold (DT) term in the sonarequation describes the signal-to-noise ratio (SNR) required toachieve a specified probability of detection (Pd) for a givenprobability of false alarm (Pfa). Direct evaluation of DT re-quires obtaining the detector threshold (h) as a function ofPfa and then using h while inverting the often complicatedrelationship between SNR and Pd. However, easily evaluatedapproximations to DT exist when the background additive noiseor reverberation is Gaussian (i.e., has a Rayleigh-distributedenvelope). While these approximations are extremely accuratefor Gaussian backgrounds, they are erroneously low when thebackground has a heavy-tailed probability density function. Inthis paper it is shown that by obtaining h appropriately fromthe non-Gaussian background while approximating Pd for atarget in the non-Gaussian background by that for a Gaussianbackground, the easily evaluated approximations to DT extend tonon-Gaussian backgrounds with minimal loss in accuracy. Bothfluctuating and non-fluctuating targets are considered in Weibull-and K-distributed backgrounds. While the Pd approximationfor fluctuating targets is very accurate, it is coarser for non-fluctuating targets, necessitating a correction factor to the DTapproximations.

Index Terms— sonar equation, detection threshold, non-Gaussian, non-Rayleigh, clutter, Weibull distribution, K distri-bution

I. INTRODUCTION

In sonar and radar, performance prediction is typicallyaccomplished through a second-order analysis termed thesonar and radar equations [1, Ch. 2], [2, Ch. 4]. It is commonto use simple, approximate models for the various terms inthe equations rather than highly accurate, but computationallyintensive alternatives (e.g., using 10 log r for transmission lossassuming 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 signal-to-noise power ratio (SNR) and the detection threshold (DT).The DT term is the SNR required to achieve a specific operat-ing point on the receiver operating characteristic (ROC) curvein terms of the probability of detection (Pd) and probability offalse alarm (Pfa); for example, Pd = 0.9 and Pfa = 10−6. Inorder to obtain DT, it is necessary to first obtain the detectorthreshold (h) as a function of the desired Pfa [see inset boxdescribing the difference between the “detection threshold”

Submitted to IEEE Journal of Oceanic Engineering March 11, 2009, revisedNovember 10, 2009 and February 9, 2010. This work was sponsored by theOffice of Naval Research under contract numbers N00014-07-C-0092 andN00014-09-C-0318.

and a “detector threshold”]. The often complicated functionbetween SNR and Pd for the given h is then inverted to findthe value of SNR achieving the specified Pd. Until the early1980’s, this process was most easily accomplished by manualevaluation of curves or nomograms relating Pfa, Pd and DT(e.g., [3]–[5]). However, in 1981 Albersheim [6] presented anaccurate and easily evaluated approximation to DT. Since thenthere have been quantifications, extensions and refinementsfor the various Marcum and Swerling target models in addi-tive Gaussian backgrounds [7]–[10]. Despite the advances indesktop computational power since Albersheim first presentedhis empirically-derived approximation to DT, the simplicityof the approximations continues to make them popular insonar- and radar-equation-level analysis (e.g., see the recenttexts [11, Ch. 6], [12, Ch. 7]) where ease of implementationis favored over extreme accuracy. Such approximations arealso imperative for performance prediction in tactical decisionaids implemented using in situ measurements [13, pg. 30][14] where computational resources are limited. In this paper,an approach for extending these approximations to accountfor non-Gaussian backgrounds (equivalently, non-Rayleigh-distributed envelopes) is described and evaluated.

Detection threshold or detector threshold? Thedetection threshold term in the sonar equation (DT) isthe SNR required to achieve a specified Pd and Pfa.It is a metric used in the analysis of sonar or radarperformance. The detector threshold (h), however, isthe quantity the detection statistic (T ) must exceed inorder for an alarm to be declared and depends onlyon Pfa. It is implicitly used in performance analysisand is necessary to implement the detector.

When the additive background noise, interference or rever-beration is non-Gaussian, the tails of the probability densityfunction (PDF) of the envelope are often heavier than theRayleigh distribution, leading to a larger Pfa than mightbe expected. In active sonar, this is often termed clutterwith many alternative distributions (e.g., Weibull, log-normal,Rayleigh mixture, and the K distributions [15], [16]) capableof representing the departure from the traditionally assumedRayleigh-distributed matched filter envelope. With the focusof the present analysis on clutter in active sonar systems,consideration is limited to the detector formed by comparing

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the envelope or intensity of a single sonar resolution cell to athreshold (h). The value of h required to achieve the desiredPfa in the non-Gaussian background is typically larger thanthat for the Gaussian background. As illustrated in [17], this inturn causes a reduction in Pd for a given SNR or, equivalently,an increase in DT to maintain the same ROC curve operatingpoint. To emphasize the potential for error in assuming aGaussian background when the PDF tails are heavier, theincrease in DT for fluctuating and non-fluctuating targets(respectively, FT and NFT) in Weibull- and K-distributedbackgrounds is shown in Fig. 1 for Pd = 0.9 and Pfa = 10−6.A shape parameter β = 1 for the Weibull distribution orα→∞ for the K distribution produce a Rayleigh-distributedenvelope and therefore no increase in DT. However, as seenin Fig. 1, DT can be more than 10 dB above the value for aGaussian background for smaller shape parameters where thebackground is heavily non-Gaussian.

