PROPERTIES OF SINE WAVE PLUS NOISE 



113 



The form assumed by pi{y) as the parameter a becomes large is examined 

 in the latter portion (from equation (1.12) onwards) of the section. 



Series which converge for all values of a but which are especially suited 

 for calculation when a < 1 may be obtained by inserting the Taylor's series 

 (in powers of x) for <p(y + x) and <p^i{y + x), x = -a cos 0, in (1.6) and 

 (1.7) and integrating termwise. When we introduce the notation* 



<p'''\y) 



dy" 



<p(y) 



-\/27r dy 



6 



(1.9) 



99.99 



99.95 



uJ 99.8 

 tr 



50 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 I 



l/(RMS TOTAL CURRENT) 



Fig. 2 — Cumulative distribution of sine wave plus noise 



2.0 2,2 2.4 2.6 



Ordinate = 100 



/ P\{y^ dyi 



J — 00 



See Fig. 1 for notation. 



we obtain 



Pi(y) 





(2n) 



W 



[y(y.)dy. = <P-^{y) + t^,(^^ <."-" 



(y) 



(1.10) 



The second equation of (1.10) may be shown to be valid by breaking the 

 interval {— ^ , y) into (— ■» , 0) and (0, y). In the first part, 



/ piiyi) dy\ = ¥'-i(0) 



