642 BELL SYSTEM TECHNICAL JOURNAL 



In a table of Fourier integrals, every integral expression would then 

 contain, in addition to the arbitrary function F{f), the same oscillat- 

 ing function cis(27r//), the same integral sign with limits — qo, +qo 

 and the same differential df. To repeat any such group of a dozen 

 characters in each of several hundred entries seems quite unnecessary. 

 It is, therefore, proposed merely to tabulate the arbitrary function 

 F{f) and the value G{t) for the evaluated integral expressed as a 

 function of the time. The table is thereby reduced to two parallel 

 columns of associated functions, one of which is employed as the 

 coefficient of the elementary cisoidal function while the other is 

 a function of the independent time variable. The table would, 

 however, be more symmetrical if both of the associated functions 

 could be regarded as coefficients of an elementary function. This 

 may be done by introducing the unit impulse as an elementary func- 

 tion, the impulse occurring at the epoch g at which instant it presents 

 a unit area whereas its value is zero for all time before and all time 

 after the epoch g. This is an essentially singular function and to 

 recognize this fact it will be designated by ^o(^ — g) which is intended 

 to emphasize the singularity. The time function may now be replaced 

 in the table by the same function of the parameter g, since the time 

 function G{i) is equal to the integral with respect to g between infinite 

 limits of the product G(g)^o{t — g). 



The table of Fourier integrals has now become also a table of paired 

 coefficient functions. This means that if the coefficient F{f) is em- 

 ployed with the cisoid, and the coefficient G(g) is employed with the 

 unit impulse, and both products are summed for the entire infinite 

 range of their parameters / and g, the same identical resulting time 

 function is obtained.^ Taken in connection with their respective ele- 

 mentary functions, the two associated coefficient functions are, there- 

 fore, equivalent, alternative ways of representing a particular time 

 function. This is the fundamental geometrical or physical point of 

 view which is needed in connection with the practical application of 

 the Fourier integral theorem. For this reason the table has been 

 headed a table of Paired Coefficients; as explained above, however, 

 it may equally well be considered to be a table of Fourier Integrals. 



There is another fundamental reason for placing both of the func- 

 tions F{f) and G(g) on the same footing as coefficients. It is this: 



■* The use of frequency and epoch as the two parametric variables gives us many 

 symnietrical formulas where, if the radian frecjuency were employed, an unsym- 

 metrical Iw would occur. In practical applications the frequency of the coefficient 

 pairs becomes the frequency which is ordinarily employed in acoustics, in music 

 and in commercial alternating currents. The basic unit for frequency is the reciprocal 

 second; the unit for epoch is the second. 



