NAPI TR-lUUO 



A convenient way to determine the critical value of § is to 

 study both sides of the extrema equation, equation l8. Define the 

 following two functions: 



where § ^ 1, |i > 0. Then equation l8 can be restated as 



f?(/.)= 1,^/-) 



and the extrema of Yp(^^) are determined by the intersections of ffdJ-) with 

 g^CM-). It Is readily verified by substitution that M- = is always a 

 solution, but since this corresponds to zero bandwidth, it is not the 

 maximum of "Y^Cm-) we are seeking. ffCl^) is a straight line, and 

 if § < ^, it has a negative slope with an intercept of 1 at |i. = 0. 

 Temporarily restricting § to be > 0, gs{^) is a "square root" type of 

 curve, always greater than one, with a value of unity at ^ = 0. Consequent- 

 ly^ i'p(M') does not intersect gp(M') at any value of |i > 0. Thus infinite 

 bandwidth solutions are obtained for all values of § < •a'* (The negative 

 §, infinite bandwidth solutions are a trivial extension of the above argu- 

 ments . ) 



For ^ = i, fi(M') is a line of zero slope, at a height of 1, and 

 gj^d-t) is "Vl+M". Thus, in this case, there is no intersection (ignoring 



2 



the trivial n = conincidence), and the infinite bandwidth solution again 

 results . 



With % ^ 2, f=■(^i) is now a straight line with positive slope. 

 With -g- < § < 1, g-d^) is still a square root type of curve. With § -* 2'^ 

 fcdj-) is approaching <» as M^---» =° linearly with ^i with 2 < 5 < 1, gc(M') 



is approaching "" as n ^ =° as M- , which is less than linear with IJ.. 



Consequently, fp(M.) will intercept gp(M') before ^ becomes infinite. 



These concepts are indicated graphically in the accompanying 

 figures for § = 0, 0.5^ 0.7. Trial and error shows that the solution 

 for § = 0.6 occurs at a value of p. approximately equal to k^. 



C-2 



