720 BELL SYSTEM TECHNICAL JOURNAL 



Putting t — Tin place of / in place of / as the variable of integration in (12) 

 we have 



^(/) = he'''' r e-^-"'+*^('-> dt. (13) 



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Thus it is seen that the inner integral of (11) is equal to g~*'^^^(/) and the 

 equation becomes 



Fo(/)=^(/)[ H (7)6-'^"^"'^' dr. (14) 



Now the F.T. of H{t) is Z{f), i.e. 



Z(/) = r H{l)e~'''' dt. (15) 



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Therefore 



Zif + fo) = r HiOe-'^"^'"'^' di. (16) 



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This differs from the integral in (14) only in the lower limit of integration. 

 But since H{t) is the response to an impulse appHed at time / = 0, H(t) = 

 for / < and the two integrals are therefore equal. Putting (16) in (14) 

 we have, finally 



Uf) = r HO + ibm-'"' dt = *(/)z(/ + /o). (17) 



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The F.T. of the conjugate envelope function, a — ib, is 



Foi-f) = r [a(t) - ime-"" dt = *(-/)Z(-/ + /„) (18) 



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where symbols with the superbar denote the complex conjugates of unbarred 

 symbols. Since Z(f) is the F.T. of a real variable, H(t), it must assume con- 

 jugate values for positive and negative values of/, i.e., Z(f) = Z{—f) and 

 therefore Z{f + /o) = Z{— / + /o). Consequently, (18) could be written 



F\i-f) = ^{-f)Z(f + fo) (19) 



(17) and (18) are the final solutions in frequency functions corresponding 

 to the solutions (8) and (9) in time functions. The formulas in frequency 

 functions have the advantage of compactness, which makes them easy to 

 remember. 



We require also the F.T. of the voltage itself, V (/), which we shall call 

 F(f). From (6) 



F(f) = f " ViOe-'"-' dt = i r {a+ ib)e-'^''-"'^'dt 



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-hi (" {a- ih)e~'^''^'''^ dU (20) 



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