Generalizations of two known noncentral chi-square results are presented. The first generalization concerns extending the expression for the probability that one (first-order) Ricean random variable exceeds another to the expression that one nth-order Ricean random variable exceeds another nth-order Ricean random variable. This latter result is shown to be equivalent to the probability of one noncentral chi-square random variable with 2n degrees of freedom exceeding another. The form of the resulting expression is such that it can easily be evaluated by a recurrence relation. The second generalization deals with the fact that the noncentral chi-square distribution function, with d degrees of freedom, differs from the complementary probability of detection, 1-P/sub N/(X, Y), only in that the latter is restricted to even degrees of freedom. We can expand the techniques and expressions developed for the robust, efficient, and accurate calculation of the P/sub N/(X, Y), or equivalently, the generalized Marcum (1960) Q-function, to the calculation of the noncentral chi-square distribution function and its complement.

The author presents corrections to his original paper (see ibid., vol.35, no.2, p.389-400, 1989). The corrections concern computational cases using the steepest descent integration technique. It is pointed out that, for certain specific parameter ranges, the calculation error is too large to be accounted for by accumulated round-off error.>

A highly reliable, accurate, and efficient method of calculating the probability of detection, P/sub N/(X,Y), for N incoherently integrated samples, where X is the constant received signal-to-noise ratio of a single pulse and Y is the normalized threshold level, is presented. The useful range of parameters easily exceeds most needs. On a VAX/11 computer with double precision calculations, better than 13-place absolute accuracy is normally achieved. There is a gradual loss of accuracy with increasing parameter values. For example, for N=10/sup 9/, and with both NX and Y near 10/sup 7/, the accuracy can drop to ten places. The function P/sub N/(X,Y) can be equated to the generalized Marcum Q-function, Q/sub m/( alpha , beta ). The corresponding limits on alpha and beta are roughly 4500 for the 13-place accuracy and 60000 for ultimate (INTEGER*4) limit.>

A highly efficient method of evaluating probabilities of detection and the generalized\cal Q-function to within an absolute accuracy\epsilonis presented. Many unnecessary computations are avoided when it can be determined by Chernoff bounds that the desired function is within\epsilonof 0 or 1. When this is not the case, efficient algorithms for the necessary computations are also provided.

LetH[s(\bullet),t]be a functional of the functions(\bullet)and the independent variablet. The criterion functionalJ(s)is defined to be the integral ofHoverT, a set of pointst. The problem considered is the maximization ofJ(s)with respect to an amplitudeconstrained functions. Necessary conditons for the optimal solution are obtained, and a method of determining the optimumsis given. Some examples are presented to clarify the theorems and their applications.

A convenient method of evaluating theQfunction over the parameter space quarter plane is presented. TheQfunction is first expressed as an infinite series. TheNterm truncated seriesQ_{N}(a, b)is used to approximateQ(a, b)fora^{2} + b^{2} \leq Rwhere the choice ofNdepends on the accuracy desired andRis determined by considerations such as computer bit capacity, computational time, and accuracy. Fora^{2} + b^{2} > R, alternate expressions are used. Whenb - a \geq d, we approximateQby 0, and whenb - a \leq d, we approximateQby 1. The accuracy is dependent on the choice of the constantd. In the reniainder of the quarter plane,a^{2} + b^{2} > Rand| b - a < d, and an efficient expression is used, but it is of limited accuracy (from 10-5to 10-9) near the linea = b.