COMMUNICATION IN PRESENCE OF NOISE 93 



and 



I y{yi) I < £' dy jf' I c{x) I exp T-Jj [lh{^) - \ v{0 il dp\ dx. 



From b'^ qsind\v\ < l/(2q - 1) it follows that 2b(^) - \ v{^) | > 2^ - 1 

 when q > I. This and | c(x) | ^ (4 + x~)(l + x-)~V4 gives 



! Viyi) \ < [ dy [ (4 + .v-)(l + .v-)"-4-' exp [-(2^ - l)(j - x)] dx 



Jo •'0 



Stt 1 



16(2g - 1) 2q - I 



which is the result we set out to establish. The double integral may be 

 reduced to a single integral by inverting the order of integration and inte- 

 grating with respect to y. Incidentally, most of the roughness of our result 

 is due to the use of the inequality for | c(x) |. 



References 



1. C. E. Shannon, A Mathematical Theory of Communication, Bell Sys. Tech. Jour., 27, 



379-423, 623-656 (1948) See especially Section 24. 



2. C. E. Shannon, Communication in the Presence of Noise Proc. I .R.E., 37 , 10-21 (1949). 



3. \V. G. Tuller, Theoretical Limitations on the Rate of Transmission of Information 



Proc. I.R.E., 37, 468-478 (1949). 



4. N. Wiener, Cybernetics, John Wiley and Sons (1948). 



5. S. Goldman, Some Fundamental Considerations Concerning Noise Reduction and 



Range in Radar and Communication, Proc. I.R.E., 36, 584-594 (1948). 



6. G. N. Watson, Theory of Bessel Functions, Cambridge University Press (1944), 



equation (1) p. 181. 



7. R. A. Fisher, The General Sampling Distribution of the Multiple Correlation Coeffi- 



cient. Proc. Roy. Soc. of London (A) Vol. 121, 654-673 (1928). See in particular 

 pages 669-670. 



8. Reference (6), equation (4) p. 394. 



9. P. K. Bose, On Recursion Formulae, Tables and Bessel Function Populations Asso- 



ciated with the Distribution of Classical D^ — Statistic, Sankhya, 8, 235-248 (1947). 



10. Compare with §8.4 of reference (6). 



11. Reference (6), p. 236. 



12. Reference (6), p. 227. 