Fig. 1. Increase in DT over DT for a Gaussian background for FT and NFTin (a) Weibull background and (b) K-distributed background for an operatingpoint of Pfa = 10−6 and Pd = 0.9.

Computing DT for non-Gaussian backgrounds requires thesame steps as those for Gaussian backgrounds (described indetail in Sect. II), but with the appropriately defined relation-ships, first between h and Pfa and then between SNR, h andPd. Although evaluating Pfa for non-Gaussian backgroundsis often straightforward, inverting the function for h mayrequire a numerical search. Evaluating Pd, however, can bedifficult and almost certainly requires a numerical search toobtain DT. However, as shown in Sect. III, the easily evaluatedapproximations 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 tothe non-Gaussian background but then approximates Pd bythat for the Gaussian background, which enables use of theeasily evaluated approximations. In Sect. IV, the approxima-tion is evaluated for Weibull- and K-distributed backgroundsfor the FT and NFT models where the former target modelresults in impressively accurate approximations while the

latter is seen to require a correction factor but still providesreasonable accuracy over a wide range of Pfa and Pd.

II. BACKGROUND

Active sonar or radar detection is traditionally accomplishedby comparing the envelope output of a matched filter to athreshold. This may be equivalently described as comparingthe envelope-squared or intensity output of the matched filterto the square of the threshold. When the received signal iscomprised of a target amid a background of noise and rever-beration, the matched-filter intensity output can be describedas

T =∣∣∣U +Aejθ

∣∣∣2 (1)

where A and θ represent the target signal’s amplitude andphase and U represents the background noise and/or reverber-ation. The target phase is typically assumed to be uniformlyrandom on (0, 2π). The Marcum target (also known as theNFT) is obtained when A is deterministic while the SwerlingType II target (also known as the FT) arises when A followsa Rayleigh distribution, which is sometimes described as anexponentially distributed cross-section. For Gaussian noise andreverberation, which leads to a Rayleigh-distributed envelopeunder the background-only case, U is complex-normally dis-tributed with zero mean and variance σ2

0 .The detector is implemented by comparing T to a threshold

h. Typically, a normalizer is used to remove any time-varyingpower level from T ; however, it is not necessary for thepurposes of this analysis. The cumulative distribution function(CDF) of T under the background-only or null hypothesis(H0), FT (t|H0), provides Pfa as a function of the detectorthreshold h,

Pfa = 1− FT (h|H0). (2)

When the background is Gaussian, T follows an exponentialdistribution with mean σ2

0 and (2) may be inverted to obtainthe detector threshold as a function of Pfa,

h = −σ20 logPfa. (3)

The detector threshold is then used to obtain Pd usingFT (t|H1), the CDF of T under the target-plus-backgroundor alternative hypothesis (H1),

Pd = 1− FT (h|H1). (4)

For both the FT and NFT, the CDF of T depends on σ20

and the signal-to-background power ratio,

S =E[A2]σ2

0

. (5)

Consider, for example, the FT where T is exponentiallydistributed with mean σ2

0(1 + S) under H1. In this case, theSNR required to achieve a specified operating point (Pfa, Pd)is

S =h0

− logPd− 1 (6)

=logPfalogPd

− 1 (7)

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whereh0 =

h

σ20

= − logPfa (8)

is the normalized detector threshold. Converting S to decibelsresults in the DT term1 in the sonar equation for active sonar

DT = 10 log10 S (9)

= 10 log10

[logPfalogPd

− 1]

(10)

No such closed-form result exists for the NFT; however,several very accurate approximations [6], [8]–[10] have beendeveloped to describe S as a function of Pd and Pfa. Theapproximations also account for the incoherent integration ofn samples (e.g., summing the echoes from multiple pulses orincoherently combining frequency bins in broadband passivesonar detection), although the focus of this paper is on the n =1 case described by (1). The result of Hmam [10] for the NFTis repeated here as it is utilized in the approach described inSect. III that accounts for non-Gaussian backgrounds. Hmam[10] approximates S for the NFT according to

S ≈[√

h0 −(n

2− 0.25

)− sign(0.5− Pd)δ

]2−(n

2− 0.25

)(11)

≡ SH(h0, Pd, n)

where

σ =1

0.19

[√0.819025 + 1.5206Pd(0.9998− Pd)− 0.905

](12)

andδ =

√−0.85616 log σ. (13)

The approximation described by (11) is extremely accurate,with errors less than a tenth of a dB over the range Pd ∈[0.05, 0.999], Pfa ∈ [10−12, 10−3], and n ≤ 20. Hmam alsoincludes an approximation to the normalized threshold h0 forn > 1; however, in this paper (11) is only used for n = 1so (8) suffices for computing h0 from Pfa for the Gaussianbackground.

