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BELL SYSTEM TECHNICAL JOURNAL 



be obtained from pair 923.1 of 'Tourier Integrals for Practical Applica- 

 tions," by G. A. Campbell and R. M. Foster.^ 



A curve showing F{y — a) plotted as a function of >; — a is given in Fig. 3. 

 It was obtained from the relation 



where 



F{s) = 2i/V-3/2x(-V\/2) 

 Jq 



-3 



■I 



Fig. 3 — Probability density of sine wave plus noise. 

 When rms /jv < < Q and / is near Q, Pi(y) ^^ a-^i^F{y - a),y - a = (/ - Q)/(rms In). 



See Fig. 1 for notation. 



This function has been tabulated by Hartree and Johnston.* 



The probability that / exceeds Q, or that y exceeds a, is, integrating the 

 second of expressions (1.13), 



/* 1 r^a ^2 /"* 



Piiy) dy = - - / / ^(x) dx, 



u IT Jo vz(2o — z) Jz 



An asymptotic expansion may be obtained by expanding (2a - z)-^'- as in 

 the derivation of (1.15) but we shall be content with the leading term. 



• Bell Telephone System Monograph B-584. 



• Manchester Lit. and Phil. Sac. Memoirs, v. 83, 183-188, Aug., 1939. 



