Padé Approximations to Rician Statistical Functions

Approximations are derived for the statistical functions that characterize the matched filter output amplitude given a non-fluctuating point target and an arbitrary clutter model. The approximations are the inverse Hankel transform of a Pade approximation to their associated characteristic functions (CF) and can be used to calculate receiver operating characteristics (ROCs).


I. INTRODUCTION
In a paper by Drumheller [1] it was shown that the probability density function (pdf) of the matched filter output amplitude could be approximated by a sum of Bessel functions.This was done by performing an inverse Hankel transform on a Pad é (rational function) approximation to the characteristic function (CF) of the complex matched filter output's joint pdf.Integration of the pdf approximation yielded an approximation to the survival function (SF), which can be used to calculate detection probabilities.Furthermore, it was shown that if the Pad é approximations to the CFs for the target and clutter fluctuation models were known separately, then inverting the product of the two CF approximations yielded approximations for the target-plus-clutter pdf and SF.The work extended a technique for approximating statistical functions using Laplace transforms that was developed by Amindavar and Ritcey [2].
The non-fluctuating point target is a simple model that serves as a standard target in detection system analysis and can be used to approximate a spatially extended target as a collection of point targets.In this work, the results in [1] (which were formulated for arbitrary fluctuating target and clutter models) are applied to this special case of target, yielding approximations to Rician type pdfs and SFs for a non-fluctuating point target in arbitrary clutter.Three examples are presented.Two demonstrate the achievable accuracy of the approximations, the other demonstrates the derivation of receiver operating characteristic (ROC) curves.

II. APPROXIMATIONS TO STATISTICAL FUNCTIONS: BACKGROUND
Suppose the complex matched filter output is given by where r and µ are, respectively, the amplitude and phase of the matched filter output.Note that r = j`j is also referred to as the "test statistic."Under the common assumption that there is no knowledge of its likely value, the phase is modeled as a uniform random variable in the interval [0, 2¼).This implies that the joint pdf of u and v is circularly symmetric.
Using the polar notation (r, µ), this joint pdf has the following form: where f R (r) is the pdf of the amplitude r.The CF for u and v is equal to the two-dimensional Fourier transform of the joint pdf in (2) and is denoted as ©(!, »).It is possible to show that due to the circular symmetry of the joint pdf, the CF is also circularly symmetric and is only a function of the "radial frequency," ½ = p ! 2 + » 2 [1].Therefore, we only need to address the relationship between the CF and the amplitude pdf, which is given by the following Hankel transform pair [3]: where J 0 (¢) is the zeroth-order Bessel function of the first kind.Once the amplitude distribution has been determined, the pdf can be integrated to give an estimate of the SF, which can be used to calculate detection probabilities.
A Pad é approximation for the CF is defined as where L < M. In [1] it was shown that the coefficients of the numerator and denominator of (6) can be found from the associated pdf by either "moment matching" at the origin (½ = 0) or by "interpolation." In the case of moment matching, the denominator coefficients are found first by solving the matrix equation 0 where c 0 = 1, c i = 0 for i < 0 and with Efr 2k g denoting the (2k)th moment of the distribution.Once b 1 , :::, b M have been found, the numerator coefficients can be determined using Suppose that a return contains clutter and a target echo.The complex matched filter output is then equal to the sum of its responses to both the clutter and the target echo.It follows that the joint pdf for the complex matched filter output for this target-plus-clutter case is equal to the two-dimensional convolution of the joint pdfs for the separate matched filter responses to the target and clutter.Consequently, Fourier analysis implies the following relationship: ©(½ j target + clutter) = ©(½ j target)©(½ j clutter): (10) This also means that an approximation for the CF for the target-plus-clutter case is simply the product of the CF for the target and a Pad é approximation to the CF for the clutter.

III. RICIAN STATISTICAL FUNCTIONS: APPROXIMATIONS USING SIMPLE POLES
The previous section reviewed the formalism for obtaining approximations to the statistical functions of a target in clutter given arbitrary individual target and clutter models.In this section and the next, the Pad é approximation technique is used to develop analytic approximations to the pdf and SF for the special case of a non-fluctuating point target in arbitrary clutter.
To begin, assume a pdf that models the matched filter response to clutter has been used to find a Pad é approximation to its associated CF.The partial fraction expansion of this approximation is given by [1] Now consider the case of a non-fluctuating point target.Its amplitude pdf and CF are ©(½ j target) = J 0 (r 0 ½) where ±(¢) is the Dirac delta function and r 0 is the amplitude of the matched filter output in response to the target.From ( 3), ( 10), (11), and (13) it follows that Closed-form inversion of ( 14) using ( 4) is possible using the following integral [4]: where I ¹ (¢) and K ¹ (¢) are, respectively, the ¹th-order modified Bessel functions of the first and second kind.Thus, it follows that the approximation to the pdf is given by Now consider the following antiderivative [4]: and the superscripts in r ¡ 0 and r + 0 refer to the function being evaluated at r 0 when approached from below and above this point, respectively.
For the "clutter only" case, r 0 = 0, and it is easy to see that ( 16) and (18) reduce to the following forms: These equations are same as [1, eq. ( 17) and ( 19)] with m k = 1.