Table I summarizes the relationships between Pfa anddetector threshold h0 and Pd and Pfa or h0 with detectionthreshold S for the two target types in a Gaussian background.The equations and approximations developed and evaluatedin the Sects. III and IV for Weibull- and K-distributed back-grounds are also shown.

III. ACCOUNTING FOR NON-GAUSSIAN BACKGROUNDS

Often the detector threshold in an active sonar or radarsystem is chosen assuming the background has a Rayleigh-distributed envelope. In the presence of a heavy-tailed non-Gaussian background, the resulting Pfa is larger than ex-pected. To obtain a specified Pfa in such a situation, thethreshold must be increased compared with (8), which will inturn impact Pd. Generalizations to (8) may be derived for non-Gaussian backgrounds through (2) by using the appropriate

1Note that S =√d where d is the detection index.

background distribution. As seen in Sect. IV, this can resultin simple expressions or require a numerical search to obtainthe threshold. Generalizations to (7), (11) and their brethrenfor DT may be similarly derived through extensions to (4)that account for the non-Gaussian background. However, thisis a daunting task owing to the difficulty in evaluating (4) fortargets in non-Gaussian backgrounds.

Fortunately, a simpler approach exists in assuming that Pd issimilar to that obtained when the background is Gaussian. Thisapproach was used in [18] to approximate the performanceof an m-of-n fusion processor operating in a background ofdependent K-distributed data and seen to be quite accurate,especially for the FT. To understand why this is a reasonableapproximation, first consider the real part of the complexmatched-filter envelope found in (1),

X = <e{U +Aejθ

}= Ur +A cos θ (14)

where Ur is a Gaussian random variable (RV) with zero meanand variance 0.5 (so σ2

0 = 1) and the latter component isconstant for the NFT or Gaussian with zero mean and varianceequal to S/2 for the FT.

Heavier-tailed non-Gaussian backgrounds may then be rep-resented as

X =√V Ur +A cos θ (15)

where V is a positive random variable with unit mean. WhenV is gamma distributed with shape α and scale 1/α, thebackground is K distributed, following from the product formdescribed in [19]. Appropriately choosing the PDF of V willlead to other standard distributions (e.g., a Rayleigh mixtureis 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-distributionshape parameter α = 1, which represents very heavy-tailedclutter. The PDFs for the two backgrounds are very similar,with differences only noticeable in the tails at the lower valuesof SNR. Based on this, Pd for the FT in a K-distributedbackground should be very similar to that for the Gaussianbackground, even when the background PDF is significantlydifferent. Accuracy is expected to degrade, however, if Pd isso large or small that the PDF is evaluated in the tails. Asimilar effect is expected for other non-Gaussian backgrounds,implying that the SNR required to achieve a specified Pd andPfa for the FT may be approximated by simply using (6),

S ≈ hNG

− logPd− 1, (16)

where the normalized intensity threshold (hNG) is obtainedthrough a functional inversion of (2) using the CDF of theintensity under H0,

hNG = F−1T (1− Pfa|H0)/σ2

0 . (17)

Examples in the following section describe the process in moredetail for both Weibull- and K-distributed clutter in additionto evaluating the accuracy of the approximation.

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TABLE ISUMMARY 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 = − logPfa (8) hW =(− logPfa)

β−1

Γ(1+β−1)(24) hK ≈ h0 +

a1ha20

αa3h

0.20

(32)

S for FT =logPfalogPd

− 1 (7) ≈ (− logPfa)β−1

−Γ(1+β−1) logPd− 1 (25) ≈ hK

− logPd− 1 (34)

S for NFT ≈ SH(h0, Pd, 1

)(11) ≈ SH

(hW , P ′d (Pd, β) , 1

)(29) ≈ SH

(hK , P

′d (Pd, βK) , 1

)(37)

Fig. 2. PDF of the real part of the complex matched-filter envelope for FTin Gaussian- and K-distributed backgrounds.

