Extension of the Geometric Mean Constant False Alarm Rate Detector to Multiple Pulses
aa r X i v : . [ s t a t . A P ] J a n GM-CFAR WITH MULTIPLE PULSES JANUARY 3, 2019 1
Extension of the Geometric Mean Constant FalseAlarm Rate Detector to Multiple Pulses
Graham V. Weinberg(Draft created at 1.47 am)
Abstract
The development of sliding window detection processes, based upon a single cell under test, and operating in clutter modelledby a Pareto distribution, has been examined extensively. This includes the construction of decision rules with the complete constantfalse alarm rate property. However, the case where there are multiple pulses available has only been examined in the partial constantfalse alarm rate scenario. This paper outlines in the latter case how the probability of false alarm can be produced, for a geometricmean detector, using properties of gamma distributions. The extension of this result, to the full constant false alarm rate detectorcase, is then presented.
Index Terms
Radar detection; Sliding window detector; Geometric mean detector; Constant false alarm rate; Multiple pulses
I. I
NTRODUCTION
This paper is concerned with the extension of the work of [1] to the multiple pulse scenario, as initiated in [2]. A keydifference is that a novel technique is introduced, utilising the fact that the cumulative distribution function of a gammadistributed random variable can be expressed as a sum of Poisson-like terms, allowing for simple derivation of the appropriateprobability of false alarm (Pfa) for the geometric mean (GM) sliding window detector. In addition to this, the work in [2]is extended to produce a GM detector with the full constant false alarm rate (CFAR) property, in the clutter environment ofinterest.To contextualise this work the basic theory of sliding window detectors is introduced. Useful references on this include [3] -[5]. Sliding window detectors assume the existence of a series of non-negative clutter measurements, denoted Z , Z , . . . , Z N ,which are assumed to be independent and identically distributed. These are referred to as the constituents of the clutter rangeprofile (CRP). The context for this work is X-band maritime surveillance radar, and so it will be assumed that these have aPareto Type I distribution. Hence for all j ∈ { , , . . . , N } , F Z j ( t ) = IP( Z j ≤ t ) = 1 − (cid:18) βt (cid:19) α , (1)for t ≥ β , and is zero otherwise. Here α > is the shape and β > is the scale parameter. This distribution function has asupport not beginning at zero. The justification of a Pareto Type I model, for the scenario of interest, has been documented in[5]. To summarise the latter, the original fits to real X-band maritime surveillance radar clutter showed that a Pareto Type IImodel is appropriate. Since in many cases the Pareto scale parameter β << it follows that the Pareto Type I model can beused as a basis for detector design.Next a cell under test is taken, which is assumed to be independent of the CRP, and denoted by Z . This is also a non-negative random variable. Sliding window detectors apply some function f = f ( Z , Z , . . . , Z N ) to the CRP to produce asingle measurement of the clutter level. This is then normalised by a constant τ > , called the threshold multiplier. Supposethat H is the hypothesis that the CUT does not contain a target, while H is the hypothesis that it contains a target embeddedin clutter. A typical test can be written Z H >< H τ f ( Z , Z , . . . , Z N ) , (2)where the notation in the above means that H is rejected only if Z > τ f ( Z , Z , . . . , Z N ) .The Pfa of test (2) is given by P FA = IP( Z > τ f ( Z , Z , . . . , Z N ) | H ) . (3)For a given Pfa and function f , one can solve for τ for application in (2). If this can be done in such a way that the Pfa doesnot vary with the clutter power, then the test is said to have the CFAR property. The importance of this property is evidentfrom the fact that if there is variation with the resulting Pfa, this can cause series problems when the detector outputs areapplied to a tracking algorithm. Hence sliding window detectors, with the CFAR property, are highly desirable in practicaldetector design. M-CFAR WITH MULTIPLE PULSES JANUARY 3, 2019 2