IV. RICIAN STATISTICAL FUNCTIONS: APPROXIMATIONS WITH REPEATED POLES
A more general form for the CF approximation is presented in this section.This approximation accounts for the case of repeated poles of the same value occurring in the Pad é approximations to the clutter CF.Even though no case of repeated poles has been encountered by the authors, there may be instances where these results could be useful for future reference, and hence, the work of this section is presented for completeness.
The CF approximation in ( 14) can be generalized to where m k is the order of the kth pole p k and is equal to an integer greater than 0. Approximations to the pdf and SF can be found using the same approach used in the previous section.The approximation for the pdf requires the following definite integral: This integral can be evaluated in closed form by differentiating g 0 (r j r 0 , a) with respect to y = a2 and interchanging the order of differentiation and integration. 1Thus, from (15) it follows that Now consider the following antiderivative: T n (r j r 0 , a) = ¡a Z g n (r j r 0 , a) dr: From (24), it follows that This allows the following definite integral to be defined Using ( 23)-( 27), inversion of the more general approximation to the CF in ( 22) can be performed.This leads to the following generalized approximations for the pdf and SF: S R (r j target + clutter) Note that the above approximations to the pdf and SF in (28) and ( 29) have similar functional forms to their counterparts in the original general approximations of [1, eq. ( 17) and ( 19)].This is not unexpected since the "clutter only" case is the special case of a non-fluctuating point target in arbitrary clutter with r 0 = 0. Thus, as r 0 !0, pdf: (¡1) m k ¡1 g m k ¡1 (r j r 0 , Clearly a non-zero value of r 0 considerably complicates the analytic form of the statistical function approximations.
V. EXAMPLES

A. Approximating the Rayleigh-Rice Fluctuation Model
The Rayleigh-Rice pdf describes the case of a non-fluctuating point target in Rayleigh clutter.The statistical functions of the Rayleigh clutter model are given by When a non-fluctuating point target is added to Rayleigh clutter, the resultant pdf, referred to as the Rayleigh-Rice distribution, has the following form: It is easy to see that as r 0 !0, the pdf in (35) converges to a Rayleigh pdf.The signal-to-clutter ratio (SCR) for this model is Although no simple formula exists for the SF, it is sometimes written in the compact notation of Marcum's Q function [5], where and is the incomplete gamma function.The series in (38) was first reported by Brennan [6].
To demonstrate the derivation of an approximation to the Rayleigh-Rice statistical functions, consider [1, example A].It was shown that for ¾ 2 = 1 the CF (11) for this model can be approximated as   To assess the quality of the approximations, the relative error was calculated, which is defined as RE = log 10 ¯1 ¡ approximate equation exact equation ¯: (41) Its negative value is equal to the number of significant digits of agreement between the exact equation and its approximation.It is also logarithmic, so it may also be interpreted as a percent error expressed as an order of magnitude, e.g., RE = ¡1 is a ten percent error, RE = ¡2 is a one percent error, etc.The relative errors of the Rayleigh-Rice pdf and SF approximations are shown in Fig. 3.The "exact" value of the SF was calculated from a truncated form of the series in (38) that contained no more than 17 terms.Using more terms did not significantly change the character of the relative error curve for this example.Note that the relative error of the SF approximation is less than ¡2 for 0 < r < 6:58, implying that in this interval there are at least two significant digits of agreement between the SF and its approximation.Detection probabilities as small as 4:32 £ 10 ¡6 can be accurately calculated. 2

B. Approximating the K-Rice Fluctuation Model
For the case of a non-fluctuating point target in K-type clutter the amplitude distribution is described by the K-Rice fluctuation model.The K-type clutter and K-Rice models have been found to be useful in modeling various physical processes.For example, the K-type clutter model has been used to model the backscatter of microwave radar from the sea surface at low grazing angles [8][9] and the K-Rice pdf has been used to model ultrasonic backscatter in medical imaging systems [10].
The K-type clutter model is given by When a non-fluctuating point target is present in K-type clutter the amplitude distribution follows the K-Rice pdf given by The derivation of this pdf is outlined in the Appendix.
The SCR for this model is SCR = (br 0 ) 2 4(º + 1) : No simple formula exists for the SF, but it can be written in terms of Marcum's Q function: The integral can be expressed as the sum of two infinite series of Bessel functions.The specific forms are presented in the Appendix.Note that the above form of the SF follows from the representation of K-type clutter as a "compound Rayleigh process" where the expected power of Rayleigh clutter is also a random variable, one that follows a gamma distribution [11].
To demonstrate deriving an approximation to a K-Rice model consider the case of º = 0:5 and b = p 3, for which the pdf and SF possess tails that decay less rapidly than that of the Rayleigh model.In [1, example B] it was shown that the CF for this model can be approximated as follows: Using ( 15)-( 21) with r 0 = 2, approximations to the K-Rice pdf and SF were derived from the poles and partial fraction expansion coefficients of the CF approximation in (48).These approximations are also shown in Figs. 1 and 2. Since Efr 2 g = 2 for the pdf in (42), this describes the SNR = 2 (3 dB) case.
The relative errors of the K-Rice pdf and SF approximations are shown in Fig. 4. Note that the relative error of the sf approximation is less than ¡2, implying that there are at least two significant digits of agreement between the SF and its approximation for 0 < r < 14.Detection probabilities as small as 5:01 £ 10 ¡9 can be accurately calculated in this interval.