As noted in [18], the variability of the Rayleigh-distributedA in (15) for the FT dwarfs the additional randomness con-tributed by V being random, but constrained to have unit mean.Unfortunately, A is constant for the NFT so the variability inV plays a greater role. The PDF of the real part of the complexenvelope again illustrates this impact, as shown in Fig. 3 forα = 1, 2, and 10 with an SNR of 20 dB and zero phaseerror (i.e., θ = 0). Noting the form of (15), changing the SNRfor the NFT simply shifts the PDFs seen in Fig. 3 to the leftor right, leaving the shape unchanged. As expected, when αincreases, the K-distribution cases tend toward the GaussianPDF. However, for small α, the PDF tails are clearly heavierthan those of the Gaussian PDF, indicating that Pd for theNFT in a non-Gaussian background will be somewhat differentthan in a Gaussian background. Despite this difference, theapproximations to S found in [6], [8]–[10] may still be utilizedto obtain DT when the background is non-Gaussian. A firstorder approximation, using Hmam’s result from (11), takes theform

S ≈ SH

(hNG, Pd, 1

)(18)

Based on the relationships illustrated in Fig. 3, this approxima-tion should be good for near-Rayleigh backgrounds or whenthe SNR is high, but become increasingly worse as the clutter

PDF tails increase, SNR decreases, or for very small or largevalues of Pd. As shown in the following section, the structuredmanner in which the PDFs depart from those for the Gaussianbackground may be exploited to form a simple correctionfactor adjusting the value of Pd used in (18) and resultingin improved accuracy for heavy-tailed clutter.

Fig. 3. PDF of the real part of the complex matched-filter envelope for NFTin Gaussian- and K-distributed backgrounds.

A. Evaluating the accuracy of the DT approximationsIn order to evaluate the approximations to S, the detector

threshold must be obtained as a function of Pfa by inverting(2) as described in (17). Given the detector threshold, the CDFof T under H1 is then utilized to find the value of S achievingthe desired value of Pd. As a result of the complicated formof the CDF for targets in most non-Gaussian backgrounds,an iterative search is required. Unfortunately, these CDFs donot usually have simple analytical forms; it is common tosee them described as integrals, infinite summations, and evenHankel transforms of characteristic functions [17], [20]–[22],[23, Sect. 8.5]. Evaluating Pd, can therefore be numericallyintensive and at times provide inaccurate results, particularlyfor the NFT in very heavy-tailed clutter. These issues simplyin evaluating Pd for common target models in non-Gaussianbackgrounds emphasize the utility of the approximations toDT developed in this paper.

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To avoid the issues of varying (and potentially unknown)accuracy associated with direct evaluation of Pd, S is insteadobtained through a Monte-Carlo-integration (i.e., simulation)evaluation of Pd [24, Sect. 5.1]. By generating samples

Ti =∣∣∣√ViUi +

√SAiejθi

∣∣∣2 (19)

for i = 1,. . . , ns, Pd can be approximated by counting thenumber of threshold crossings and dividing by ns,

Pd =1ns

ns∑i=1

I(Ti ≥ hNG

)(20)

where I(Ti > hNG) is an indicator function returning onewhen the argument is true and zero otherwise. In (19), Vi andUi are generated so that the background is either Weibull-or K-distributed but with unit power (i.e., E[Vi|Ui|2] = 1).The normalized target amplitude is a constant for the NFT,Ai = 1, and Rayleigh distributed for the FT with unit power,E[|Ai|2] = 1. θi is uniformly random on (0, 2π). An estimateof DT may be formed by varying S in (19) until the right-hand-side of (20) equals the desired Pd. Although the resultingquantity is random, it is an unbiased estimate with predictableerror variance.

The number of simulation trials (ns) is chosen to limit theerror variance in the estimate of DT (call this estimate SdB).The variability of SdB may be inferred by describing it as afunction of Pd, which is a binomial random variable divided byns. Approximating the function by a first-order Taylor series,the standard deviation of SdB is approximately

Std[SdB] ≈∣∣∣g′(Pd)∣∣∣ Std[Pd]

≈ |g′(Pd)|

√Pd(1− Pd)

ns(21)

where g(Pd) relates Pd to DT (in dB), essentially 10 log10 ofthe functional inverse of (4) for S.

Setting ns = 106 and approximating g(Pd) by 10 log10 of(16) or (18), the maximum standard deviation of SdB over theshape parameter range shown in Table II was less than 0.01dB for the NFT for all cases, except for the combination ofPfa = 10−2 and Pd = 0.1 where the standard deviation was0.011 dB. For the FT, the standard deviation was similar in theNFT exception at Pd = 0.1. However, owing to the sharpnessof g(Pd) for the FT at high Pd, the standard deviation roseabove 0.01 dB, starting at approximately Pd = 0.8, to anunacceptably high level. Thus, for values of Pd ≥ 0.8,ns = 107 samples were used for the FT, which kept the errorstandard deviation below 0.01 dB except for Pd = 0.99 forwhich the standard deviation of the error rose to 0.014 dB.The results were nearly identical for both the Weibull- andK-distributed backgrounds. With ns ≥ 106, the Pd estimate isessentially Gaussian, so it is very unlikely that the errors wouldrise above three standard deviations. Thus, the simulationapproach is expected to provide results with accuracy betterthan one twentieth of a dB in all of the cases articulated inTable II.

