Fii. 3. Falsealarm exponent 9 and the probability. of detection PD versus average SIN for chi-square fading signal in Gaussian noise. Both CFAR &=3) and minimumuxt thresholding. K =0.0187 Urnin =3). are shown.system overall false-alarm rate not the false-alarm rate a t a particular point in parameter space. Minimumtost thresholding has been used with good results in computer simulations of track-whilee n radar, although no detailed comparison with other thresholding I t would be desirable to relate the thresholding level K to a more general measure of system performance such as total system falsealarm rate. Without this relation the system designer must choose K by the somewhat subjective method of plotting families of PFA and PD versus SIN curves and evaluating these in the light of system requirements. I t should be remembered, however, that choosing a CFAR threshold level is also a subjective decision. strategies was performed. REFERENCES 111 H. M. Finn, 'Adaptive detection with regulated error probabilities." RCA M.. 121 C. W. Helstrom. Sfalislical Thcory of S i g d Dcfcclion. 2nd ed. Oxford, En-[3] R. S. Berkowitz. M&n Radar: Analysis, Evaluation and System Design. pp. 653-678. Dec. 1967. gland: Pergamon. 1968.Abstract-A proof of the Weierstrass approximation theorem is obtained using the Fourier transform and band-limited functions.A band-limited function h(t) is a complex valued function of a real variable t defined for all t in (-m, m), such that h(t) is the inverse Fourier transform of an integrable function H ( u ) , where H(u) has compact support on the o axis, that is, and H=O outside a compact set.The first Weierstrass approximation theorem is proved for f ( t )real and continuous on [a, b I, by first approximating f(t) by a polygonal function; then approximating the polygonal function by a band-limited one; and, last, approximating this band-limited function by its uniformly convergent partial sums.Theorem ( Weierstruss Approximution) e>O, there exists a polynomial with real coefficients p ( t ) such that If f(t) is a real valued continuous function of [a, b 1, then for every for all t in [a, b ] . Proof: Since f(t) is continuous on a compact set and therefore uniformly codtinuous there, we can find a polygonal function q*(t) Manuscript C. R. Giardina is with FairIeigh Dickinson University, Teaneck. N. J. P. M. C W h n is with Stevens Institute o f Technology. Hoboken. N. J. such that for all t in [a, b ] , where by a polyg~nal function on [a, b ] we mean a function that is piecewise linear and continuous on [a. b ] and zero elsewhere. In either case, q(t) is a polygonal function on the interval [c, d l , where c = (3a -b ) / 2 and d = (3b -a)/2, and is continuous for all t. Note that q=p* on [u, b ] and If-ql < r / 3 for all t in [a, b ] . The extra interval has been added to eliminate discontinuities at the ends of [a, b ] . Assume that p has a derivative on [c, d 1. except possibly a t to, h, . , are the 'break times" of the polygonal function.) Then k , where c = t r < t l < k < --. < L < t . = ...
