MATHEMATICAL ANALYSIS OF RjiNDOM NOISE 149 



which shows that the agreement is perfect in this case. Next we let Q = 0. 

 The two expressions then give 



i+OO 



Wc(f) = ./ . / w(x)w(f — x) dx 



1 /• 



=^) = To~r / '^ix)wU - x) 

 AZttxI/o J-.oo 



where .1 = x for Ragazzini's formula and A = 4 for (4.10-16). Thus the 

 agreement is still quite good. The limiting value for (4.10-16) may also 

 be obtained from (4.7-8). 



Even if the index of modulation r is not negligibly small it may be shown 

 that when Q —^ cc Wdf) still approaches the value given by (4.10-19). 

 Ragazzini's formula gives a somewhat larger answer because it includes the 

 additional terms, shown in (4.5-17), which contain k /4, but this difference 

 does not appear to be serious. If the Q + Ixpo in the denominator of (4.10- 

 13) be replaced by Q' -\- ^Q k" -\- 2\f/o the agreement is improved. 



APPENDIX 4A 

 T.4BLE OF Non-linear Devices Specified by Integrals 



Quite a number of non-linear devices may be specified by integrals of the 

 form 



1 = ^1 F(iu)e''''' du (4A-1) 



Zir J c 



where the function F(iu) and the path of integration C are chosen to fit the 

 device.* The table gives examples of such devices. Some important cases 

 cannot be simply represented in this form. An example is the limiter 



I = - aD, F < -D 



I = aV, -D < V < D 



I = aD, D <V (4A-2) 

 which may be represented as 



^2a r 



T Jo 



sin Vu sin Du — - 



= — aD + -—. / e sm Du —r 

 2in J c u^ 



(4A-3) 



where C runs from — x to + x and is indented downward at the origin. 

 This is not of the form assumed in the theory of Part IV. However it 

 appears that it would not be difficult to extend the theory in the particular 

 case of the limiter. 



* Reference 50 cited in Section 4.9. 



