742 BELL SYSTEM TECHNICAL JOURNAL 



and interchanging rows and columns. Thus, the solution of the cy is ex- 

 pressed directly in terms of the Ck and becomes 



n 



[a^ = [Pik]-' X [C/,.] or ay = Z PlkCu . (29) 



3=0 



The sufficiency of this procedure is established when it is proved that the 

 determinant || Pjk \\ is different from zero for the particular value of s con- 

 sidered. Since 5 is a rational multiple of tt in this case and all non-diagonal 

 entries are algebraic numbers, tt cannot satisfy an equation with algebraic 

 coefficients to make || Pjk \\ = 0. Thus, the system of eq. (29) is a unique 

 solution, and this solution gives the absolute minimum in the sense that 

 no other set of ay will produce a smaller mean-square error over the range 



Otof. 



However, for some values of n the determinant of coefficients becomes 

 extremely small. This condition produces very large numerical values of the 

 elements of [FyA-]~^ Since the ay and Ck are usually small compared with 

 these elements, the accuracy of the solution is impaired. Hence, the system 

 of eq. (29) in some cases represents a set of nearly dependent equations 

 with a fairly wide range of solution. This practical limitation on the unique- 

 ness of these equations may be overcome quite readily by arbitrarily chang- 

 ing one of these equations to produce, for calculation purposes, a dependent 

 set of equations. It turns out that the most expedient choice of this change 



is to replace the Poo = ^ of [P^i] by Poo = |. This, in effect, modifies the 



weighting of ao in these equations and does not, in general, limit the useful- 



TT TT 



ness of the result. Hence, the system of eq. (28) with - replaced by - de- 

 termines a set of coefficients, ao • • • On , which are reasonably close to the 

 optimum tor 5 = - . 



It is appropriate at this point to indicate a practical modification in the 

 approximation method which serves, incidentally, to clarify the reasons for 

 accepting as suitable a set of coefficients that are not the optimum aj over 

 the useful band in the <p domain. 



This modification arises since the foregoing method has considered only 



the average error over the range to - . However, an anah'sis of the i)er- 



centage error in f(x~), and of the corresponding deviation in a over this 

 range, shows that the approximation to a((p) is most critical at high fre- 

 quencies and becomes decreasingly critical as lower frequencies are reached. 

 Thus, in any design, it is necessary to make a slight adjustment of the 