Detectors of the form (2) do not have the full CFAR property in the Pareto case, as can be observed in [6]. In order toaddress this, a transformation approach for the design of detectors was introduced in [1], which was generalised in [7]. Themain detector to be considered in the current work takes the form Z H >< H β − Nτ N Y j =1 Z τj , (4)which when applied to the Pareto Type I case results in the Pfa given by P FA = 1(1 + τ ) N . (5)This will be referred to as the GM detector throughout. In view of (5), it follows that the detector (4) is CFAR with respectto the Pareto shape parameter, but requires a priori knowledge of the Pareto scale parameter.Further analysis revealed that the CRP minimum is a complete sufficient statistic for β , and consequently one can considerthe alternative detector Z H >< H Z − Nτ (1) N Y j =1 Z τj , (6)where Z (1) = min { Z , Z , . . . , Z N } . Then it is shown in [8] that (6) has Pfa P FA = NN + 1 1(1 + τ ) N , (7)proving that the detector (6) is completely CFAR.The main idea in [2] is to extend (4) to allow for multiple pulses, or equivalently multiple CUTs. In the next section thisdetector is specified and its Pfa produced using some properties of gamma distributed random variables. The approach is thenused to produce a variant of (6) in the multiple pulse scenario, with the full CFAR property.II. C ASE
1: P
ARTIAL
CFAR D
ETECTOR
The multiple pulse based detector assumes that there are a series of N CUTs available, which will be denoted X , X , . . . X N .The first version to be considered takes the form N Y i =1 X i H >< H β N − Mτ M Y j =1 Z τj , (8)where the CUT variables are assumed to be independent and identically distributed under H , N is a positive natural number,and the CUT statistics are assumed to be independent of the CRP. In [2] a more general form is taken, where multiple CRPsare assumed. The analysis below extends easily to this setting, and is hence omitted for brevity.In order to derive the Pfa of (8), introduce random variables X ∗ i and Z ∗ j which are the Pareto duals. These are exponentiallydistributed random variables with the property that X i = βe α − X ∗ i (under H ) and Z j = βe α − Z ∗ j , and so generate the relevantPareto variables. Then it can be shown that P FA = IP N X i =1 X ∗ i > τ M X j =1 Y ∗ j . (9)Since the duals in (9) have exponential distributions with parameter unity, introduce random variables W and W which havegamma distributions: W d = γ ( N, and W d = γ ( M, . Then the Pfa of (9) can be written P FA = Z ∞ f W ( t )IP ( W > tτ ) dt, (10)where f W is the density of W . It can be shown that since N is a natural number, the distribution function of the gammavariable in (10) can be written in the form IP ( W > tτ ) = N − X l =0 l ! w l e − l , (11)which is a sum of Poisson point probabilities (see [9]). Hence, since f W ( t ) = 1( M − t M − e − t (12) M-CFAR WITH MULTIPLE PULSES JANUARY 3, 2019 3 it follows by applying (11) and (12) to (10), the Pfa reduces to P FA = N − X l =0 (cid:18) M + l − l (cid:19) τ l ( τ + 1) M + l . (13)This expression is consistent with the corresponding result in [2]. In the next section it is shown how a similar line of analysiscan be used to produce a multiple pulse version of (6) with the full CFAR property.III. C ASE
2: F
ULL
CFAR D
ETECTOR
Replacing β with the CRP minimum, one arrives at the detector N Y i =1 X i H >< H Z (1) N − Mτ M Y j =1 Z τj . (14)In order to derive the Pfa of (14), one applies the Pareto duals as before, which yields P FA = IP N X i =1 X ∗ i > ( N − M τ ) Y ∗ (1) τ + M X j =1 Y ∗ j , (15)where Y ∗ (1) is the minimum of the duals Y ∗ j , and so has an exponential distribution with parameter M . By conditioning onthis minimum, the Pfa becomes P FA = Z ∞ f Y ∗ (1) ( t )IP N X i =1 X ∗ i > N t + τ M X j =1 (cid:0) Y ∗ j − t (cid:1) | Y ∗ (1) = t dt. (16)Let W = P Ni =1 Z ∗ i d = γ ( N, and define W = P Mj =1 (cid:0) Y ∗ j − t (cid:1) | Y ∗ (1) = t . In [8] it is shown that W d = γ ( M − , . Henceit follows that P FA = Z ∞ M e − Mt IP( W > N t + τ W ) dt = Z ∞ Z ∞ M e − Mt f W ( w )IP( W > N t + τ w ) dtdw = Z ∞ Z ∞ M e − Mt M − w M − e − w N − X l =0 l ! [ N t + τ w ] l e − Nt − τw dtdw, (17)where the appropriate densities have been applied, in addition to (11). By applying the binomial expansion [ N t + τ w ] l = l X n =0 (cid:18) ln (cid:19) ( N t ) l − n ( τ w ) n (18)it can be shown that the Pfa reduces to P FA = N − X l =0 M ( M − l ! l X n =0 (cid:18) ln (cid:19) τ n Z ∞ t l − n e − [ N + M ] t dt Z ∞ w M + n − e − [ τ +1] dw. (19)Finally, by evaluating the gamma function integrals, the Pfa can be shown to reduce to P FA = M N − X l =0 l X n =0 (cid:18) M + n − n (cid:19) [ N + M ] − ( l − n +1) τ n [ τ + 1] M + n . (20)Thus (20) shows that (14) is a CFAR decision rule, for application to the context of [2].It is worth observing that in the scenario where N = 1 , which requires l = 0 and n = 0 , the Pfa in (20) reduces to anexpression analogous to (7), showing that the multiple pulse detector’s Pfa is consistent with the single pulse case .IV. C ONCLUSIONS AND F URTHER W ORK