156 BELL SYSTEM TECHNICAL JOURNAL 



The expression for G2(/) given in Table 2 corresponding to Filter c is 

 exact. The expressions for Filters a and b give good approximations around 

 / = and/ = 2/o where Gi{f) is large. However, they are not exact because 

 terms involving / + 2/o have been omitted. It is seen that all three G^?. 

 behave in the same manner. Each has a peak symmetrical about 2/o whose 

 width is twice that of the original iv{f), is almost zero between and 2/o, 

 and rises to a peak at whose height is twice that at 2/o . 



GsU) is obtained by cubing the i/'(t) given in Table 1 and using 



cos 2x/oT = f cos 27r/oT + \ cos ^ttJqt. 



From the way in which the cosine terms combine with cos 27r/T in (4C-1) we 

 see that Gz{f), for our relatively narrow band pass filters, has peaks at /o 

 and 3/o , the first peak being three times as high as the second. The ex- 

 pressions given for Gz{f) and Gi{f) are approximate in the same sense as are 

 those for G^if). It will be observed that the coefhcients within the brackets, 

 for Filters a and b, are the binomial coefficients for the value of n concerned. 

 Thus for w = 2, they are 2 and 1, for » = 3 they are 3 and 1, and for n = 4 

 they are 6, 4, and 1. 



The higher GnifYs for Filters a and b may be computed in the same way. 

 The integrals to be used are 



I e cos 27r/r dr 



Jo 



I 



e cos 1-KjT dr = 



2a\/2mr 

 na 



'o •' 2xw2«2_|_y2 



In many of our examples we are interested only in the values G„(/) for 

 / near zero, i.e., only in that peak which is at zero. It is seen that G„(/) 

 has such a peak only when n is even, this peak arising from the constant 

 term in the expansion 



cos'^^ = _!_ [cos 2kx + 2k cos 2{k - \)x + (^^)(^^ " ^) cos 2{k - 2)x 



+ ...+ ^_(?«i-^ cos 2x + Mil 



