SPECTRA OF QUANTIZED SIGNALS 463 



correlation function of the error. Let 



F{]\ , ^2) = 6(/)e(/ + r) = (Fi - mEo)(V2 - nEo), 



':^^«<^'^<'^'^-'^ (2.5) 



m, w = 0, zb 1, ±2, • • • 



Eq. (2.5) defines F{Vi, V2) as a definite constant value in each square of 

 width £0 in the F] F2-plane. By elementary statistical theory, the correla- 

 tion function ^r of the error wave is now 



^r = F{y\7V^ =11 ^(^'1 ^ y^)P^y^ . ^'2) dV\ dV2 (2.6) 



J — 00 J — 00 



The correlation may therefore be calculated since F and p are known 

 functions. The power spectrum 12/ of the error wave is then equal to the 

 right-hand member of (2.1) with ^t substituted for xf/r. 



We are interested in the case in which the signal voltage has a smoothly 

 varying spectrum over a specified band. This is a property of a random 

 noise function which has a normal distribution of instantaneous voltages. 

 The two-dimensional probability density function of such a wave is known^^^ 

 It is 



,,, . ,. . 1 [UVI + Vl) - 2^. Fi F2l 



(2.7) 



By inserting this value and that of F{Vi, V2) from (2.5) in (2.6), making the 



change of variable: 



Vl - mEo = Eox/2] 



(2.8) 

 F2 - nEo = Eoy/2] 



and adopting the notation, 



k = El/rPo, a = h/^|^o, G{a) = ^./i^o, (2.9) 



we obtain the following integral determining ^r , 



^, , _ k^ 



^^""^ 32^(1 - a2)i/2 



11 .22 . (2.10) 



f f n( \ -k{x + y - 2axy) , , 



00 00 



H{x, y)= T, T, 



~— ^ (2.11) 



— k\m^+mix— ay) + w -{- n{y — ax) — lamn] 

 ' ^""P " ~ 2(1 - a2) 



