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# Approximation of integrals via Monte Carlo methods, with an application to calculating radar detection probabilities

DOI:AIR 01/217
Source: OAI

ABSTRACT The performance analysis of a radar detection scheme requires estimation of probabilities of false alarm and detection, under various clutter scenarios. These probabilities, which often appear as definite integrals, are frequently analytically difficult to evaluate. Hence, numerical approximation schemes are employed. Monte Carlo estimators use statistical simulation to evaluate such integrals. As with any approximation scheme, there are limitations and drawbacks in its application. One of the major difficulties with Monte Carlo estimators is that very large sample sizes may be required, in order to achieve a reasonable estimate. This is especially true in the context of estimating probabilities of rare events, such as radar false alarms. The purpose of this report is to examine the Monte Carlo estimation of integrals in general. After formulating the scheme, applications to the evaluation of two special functions are considered. The success of an estimator will be decided on its performance in terms of providing a reasonable estimate for the smallest sample size possible. The major application in this report will be to obtain estimates for a detection probability in a binary integration context. Under the assumption of a Swerling target model, an expression for the binary integrated probability of detection is obtained in Shnidman's 1998 paper entitled Binary integration for Swerling target fluctuations (IEEE Transactions on Aerospace and Electronic Systems, Volume 34, pp. 1043-1053). We apply Monte Carlo simulations, together with some functional approximations, to estimate this probability for Swerling 1 and 3 target models. The approximation of definite integrals using Monte Carlo simulations is the focus of the work presented here. The methodology methodology of estimation by sampling is introduced, and is applied to the approximation of two special functions of mathematics: the Gamma and Beta functions. A significant application, in the context of radar detection theory, is based upon the work of [Shnidman 1998]. The latter considers problems associated with the optimal choice of binary integration parameters. We apply the techniques of Monte Carlo simulation to estimate binary integration detection probabilities. CDR MPG

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### Keywords

analytically difficult

binary integration context

CDR MPG

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definite integrals

detection probability

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estimate binary integration detection probabilities

false alarm

IEEE Transactions

large sample sizes

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numerical approximation schemes

rare events

reasonable estimate

Swerling target fluctuations

various clutter scenarios