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BELL SYSTEM TECH NIC A L JOURNAL 



maximum value of P(r;b). However, as shown in Appendix C, curves of 

 these quantities versus b can be computed, in an indirect manner, by 

 temporarily taking b as the dependent variable and taking T, defined by 

 (4.6), as an intermediate independent variable. For let Tc denote the 

 critical value of r, that is, the value of r at which P(r;b) has its maximum 

 value; and let Tc denote the corresponding value of T, whence, by (4.6), 



Tc= b/{\-b'-)r\ 



(4.9) 



Then, computed by means of the formulas derived in Appendix C, Fig. 4.3 

 gives a curve of Vc and a curve of Max P{r;b), each versus b. From these 

 curves it is seen that re and Max P{r\b) do not vary greatly with b, as also 

 is seen from inspection of Fig. 4.1 giving curves of P{r\b) as function of r 

 with b as parameter. 



< 



g 0.4 



to 



z 



2 0.2 



t- 



u 



z 



£ 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



PARAMETER, b 



Fig. 4.3 — Functions relating to the maxima of the distribution function for the reciprocal 



of the modulus. 



5. The Cumulative Distribution Function for the Modulus 



The cumulative distribution function Q{<R,di2) = Q{R) for the 

 modulus R of any complex variate W = R{cos 6 + i sin 6) is defined by 

 equation (1.11) on setting p = R, a = 6, pi = Ri ~ 0, ai = 6i — and 

 (72 = 6-. = Itt; thus 



QiR) = p{{)<R'<RA)<d'<2Tr). (5.1) 



Similarly, from (1.12), the complementary cumulative distribution function 

 Q{>R,di2) = Q*{R) is defined by the equation 



Q*{R) - p(R<R'<-^^,{)<e'<2Tr). 



(5.2) 



Q*iR) is usually more convenient than Q{R) for use in engineering ap- 

 plications, because it is usually mor? convenient to deal with the relatively 