IV. EXAMPLES

In this section, the accuracy of the approximations to Sdescribed by (16) and (18) is evaluated for the FT and NFTin Weibull- and K-distributed backgrounds. The range of Pfaand Pd evaluated are found in Table II, along with the rangeof shape parameters for the Weibull and K distributions. Notethat extremely low values of shape parameter (i.e., very heavy-tailed background PDFs) are evaluated. Although there maybe some doubt as to the physical realism of such heavy-taileddata, the intent in this paper is to determine the extent to whichthe proposed approximations to DT are accurate.

TABLE IIPARAMETER RANGES FOR EVALUATING S .

Parameter RangePfa 10−10 to 10−2

Pd 0.1 to 0.99

Weibull shape β ∈ [0.25, 1]

K shape α ∈ [0.1, 100]

Simulation trials:FT for Pd ≥ 0.8 ns = 107

all other cases ns = 106

A. Weibull-distributed background

The matched-filter-intensity PDF and CDF under H0 for aWeibull-distributed background are

fT (t|H0) =βtβ−1

λβe−(t/λ)β (22)

andFT (t|H0) = 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-lawtransformation (i.e., if W is Weibull distributed, then so willbe 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 Pfa and β,

hW =(− logPfa)β

−1

Γ (1 + β−1). (24)

1) Fluctuating target: Using (24) in (16) results in thefollowing approximation for a FT in a Weibull-distributedbackground,

S ≈ (− logPfa)β−1

−Γ (1 + β−1) logPd− 1 = S. (25)

To evaluate the accuracy of the approximation, consider themaximum absolute error in decibels between the approxima-tion of (25) and that obtained from the simulation analysis,

εdB = maxPd,β

∣∣∣SdB − SdB

∣∣∣ (26)

where SdB = 10 log10 S and the maximization occurs overall the combinations of Pd and β evaluated or as otherwise

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specified. Table III displays εdB for each of the Pfa levelsevaluated. The approximation of (25) is surprisingly accuratewith errors less than 0.1 dB for all values of Pfa except 10−2.As might be expected, the maximum error for Pfa = 10−2

occurred for Pd = 0.1 and β = 0.25. For the smaller values ofPfa, the maximum error occurred for Pd = 0.99. An exampleset of DT curves is shown in Fig. 4 for Pfa = 10−6 andvarious values of Pd. On this scale, it is not possible to seethe difference between the approximation and the simulatedresult.

TABLE IIIMAXIMUM ABSOLUTE ERROR IN DT APPROXIMATION (εdB) FOR FT AND

NFT MODELS IN WEIBULL-DISTRIBUTED BACKGROUND OVER

β ∈ [0.25, 1] AND Pd ∈ [0.1, 0.99].

Pfa FT NFTno corr. w/corr.

1e-2 0.13 1.44 0.491e-3 0.05 0.57 0.241e-4 0.05 0.42 0.171e-5 0.05 0.36 0.151e-6 0.05 0.32 0.141e-7 0.05 0.28 0.131e-8 0.05 0.26 0.121e-9 0.05 0.23 0.111e-10 0.05 0.22 0.10

Fig. 4. DT for FT in Weibull-distributed background for Pfa = 10−6

and various values of Pd. The DT approximation is nearly identical to thesimulation results.

2) Non-fluctuating target: Direct application of Hmam’sDT approximation for the NFT using the Weibull-backgrounddetector threshold (i.e., (18) using (24)) achieves a reasonableaccuracy for smaller values of Pfa, as seen in Table III(column labeled ‘no corr.’ under NFT). The smallest errorsare around one quarter dB and occur for the smallest Pfawhile the errors range above one decibel at the largest Pfa.The maximum error typically occurred at either Pd = 0.1(larger values of Pfa) or at Pd = 0.99 (smaller values of Pfa)

with the errors generally smaller away from these boundaries.For example, for Pd = 0.5, the errors were nearly identicalto those reported for the FT. The magnitude of these errorscompared with that achieved for the FT and the structuredmanner in which Pd for the NFT varies when progressingfrom a Gaussian background to a heavier-tailed backgroundmotivates the development of a correction factor to improvethe accuracy of the approximation.

An examination of Fig. 3, illustrates that Pd for a non-Gaussian background could be above or below that for aGaussian background with the same SNR, depending on thethreshold. This has an inverse effect on DT; that is, when Pd isslightly higher for the non-Gaussian background, the DT willbe slightly smaller than (18) using (24). However, the disparityshould disappear as β → 1 and may be small near thresholdsproducing Pd = 0.5. Empirically, it was determined that bymodifying the value of Pd used in (18) according to

P ′d(Pd, β) = P βc(Pd)

d (27)

where the exponent on β is

c(Pd) = [0.5− Pd] ·min{

6Pd + 0.2, 2.6− 118(1− Pd)

},

(28)the approximations, now described by

S ≈ SH(hW , P

′d (Pd, 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(Pd, β) =Pd) when β = 1 or Pd = 0.5.

