740 



BELL SYSTEM TECHNICAL JOURNAL 



linear equations becomes 



2^ Pjk aj = Ck . 



3=0 



(i = 0,1,2, ■■■,n) 



(25) 



Therefore, the procedure for determining the optimum coefficients for the 

 range to 5 in the ip domain is as follows: First, compute the Ck which 

 depend on the approximated function a{ip). 



Ck = I [aicp) cos k(p] dip. 

 Jo 



(26) 



Next, compute the elements of Pjk given by 



Pik — 



_ sin ij - k)s sin (j -j- k)s ,^ .. 



2(7 - k) 



^" 2' 



2(i + k) 

 Poo = -y. 



(27) 



These elements depend only on the range 5 and termmate with the desired n 

 in any design. For convenience, these numbers may be arranged in the form 

 of a symmetrical matrix [Pjk]. Hence, the optimum coefficients are found by 

 solving the matrix equation, 



[Pjk] X k] = [Ck]. (j,k = 0,l,2,'--, n) (28) 



In this problem of approximating B(x^) to a high degree of precision over 

 the useful frequency range, the range in the cp domain of most interest is 



TT 



to - . However, before the approximation over only part of the frequency 



range is considered, it is helpful to set down the relations which apply when 

 a{(p) is approximated over the whole frequency range, 5 = tt. In this case, 

 the matrix [Pjk] takes on a form in which all non-diagonal entries are zero. 

 Thus, 



"'tt • • 



[P,A.] = 



Poo 

 Pw 



Poi 

 Pn 



PnO 



Pon 



Pm 



■t^nn 



0^00 



0^0 

 J 