C. ROC Curves
From (36) and (46) it can be seen that changing the SCR amounts to only changing r 0 .Since the approximations to the pdf and SF for the target-plus-clutter case are parameterized by r 0 , this allows detection probabilities to be easily determined.
First, given the probability of false alarm P fa and the "exact" form of the clutter SF, the detection threshold °can be found by solving the equation S R (°j clutter) = P fa using Newton's method.Next, the detection probability P d for a given r 0 is found by evaluating the approximation to the target-plus-clutter SF at the threshold value.This process for finding P d can be repeated over a range of values of r 0 to produce a ROC curve.This procedure described above was used to derive ROC curves for detecting a non-fluctuating point target in Rayleigh and K-type clutter.Curves were calculated for a SCR range of 0 to 20 dB and three common values of P fa .The curves are shown in Fig. 5.The figure clearly shows that the statistical nature of clutter fluctuations have a profound effect upon P d .In particular, it reveals that the detection problem examined here is more difficult than that under the Rayleigh assumption.This is often the case when non-Rayleigh statistics apply because the clutter pdfs have larger tails.

VI. CONCLUSIONS
The work presented in this paper is an extension of that presented in [1].The Pad é approximation technique from that work is used to derive analytic expressions of the statistical functions for the case of a non-fluctuating point target in arbitrary clutter.Examples using the well known Rayleigh and K-type clutter models illustrate the accuracy and application of this technique.In addition, given that the expressions for the Rayleigh-Rice SF and K-Rice pdf and SF are infinite series, it can be said that the approximation technique presented here offers a computationally convenient method of calculating Rician statistical functions and detection probabilities.This is because the Pad é-based computation generally requires far fewer terms than a direct summation of the corresponding infinite series representation.This is especially evident in the example of the K-Rice distribution where the convergence of the series is relatively slow.
Collectively, this paper and [1] document a comprehensive technique for accurately approximating matched filter detection probabilities for arbitrary target and clutter models.

APPENDIX. THE K-RICE PDF AND SF
The following is an outline of the derivation of the K-Rice pdf and SF.
Consider first the K-Rice pdf.By multiplying the CFs from ( 13) and ( 14) the CF for the target-plusclutter case is Substituting this in (4) then yields The integration can be performed as follows.Write the the denominator of the integrand in (50) so that and interchange the order of integration.Integration with respect to ½ can then be done using Finally, by using a series expansion of the modified Bessel function, and evaluating the remaining integral with This result is the two-dimensional special case of the "homodyned K distribution" given by [11, eq. (4.13)].The result given here is apparently different from the corresponding case in that reference.We were unable to get a numerically convergent series (and hence physically sensible results) using eq.(4.13).Furthermore, in as r 0 !0, eq.(4.13) fails to converge to the expected K-type pdf, and instead diverges.Now consider the K-Rice SF.Substituting the series in (38) into (47) and interchanging the order of integration yields S R (r j target + clutter) Numerical studies revealed that this form of the K-Rice SF converges rapidly for r À r 0 with the number of required series terms decreasing as r increases, but it was found to converge slowly for r < r 0 .In this case, an alternate form for the SF was used that was derived by first realizing that [12] Q(®, ¯) = 1 ¡ e ¡® This form of the K-Rice SF converges more rapidly for r < r 0 than does the form in (57), but it still requires a large number of terms for convergence.Summation of the series for the pdf and both forms of the SF was halted when the last term included a series that was less than 0.0001 percent of the accumulated sum.

Fig. 3 .
Fig. 3. Relative error curves for approximations to Rayleigh-Rice pdf and SF for SCR = 3 dB.Vertical axis is logarithmic.

Fig. 4 .
Fig. 4. Relative error curves for approximations to K-Rice pdf for SCR = 3 dB.Vertical axis is logarithmic.

Fig. 5 .
Fig. 5. ROC curves for point target in Rayleigh and K-type clutter.Curves calculated for 3 common false alarm probabilities.
55)Using the identity¡ (k + 1,z) = k! e ¡z k procedure for deriving (57), it can be shown that S R (r j target + clutter)