As seen in Table III, the errors with the correction of (27)–(29) are less than half of those without the correction, rangingdown to one tenth of a decibel at the lowest Pfa. As seen in theexample set of DT curves found in Fig. 5 for Pfa = 10−6 andvarious values of Pd, the errors are greatest for the smallestand largest values of Pd. However, in a wide region betweenthese extremes, the errors are even smaller; for example, forPfa ≤ 10−3 and Pd ∈ [0.2, 0.9], the errors are all less than onetenth of a decibel. The maximum error over Pd is shown in Fig.6 as a function of β for various values of Pfa. As expected,the errors approach the simulation resolution as β → 1, butcan also be seen to decrease somewhat as β decreases belowabout 0.5. Without the correction factor, these errors wouldrise as β decreases.

With or without the correction, the approximation accuracyimproves as Pfa decreases. This arises from the resultinghigher values of detector threshold and therefore S. In thisregime, the improved fit of the Pd approximation (i.e., assum-ing a Gaussian background) indicates that the approximationsmay retain accuracy for Pfa < 10−10. However, the same isclearly not true for larger values of Pfa.

B. K-distributed background

The matched-filter-intensity PDF and CDF under H0 for aK-distributed background are

fT (t|H0) =2

λΓ(α)

(t

λ

)α−12

Kα−1

(2

√t

λ

)(30)

7

Fig. 5. DT for NFT in Weibull-distributed background for Pfa = 10−6

and various values of Pd.

Fig. 6. Maximum absolute error in DT approximation (εdB(β)) for the NFTin Weibull-distributed background over Pd as a function of β for various Pfa.

and

FT (t|H0) = 1− 2Γ(α)

(t

λ

)α2

Kα

(2

√t

λ

)(31)

where α is the shape parameter and λ is the scale parameter.The normalized detector threshold is obtained by setting λ =σ2

0/α and using (31) in (17). In contrast to the Weibullbackground, there is no closed-form solution for the detectorthreshold in a K-distributed background, which forces theuse of numerical routines. However, in the spirit of Hmam’s[10] development of approximations to the detector thresholdfor the Gaussian background with incoherent integration ofmultiple samples, an approximation of the form

hK ≈ h0 +a1h

a20

αa3h0.20

(32)

is proposed for a K-distributed background with the parame-ters a1, a2 and a3 chosen according to Pfa as found in the firsttwo rows of Table IV. Equation (32) was derived empiricallyand is seen to (i) be greater than the intensity threshold for aRayleigh-distributed envelope (h0) when a1 > 0, illustratingthe increase necessary to account for the heavier tails inthe K distribution, and (ii) tend to h0 when a3 > 0 andα→∞, where the K distribution simplifies to the Rayleigh.The accuracy of the approximation over a wide range of αand Pfa is shown graphically in Fig. 7 where the coefficientsfound in the first two rows of Table IV are used.

The values of the parameters shown in Table IV wereobtained by minimizing the maximum error in the specifiedregion of Pfa (evaluating each order of magnitude) and forα ∈ [0.2, 100]. The maximum error over each region aslisted in the table is less than 0.2 dB. As seen in Fig. 8, theerror is also less than 0.2 dB when α ∈ [0.1, 0.2] except forPfa = 10−2 and 10−5 where the approximation fails belowα = 0.2. Clearly, using the approximation of (32) outside ofthe regions in which it has been evaluated could result in largererrors than those reported herein.

Fig. 7. Detector threshold for K-distributed background compared with theapproximation using first two rows of Table IV as a function of α for variousPfa.

TABLE IVPARAMETERS FOR K-DISTRIBUTION DETECTOR THRESHOLD

APPROXIMATION.

Pfa range a1 a2 a3Maximum error (dB)

when α ≥ 0.2

1e-2 to 1e-5 0.099 2.367 0.502 0.1791e-6 to 1e-10 0.202 2.070 0.479 0.138

1e-6 0.120 2.261 0.500 0.042

When greater accuracy is necessary, parameters can befound for (32) that minimize the error for a specific Pfa. Theparameters for Pfa = 10−6 are shown in Table IV as anexample. The parameters optimizing each order of magnitude

8

over the full range of Pfa considered are listed in the Ap-pendix. For example, as seen in Fig. 8, restricting applicationto one Pfa can reduce the error to less than one twentieth ofa decibel over α ∈ [0.2, 100]. Such an approximation is usefulwhen implementing an adaptive detector threshold (similar toa statistical normalizer [25], [26]) to approximate a constantfalse-alarm rate; that is, obtaining a local estimate of α andincreasing the detector threshold above h0 according to αusing (32). The approximation of (32) is also useful to start aNewton-Raphson iteration to increase the accuracy for a givenPfa and α,

hK := hK −FT

(σ2

0hK |H0

)− (1− Pfa)

fT

(σ2

0hK |H0

) . (33)

For all of the cases considered, starting (33) with (32) resultedin six decimal places of accuracy (for the threshold in dB)within five iterations.

