314 BELL SYSTEM TECHNICAL JOURNAL 



where the last integral is to be regarded as a Stieltjes' integral. When the 

 expression (2.2-3) for \P{t) is placed in the first formula of (2.2-4) we get 



^ (A- when < f < fo 



i-«''« = |^=+f, " />/o (^-^-^ 



When this expression is used in the second formula of (2.2-4), the increments 

 of the differential are seen to be yl at/ = and C /2 at/ = /o . The re- 

 sulting expression for \j/{t) agrees with the original one. 



Here we desire to use a less rigorous, but more convenient, method of 

 dealing with periodic components. By examining the integral of wif) as 

 given by (2.2-5) we are led to write 



u>{f) = 2A'8if) + y 5(/ - /o) (2.2-6) 



where d{x) is an even unit impulse function so that if e > 



[ d{x) dx=\ I Kx) dx= \ (2.2-7) 



Jo 2 J-t 



and b{x) = except at x = 0, where it is infinite. This enables us to use 

 the simpler inversion formulas of section 2.1. The second of these, (2.1-6), 

 is immediately seen to give the correct expression for i/'(r). The first one, 

 (2.1-5), gives the correct expression for w(f) provided we interpret the in- 

 tegrals as follows: 



/ cos 27r/r dr = 18(f) 

 Jq 



(2.2-8) 



/ cos IrfoT cos 27r/T dr = j5(/ — fo) 

 Jo 



It is not hard to show that these are in agreement with the fundamental 

 interpretation 



f '" e-'"^'dt = f '^ e'^'^'dt = b{j) (2.2-9) 



J— 00 J— 00 



which in its turn follows from a formal application of the Fourier integral 

 formula and 



6(/)e'''^' df = \ 5(/)e-''"^' dj = 1 (2.2-10) 



00 J— CO 



We must remember that /o > and/ > in (2.2-8) so that 5(/ + /o) = 

 for / > 0. 