Fig. 8. Error in the detector-threshold approximation for K-distributedbackground as a function of α for various Pfa.

1) Fluctuating target: For the FT, (16) approximates DTwhere the normalized threshold is obtained from (32) as anapproximation or starting with (32) and iterating (33) for theprecise result,

S ≈ hK− logPd

− 1. (34)

The maximum error of the DT approximation over Pd andα (i.e., (26) where the maximization is over α instead of β)is found in Table V as a function of Pfa where it is seento be approximately one twentieth of a decibel for the exactthreshold and, excepting Pfa = 10−2, between one tenthand one quarter of a decibel for the approximate hK . WhenPd ≤ 0.97 and Pfa ≤ 10−3, the maximum error for theexact threshold drops to 0.016 dB. Note that restricting theerror evaluation to α ≥ 0.2 for Pfa = 10−2 when using theapproximate hK , as might be suggested by the above analysis,more than halves the error from 0.72 to 0.35.

An example set of DT curves is shown in Fig. 9 for Pfa =10−6 and various values of Pd where the approximation to DTuses the exact threshold. Similar to the Weibull-backgroundresults, it is not possible to see the difference between theapproximation and the simulated result on this scale.

TABLE VMAXIMUM ABSOLUTE ERROR IN DT APPROXIMATION (εdB) FOR FTMODEL IN K-DISTRIBUTED BACKGROUND OVER α ∈ [0.1, 100] AND

Pd ∈ [0.1, 0.99].

Pfa Exact hK Approximate hK1e-2 0.06 0.721e-3 0.05 0.231e-4 0.05 0.171e-5 0.05 0.241e-6 0.05 0.231e-7 0.05 0.181e-8 0.05 0.111e-9 0.05 0.12

1e-10 0.05 0.19

Fig. 9. DT for FT in K-distributed background for Pfa = 10−6 and variousvalues of Pd. The DT approximation is nearly identical to the simulationresults.

2) Non-fluctuating target: Direct application of Hmam’sDT approximation for the K-distributed background (i.e., (18)using (32) or (33)) leads to similar errors to the Weibull-background. As seen in Table VI, without a correction theerrors are over one decibel for the largest Pfa and range downto one quarter of a decibel for the smallest Pfa. Ideally, thecorrection factor would be similar to that used for the Weibulldistribution. A reasonable approach is to obtain a value ofthe Weibull shape parameter (call it βK) that produces a PDFsimilar to the K PDF for a given α. By equating the first twointensity moments of the distributions, it is possible to relateβK to α according to

α =2Γ2(1 + 1/βK)

Γ(1 + 2/βK)− 2Γ2(1 + 1/βK). (35)

9

Inverting (35) to obtain βK as a function of α clearly requiresa numerical solution or table look-up. However, owing to thesmooth nature of the Pd-correction found in (27) with respectto β and the secondary nature of the correction, the followingapproximation was found to be adequate over the range ofshape parameters being considered,

βK = 0.3 + 0.7 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 ≥ 0.25.

DT for the NFT in a K-distributed background is thenapproximated by using (36) in (29),

S ≈ SH(hK , P

′d (Pd, βK) , 1

). (37)

Using the correction factor of (27) with the exact detectorthreshold, as seen in Table VI, results in a similar reductionin error to the Weibull case: the errors are halved for all valuesof Pfa considered and range from 0.5 dB at the highest Pfadown to approximately one tenth of a decibel at the smallestPfa. Although the errors decrease with Pfa when the exactthreshold is used, the approximate threshold of (32) resultsin a slight oscillation in the errors. The errors for Pd = 0.99generally exceed those for smaller values; as previously noted,the approximations are expected to lose accuracy at such highvalues of Pd. The example DT curves found in Fig. 10, whichutilize the exact detector threshold and the correction notedin (37), illustrate that the approximations degrade as Pd tendstoward high or low values and as α decreases, especially belowone. The maximum errors over Pd seen in Fig. 11 as a functionof α and Pfa are similar to those for the Weibull distribution;however, the reduction in error observed for very heavy tailedWeibull backgrounds seems to be just starting at α = 0.1.

TABLE VIMAXIMUM ABSOLUTE ERROR IN DT APPROXIMATION (εdB) FOR NFT

MODEL IN K-DISTRIBUTED BACKGROUND OVER α ∈ [0.1, 100] AND

Pd ∈ [0.1, 0.99].

Pfa Exact hK Approximate hKno corr. w/corr. no corr. w/corr.

1e-2 1.25 0.50 1.59 0.991e-3 0.67 0.29 0.76 0.341e-4 0.47 0.20 0.51 0.231e-5 0.39 0.17 0.53 0.331e-6 0.34 0.16 0.47 0.311e-7 0.30 0.14 0.38 0.241e-8 0.27 0.13 0.29 0.161e-9 0.24 0.12 0.22 0.09

1e-10 0.22 0.11 0.28 0.19

Fig. 10. DT for NFT in K-distributed background for Pfa = 10−6 andvarious values of Pd.

Fig. 11. Maximum absolute error in DT approximation (εdB(α)) for theNFT in K-distributed background over all values of Pd as a function of αfor various Pfa.

V. CONCLUSIONS

An approach was described to approximate the DT term inthe sonar or radar equation (i.e., the SNR required to achieve aspecified Pd for a given Pfa) for non-Gaussian backgrounds.By approximating Pd for targets in non-Gaussian backgroundsby that for Gaussian backgrounds, the DT approximationsdeveloped for Gaussian backgrounds may be used along withan appropriately derived detector threshold to approximateDT in a non-Gaussian background. The approximations, assummarized in Table I, were seen to be easily evaluatedand quite accurate over a wide span of Pfa, Pd, and forWeibull- and K-distributed backgrounds ranging from near-Rayleigh envelope distributions to very heavy-tailed PDFs.The maximum errors (over the evaluated range of Pd and

10

background shape parameter) for fluctuating targets were ap-proximately one-twentieth of a decibel, while non-fluctuatingtargets required a correction factor to produce coarser, but stilladequate, approximations with errors ranging from 0.5 dB atthe largest value of Pfa = 10−2 down to one tenth of a decibelat the smallest Pfa = 10−10. Although the approximationaccuracy for the NFT is somewhat worse than that for theFT, it is certainly adequate for the intended use as a theo-retical or hypothetical target model representing benchmarkperformance for consistent (i.e., deterministic) target echoesin sonar-equation analysis, particularly in light of the errorscommon in other terms in the sonar equation (e.g., see [27]where reverberation modeling errors dwarf those illustratedhere). An approximation to the detector threshold for the Kdistribution was proposed, providing errors less than one fifthof a decibel over a wide range of Pfa and α.

These results will be useful in predicting the performanceof sonar or radar systems when the background is non-Gaussian. The idea of approximating the PDF of a target in anon-Gaussian background by that for a Gaussian backgroundmay be particularly useful in areas such as target tracking,classification or image segmentation. While the present resultsfor the K and Weibull distributions cover a wide range ofnon-Gaussian background PDFs, the basic DT approximationsshould 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 thatpresented for the Weibull background where an equivalentWeibull shape parameter is obtained via moment matching,as illustrated for the K distribution.

As the present analysis is limited to the detector formedby comparing a single sonar resolution cell to a threshold, itis most appropriate for use in active sonar systems. However,noting the efficacy of the approach as it was applied to theanalysis of an m-of-n fusion processor operating on dependentK-distributed data [18], these results are expected to extendto detectors incorporating incoherent integration (e.g., post-matched-filter integration in active sonar or broadband energydetection in passive sonar). The primary difficulty, however,lies in accurate evaluation of the true values of both detectorthreshold and DT. Of course any extrapolation beyond theconditions evaluated in this paper requires an appropriate erroranalysis.

ACKNOWLEDGMENT

The author thanks Dr. J. R. Preston (ARL/PSU) for pointingout the work of Pryor [5] and the reviewers for their construc-tive comments.

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[10] ——, “SNR calculation procedure for target types 0, 1, 2, 3,” IEEETransactions on Aerospace and Electronic System, vol. 41, no. 3, pp.1091–1096, 2005.

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[20] G. V. Trunk, “Non-Rayleigh sea clutter: Properties and detection oftargets,” Naval Research Laboratory, Report 7986, 1976, reprinted inAutomatic Detection and Radar Data Processing, D. C. Schleher, Ed.,Artech House, Dedham, 1980.

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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 Pfaand α provides an easily evaluated alternative to numerically

11

inverting the K-distribution intensity CDF. Parameters for (32)providing minimal error for specific values of Pfa are listedin Table VII. By articulating the parameters for a specificPfa, the error can be limited to less than one twentieth ofa decibel for Pfa ≤ 10−6 and less than one tenth of decibelfor Pfa ≤ 10−3.

TABLE VIIPARAMETERS FOR K-DISTRIBUTION DETECTOR THRESHOLD

APPROXIMATION.

Pfa a1 a2 a3Maximum error (dB)

when α ≥ 0.2

1e-2 0.112 2.256 0.516 0.1301e-3 0.098 2.376 0.525 0.0961e-4 0.128 2.245 0.519 0.0701e-5 0.149 2.179 0.509 0.0541e-6 0.145 2.189 0.500 0.0421e-7 0.116 2.267 0.491 0.0351e-8 0.234 2.018 0.481 0.0321e-9 0.236 2.018 0.473 0.0331e-10 0.274 1.972 0.466 0.040